Dr Christophe Prieur

Pr Pr Delphine Bresch-Pietri

Laurentiu Floquet

Christophe Hetel

Ying Fiter

Mihaly Tang

Petreczky) for their great support and help. A special thanks to Laurentiu who has a pillar of support and help to all the Ph.D.

students, including me

R denotes the set of real numbers; R + denotes the set of non-negative real numbers;

N denotes the set of natural numbers including zero (non-negative integers); N * denotes the set of natural numbers excluding zero (positive integers); Z denotes the set of all integers including zero;

R n denotes the real coordinate space of dimension n ∈ N * , that is the set of the n-tuples of real numbers, with the canonical basis {e 1 , • • • , e n } where e i = [δ 1,i , δ 2,i , • • • , δ n,i ] , with δ ij is the Kronecker delta dened by δ i,j = 1, if i = j,

0 if i = j;
R n + denotes the non-negative real coordinate space of dimension n ∈ N * , that is the set of the n-tuples of non-negative real numbers; | • | denotes the absolute value of a real number;

• 2 denotes the Euclidean vector norm and is dened for any z ∈ R n as

z 2 = n i=1 |z i | 2 ;
sign(•) denotes the sign of a real number; For all a ∈ R + and all x ∈ R we dene the signed power a of x by {x} a = sign(x)|x| a ;

We denote by 1 {x>0} : R → {0, 1} the function dened by 1 {x>0} (x) := 1 if x > 0, Let us also denote by f t (t, x) (resp. f x (t, x)) the partial derivative of a function f with respect to the time (resp. space) variable t (resp. x);

0 if x ≤ 0,
Let us also denote by f xx (t, x) the second partial derivative of a function f with respect to the space variable x;

A function γ : R + → R + is said to be a class-K function if it is continuous, zero at zero, and strictly increasing;

A function γ : R + → [0, 1) is said to be a class-K 1 function if it is continuous, zero at zero, strictly increasing, and lim r→+∞ γ(r) = 1;

A class-K function γ : R + → R + is said to be a class-K ∞ function if it is is unbounded with its argument;

A continuous function β : R + × R + → R + belongs to the class-KL if β(•, t) ∈ K ∞ for each xed t ∈ R + , and β(r, •) is decreasing and lim t→+∞ β(r, t) = 0 for each xed r ∈ R + ;

A continuous function β : R + × R + → R + is said to be a generalized class-KL function (GKL-function) if i) the mapping r → β(r, 0) is a class-K function;

ii) for each xed r ≥ 0 the mapping t → β(r, t) is continuous, decreases to zero and there exists some T (r) ∈ [0, +∞) such that β(r, t) = 0 for all t ≥ T (r);

Résumé long

Cette thèse est consacrée à l'étude des problèmes d'estimation d'états non-asymptotique et de

     z t (t, x) = z xx (t, x) + λz(t, x), t ≥ t0 , x ∈ [0, 1], z(t, 0) = 0, t ≥ t0 , z(t, 1) = U (t), t ≥ t0 , (4) (5) (6)
où t0 ≥ 0 est le temps initial, λ ∈ R est le terme de réaction, z(t, x) ∈ R représente l'état du système, U (t) ∈ R est la commande.

Pour résoudre ce problème, nous utiliserons une approche basée sur les Fonctions de Lyapunov de Contrôle (CLF). L'idée de l'approche consiste à utiliser la norme L

ż(t) = Az(t) + D 0 B(D -σ)U (t -σ)dσ, t ∈ [ t0 , t0 + T ), (7)
           z t (t, x) = z xx (t, x) + λz(t, x), t ∈ [ t0 , t0 + T ), x ∈ [0, 1], z(t, 0) = 0, t ∈ [ t0 , t0 + T ), z(t, 1) = U (t -D), t ∈ [ t0 , t0 + T ), Y (t) = z x (t, 1), t ∈ [ t0 , t0 + T ), (8) (9)
(10) [START_REF] Mazenc | Reduction model approach for linear timevarying systems with delays[END_REF] où t0 ≥ 0 est le temps initial, T > 0 est un temps prescrit à l'avance, λ ∈ R est le terme de réaction, z(t, x) ∈ R représente l'état du système, U (t) ∈ R est la commande (avec U ( t0 + s) = 0 pour tout s ∈ [-D, 0]), Y (t) ∈ R est la partie mesurée, et D > 0 est un retard constant connu à l'avance.

Pour résoudre ce problème, nous commencerons par reformuler le système sous la forme d'un système en cascade EDP-EDP, consistant en une cascade d'une EDP de transport linéaire avec une EDP de réaction-diusion linéaire. Ensuite, nous appliquerons une transformation backstepping linéaire dépendant des gains variant dans le temps et des prédicteurs (généralisés pour le cas de dimension innie) pour convertir le système en cascade en un système cible stable en temps prescrit. L'analyse de stabilité sera eectuée sur le système cible, et Introduction

Context and Motivation

Science has always been driven by the desire to not only understand and describe complex phenomena through mathematical models but also to inuence and control them. In engineering applications, the need for suitable mathematical models becomes more critical. Indeed, by expressing the system dynamics through mathematical equations, one can analyze and understand how inputs, outputs, and states interact, allowing for a deep understanding of how a system behaves and how it responds to dierent inputs. This understanding can be later used for designing control strategies that eectively manipulate the system to achieve desired outcomes. Control theory provides then suitable tools for the modeling and control of complex engineering systems.

To better understand this, let us take the vehicle dynamics system as an example. In this context, we aim to move a vehicle (a car, for instance) from one location to another smoothly and safely.

To control the car's movements, we use the gas pedal, the brake, and the steering wheel. From an engineering point of view, these variables serve as control inputs. As the goal is to move the car, the variables of interest -or the states-are the position of the car, its velocity, the orientation of the car, and its associated velocity. Additionally, the vehicle's position can be measured using, for example, a GPS. This measurement serves as the output. In this particular application, without a mathematical model, it would be suciently easy to control the movement of the car using the inputs. However, having a mathematical model provides a precise description of the relationship between the dierent variables of the system which allows us to anticipate and examine the behavior of the vehicle under dierent scenarios without risks and facilitate the design and optimization of control strategies to reach the desired outcome. To further elaborate on the role of mathematical models, let us consider another example, this time, related to option pricing in nancial markets where an option is a nancial contract that gives the holder the right to buy or sell an underlying asset at a predetermined price. In this complex scenario, mathematical models (e.g.

the Black-Scholes model [START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF], [2]) aim to determine the fair value of options and eectively manage their associated risks. Note that, we can draw parallels between controlling a vehicle's movements and pricing options. Instead of a gas pedal, brake, and steering wheel, the control inputs in the model consist of variables such as the underlying asset's price (the current market value of the asset), the option's strike price (the price at which you can buy or sell the underlying asset if you decide to exercise the option), the risk-free interest rate (the theoretical rate of return of an investment with zero risk), and the option's time to expiration. These inputs guide the valuation and pricing strategies for options and lead the options' prices to converge to specic reference values. Similar to the position, velocity, angle, and angular velocity of a vehicle serving as the states of the vehicle model, the states in this economical example are the option's price and other relevant market variables. These states capture the dynamics of the nancial market and provide insights into the option's value. Just as a GPS can measure the position of a vehicle, market data in the nancial model, such as the current prices of the underlying asset and the option, serve as the output. These measurements help analysts monitor the market conditions and adjust their pricing strategies, accordingly. On the one hand, ODEs are equations containing derivatives of dierent orders of one or multiple required unknown functions with respect to one independent variable (generally thought of as time). These functions are usually referred to as the states of the system, and they evolve on a nite-dimensional space (R n for instance) which means that they can be described by a nite number of degrees of freedom. Due to this fact, ODEs are generally referred to as nite-dimensional systems. Sometimes, ODEs contain some additional functions that can be freely adjusted. In control theory, these functions are referred to as inputs and they serve as a tool to adjust the systems to achieve desired outcomes or what we call outputs (e.g. stabilization of the process, attenuation/rejection of uncertainties, optimization of a performance criterion, tracking a reference trajectory • • • ). When ODEs involve a delay term in their formulation, we talk about TDS.

This class of systems is ubiquitous in physics, biology [START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF], epidemiological processes [START_REF] Cooke | Analysis of an SEIRS epidemic model with two delays[END_REF], [START_REF] Castaños | Observer-based predictor for a susceptible-infectious-recovered model with delays: An optimal-control case study[END_REF] and engineering (robotics [START_REF] Sun | Application of wave-variable control to bilateral teleoperation systems: A survey[END_REF], networked control systems [7], [START_REF] Liu | Networked control systems: A time-delay approach[END_REF], intelligent transportation systems [START_REF] Juárez | Connected cruise control of a car platoon: A timedomain stability analysis[END_REF],

[10],...) among many other disciplines [START_REF] Mazenc | Reduction model approach for linear timevarying systems with delays[END_REF], [START_REF] Zhou | Stabilization of linear systems with distributed input delay and input saturation[END_REF]. In most of these applications, particularly those related to networks, time delays occur during the transmission of information, energy, or/and material through the network which may lead to performance degradation or even instability of the system [START_REF] Ganey | Gene expression time delays and turing pattern formation systems[END_REF] [START_REF] Mondié | Instability conditions for linear time delay systems: A lyapunov matrix function approach[END_REF].

On the other hand, PDEs can model, in a much more accurate manner, vast real applications and phenomena. In particular, they are useful in modeling transport phenomena such as the propagation of waves through a medium [START_REF] Lasiecka | Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary[END_REF] or physical networks of dierent nature: hydraulic [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF], [START_REF] Bastin | Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks[END_REF], road trac [START_REF] Garavello | Conservation laws on complex networks[END_REF], gas pipeline [START_REF] Gugat | Gas ow in fan-shaped networks: Classical solutions and feedback stabilization[END_REF], electrical lines [START_REF] Magnusson | Transmission lines and wave propagation[END_REF], data/communication [START_REF] D'apice | Packet ow on telecommunication networks[END_REF] networks.

Besides, PDEs can model electrochemical systems including fuel cells or batteries, chemical or biochemical tubular bioreactors [START_REF] Jakobsen | Chemical Reactor Modeling Multiphase Reactive Flows[END_REF], thermal systems [START_REF] Baehr | Heat and Mass Transfer[END_REF], exible structures in aerospace applications [START_REF] Stanewsky | Adaptive wing and ow control technology[END_REF]), uid dynamics ( [START_REF] Aamo | Flow Control by Feedback[END_REF], fusion reactors including tokamak plasma [START_REF] Bribiesca | Safety Factor Prole Control in a Tokamak[END_REF], large-scale networks of Multi-Agent Systems (MAS) [START_REF] Meurer | Finite-time multi-agent deployment: A nonlinear PDE motion planning approach[END_REF], epidemiological applications [START_REF] Chalub | The SIR epidemic model from a PDE point of view[END_REF] [START_REF] Kitsos | High-gain observer design for a class of quasi-linear integro-dierential hyperbolic systems-application to an epidemic model[END_REF], and more general diusion processes (e.g. diusion of chemicals in a uid [START_REF] Grzybowski | Chemistry in motion[END_REF], the dispersal of pollutants in the atmosphere [START_REF] Cantrell | Spatial ecology via reaction-diusion equations[END_REF], the diusion of heat in a solid material, the spread of infectious diseases within a population [START_REF] Miller Neilan | Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons[END_REF], [START_REF] Grave | Modeling nonlocal behavior in epidemics via a reactiondiusion system incorporating population movement along a network[END_REF], ...) among many others. Overall, the most relevant classes of PDEs modeling those applications are Hyperbolic and Parabolic PDEsin one or higher dimension of space. Both TDS and PDEs are of innite-dimensional nature due to their solutions evolving on an innite-dimensional space 1 or in other terms having an innite number of degrees of freedom.

The dynamic operation of all aforementioned classes of systems essentially relies on incorporating suitable control and estimation strategies to inuence the system dynamics while leading the system to behave as desired. Regarding the estimation design, it is known that the knowledge of the whole state of the innite-dimensional system is in general not available nor realistic (e.g., no access to the whole state or putting sensors everywhere is expensive). We can access information through the boundaries or very point-wise locations in the domain. Therefore, there is a need to reconstruct or estimate the state from a few available measurements. For that, it is common to use observers. As for the control design of PDEs, in particular, two main ways of acting on these systems can be highlighted: boundary and in-domain control. Although in several applications, in-domain control is hard to achieve due to many reasons: rst, it is hard to, physically, access the interior of the system to control it; secondly, for in-domain control often requires signicant 1 Consider for instance the space L 2 (0, 1) := {f : [0, 1] → R : 1 0 |f (x)| 2 dx < ∞} which is an innite-dimensional space because it contains innitely many elements that are pairwise linearly independent; which means that we can not nd a nite basis that can generate the entire space.

1.1. Context and Motivation 7 nancial and maybe logistical resources to be eectively applied. On the other hand, acting on the boundaries of the systems is usually much easier and feasible. Moreover, as boundary control targets only specic parts of the domain, it requires fewer resources compared to in-domain control. Due to these factors, boundary controls are often a good physical choice.

For boundary control, the most used and powerful methods are the Lyapunov techniques ( [START_REF] Bastin | Stability and Boundary Stabilization of 1-D Hyperbolic Systems[END_REF])

and the backstepping control design ( [START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF]). The latter consists in transforming the unstable PDE, by using an invertible Volterra and/or Fredholm type transformation, into another PDE system of the same type, called the target system, satisfying a desired stability property. Then, using the inverse transformation the desired stability property is transferred back to the original PDE system. The method has been used to deal with the boundary stabilization of broader classes of PDEs such that: Slender Timoshenko Beam equations [START_REF] Krstic | Backstepping boundary controllers and observers for the slender timoshenko beam: Part i -design[END_REF], Navier-Stokes equations [START_REF] Cochran | Backstepping boundary control of navier-stokes channel ow: A 3d extension[END_REF] in 2006, Schrodinger equations [START_REF] Krstic | Boundary controllers and observers for schrödinger equation[END_REF], [START_REF] Krstic | Boundary controllers and observers for the linearized schrödinger equation[END_REF] in 2007, Burgers equations [START_REF] Krstic | Nonlinear stabilization of shock-like unstable equilibria in the viscous burgers PDE[END_REF], Euler-Bernoulli beam equations [START_REF] Smyshlyaev | Boundary controllers for euler-bernoulli beam with arbitrary decay rate[END_REF] and hyperbolic wave equations in [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF] in 2008, ... Moreover, this approach has been exploited to deal with the problem of delay compensation for LTI systems with input delay in [START_REF] Krstic | Backstepping boundary control for rst-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], where the key idea was to notice that the input delay can be expressed using a transport equation for which it is possible to apply the backstepping approach and the resulting control is the classical predictor feedback obtained by the Artstein's reduction approach [START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF]. This last result was later generalized for LTI systems with time-varying input delay in [START_REF] Bekiaris-Liberis | Stabilization of nonlinear strict-feedback systems with time-varying delayed integrators[END_REF], for nonlinear ODEs with arbitrary long constant input delays in [START_REF] Krstic | Input delay compensation for forward complete and strict-feedforward nonlinear systems[END_REF], for state-dependent delay in [START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF], for time-varying input delay saturation in [START_REF] Bresch-Pietri | Sucient conditions for the prediction-based stabilization of linear systems subject to input with input-varying delay[END_REF], [START_REF] Bresch-Pietri | Prediction-based stabilization of linear systems subject to input-dependent input delay of integral-type[END_REF] and delay-adaptive control [51][53].

This method has been widely extended to other parabolic systems (see [START_REF] Sano | Neumann boundary stabilization of one-dimensional linear parabolic systems with input delay[END_REF][60] and the references therein). In [START_REF] Deutscher | Fredholm backstepping control of coupled linear parabolic PDEs with input and output delays[END_REF], for instance, exponential stabilization of a class of coupled reaction-diusion PDEs with dierent input and output delays was solved using an observer-based boundary feedback law based on an invertible Fredholm backstepping transformation. Note that an alternative method for stabilization of parabolic PDEs with input delay is the modal decomposition method (see [START_REF] Prieur | Feedback stabilization of a 1-d linear reaction-diusion equation with delay boundary control[END_REF], [START_REF] Djebour | Feedback stabilization of parabolic systems with input delay[END_REF]). The method consists of splitting the unstable PDE into a stable innite-dimensional part and a nite-dimensional unstable part. Then, using classical predictor approaches, the unstable nite-dimensional system is stabilized. Recently, this approach was applied in [START_REF] Katz | Delayed nite-dimensional observer-based control of 1-d parabolic PDEs[END_REF], [START_REF] Katz | Boundary delayed observer-controller design for reactiondiusion systems[END_REF] to construct a new nite-dimensional observer design for a class of parabolic PDEs, which in turn was used in [START_REF] Katz | Delayed nite-dimensional observer-based control of 1-d parabolic PDEs[END_REF], [START_REF] Katz | Boundary delayed observer-controller design for reactiondiusion systems[END_REF] to construct an observed-based control that stabilizes the PDE system.

For most of the mentioned mathematical models, the control and estimation designs usually achieve convergence in an innite amount of time (asymptotically or exponentially). However, in many applications, the need to meet time constraints and increase temporal performance is crucial.

To clarify this, let us take as an example the situation of the spatial spreading of an epidemic in a population. As the disease evolves in time and space, this process can be modeled by, e.g., reaction-diusion PDEs [START_REF] Miller Neilan | Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons[END_REF], [START_REF] Capasso | Asymptotic behaviour of reaction-diusion systems in population and epidemic models[END_REF]. In this situation, the control input includes all the measures that help in controlling the spreading of the disease in the population (through control actions such as lock-downs, travel bans, and mass testing and vaccination projects...). Using asymptotic controllers (i.e., those whose action can only be appreciated over a long period) may eectively reduce the spread of the epidemic to a manageable level or eliminate it, but this may lead to prolonging the impact of the epidemic on the population and the economy. How can we deal with this situation? Can we reduce the time it takes to control the spread of the epidemic? Moreover, can we make this time to be nite?

All these questions are hard to answer, even in other general situations (or applications other than epidemics). If one considers nite-time (Non-Asymptotic) mitigation measures, it would be possible to reduce the impact of the epidemic on the population and the economy and avoid situations where the epidemic escalates and evolves to become a severe threat to the population, such as becoming a highly contagious or deadly disease. Moreover, what about if delays are present in the epidemiological process or in the control actions (e.g., a lock-down policy whose positive eects can only be perceived some weeks or months later)? Can we perfectly compensate for the delay within a nite amount of time?

All of this calls for advanced control and estimation methods that accelerate the convergence and improve the performance (e.g., transient processes to occur within a nite time) while also accounting for the eects of the perturbation and the delays when present. The controllers allowing to achieve this type of better performance will be referred to as Non-Asymptotic controllers (with some specicities to discuss later on). It is then worth recalling and further motivating Non-Asymptotic concepts for dynamical systems, particularly in the context of innite-dimensional systems, as we discuss next (as well as in Chapter 2).

Non-Asymptotic concepts

Let us rst recall that the stability notion is one of the most interesting mathematical notions for studying mathematical models including dierential equations. In fact, this notion was introduced to help in understanding the behavior of systems over time, even when the exact solutions are not known explicitly. In engineering, this notion can be used as a tool to analyze mathematical models subject to control laws to make sure that the system achieves the desired outcome. Mathematically speaking, the stability of a system means that if the initial condition of a system is close to the equilibrium point, then the system's trajectories will tend to remain close to this equilibrium point.

When such a condition holds, the system is called stable ; otherwise, it is called unstable.

Depending on the time of convergence, the stability notion can be divided into two main categories: Asymptotic and Non-Asymptotic. Asymptotic stability refers to the situation where a system's trajectories approach the system's equilibrium when time tends to innity. In other words, it means that after a suciently large amount of time, all trajectories will approach the equilibrium, but they will never reach it. When, in addition, the rate of convergence to the equilibrium is exponential, we talk about exponential stability. The asymptotic and exponential stability properties of dynamical systems have been widely studied in the control community (see [START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF], [START_REF] Malkin | Theory of stability of motion[END_REF][69],...).

Non-Asymptotic stability, on the other hand, refers to the situation where a system is stable with trajectories approaching the equilibrium in a nite amount of time (referred to as the settling time or the terminal time). It is worth mentioning that in some dynamical systems, such as the one depicted in Figure 1.1 and governed by Torricelli's law, the Non-Asymptotic stability concept appears naturally. Torricelli's law is a theorem in uid dynamics relating the speed of uid owing from an orice v to the height of uid above the opening h, assuming no air resistance, viscosity, or other hindrance to the uid ow, (see Figure 1.1) by the formula v = √ 2gh. This theorem is an application of Bernoulli's principle. In this application, the rate of change of the water level height can be described by the following equation:

A a dh dt = -2gh, (1.1)
where A and a are respectively the cross sections of the container and the tube, g is the gravitational constant, and h is the water level height that can be computed, by solving the previous equation, to get h(t) = h(0) 1 -t T

2

. By analyzing this last expression, it can be shown that the total time to empty the container (i.e., h(T ) = 0) is nite and is given by T = A a 2 g h(0) < ∞.

Non-Asymptotic concepts

Finite-time stability

When the settling time is a function of the initial condition, the stability is referred to as nitetime stability (FTS). Historically, nite-time stability was early studied by Zubov [START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF] in 1958, then by Roxin [START_REF] Roxin | On nite stability in control systems[END_REF] in 1966, and by Haimo [START_REF] Haimo | Finite time controllers[END_REF] in 1986. It was not until the end of the 90s for this theory developed with Bhat and Bernstein in [START_REF] Bhat | Lyapunov analysis of nite-time dierential equations[END_REF], [START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF], where they showed in [START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF] that under some homogeneity property, the asymptotic stability is equivalent to the nite-time stability. Later on, in [START_REF] Bhat | Continuous nite-time stabilization of the translational and rotational double integrators[END_REF], [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], they proved the rst converse theorem adding a new page to this theory. In 2003, Moulay and Perruquetti took up this work and extended the results to dierent system classes including non-autonomous systems in [START_REF] Moulay | Finite time stability of nonlinear systems[END_REF]. This last result was extended in 2019 by Zimenko et al. to output nite-time stability in [START_REF] Zimenko | On condition for output nite-time stability and adaptive nite-time control scheme[END_REF], [START_REF] Zimenko | On necessary and sucient conditions for output nite-time stability[END_REF].

It is worth mentioning the works of Amato in [START_REF] Amato | Necessary and sucient conditions for nite-time stability of linear systems[END_REF], Lazarevic in [START_REF] Lazarevi¢ | Finite-time stability of delayed systems[END_REF], Weiss and Infante in [START_REF] Weiss | Finite time stability under perturbing forces and on product spaces[END_REF],

Dorato in [START_REF] Dorato | An Overview of Finite-Time Stability[END_REF], Amato et al. [START_REF] Amato | Sucient conditions for nite-time stability and stabilization of nonlinear quadratic systems[END_REF], which considered the notion of practical nite-time stability.

Fixed-time stability

The notion of FTS can be upgraded to dene what we call xed-time stability (FxTS), which refers to the case when the settling-time function of an FTS system is bounded by a nite constant value that does not depend on the initial conditions. This property implies that all the system's trajectories will converge exactly to zero before a specied time regardless of the system's initial conditions. To the best of our knowledge, FxTS was introduced, for the rst time, in [85, Corollary 2.24] using the notion of homogeneity in bi-limit. However, this result did not allow for adjusting or even estimating the settling time of the closed-loop system. It was not until 2012, in the work of Polyakov [START_REF] Polyakov | Nonlinear feedback design for xed-time stabilization of linear control systems[END_REF], that FxTS was formally dened and a new Lyapunov-based stability result, that provides xed-time stability and overcomes the shortcomings of [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], was given using what he called a special modication of "nested (terminal) second-order sliding mode control algorithm [START_REF] Levant | Principles of 2-sliding mode design[END_REF]. In 2015, the FTS and FxTS Lyapunov stability results were extended by Polyakov et al.

in [START_REF] Polyakov | Finite-time and xed-time stabilization: Implicit lyapunov function approach[END_REF] using Implicit Lyapunov Functions (ILF) which are, as their name indicates, Lyapunov functions dened, implicitly, as solutions to some algebraic equation. In Lu et al. [START_REF] Lu | A note on nite-time and xed-time stability[END_REF] in 2016, new results were obtained concerning sucient conditions for FxTS. In 2018, Lopez-Ramirez et al.

provided in [START_REF] Lopez-Ramirez | On necessary and sucient conditions for xed-time stability of continuous autonomous systems[END_REF], [START_REF] Lopez-Ramirez | Conditions for xed-time stability and stabilization of continuous autonomous systems[END_REF] some necessary and sucient conditions for FxTS of continuous autonomous systems that take into account the regularity of the settling-time function. Recently, some sucient optimality conditions for FxTS in terms of the optimal control theory were provided by Michalak and Nowakowski in [START_REF] Michalak | New approach to xed-time stability of a nonlinear system[END_REF].

Chapter 1. Introduction

Predened-time stability

A more recent stability concept called predened-time stability (PdTS) was introduced in 2014 by Sánchez-Torres et al. [START_REF] Sánchez-Torres | A discontinuous recurrent neural network with predened time convergence for solution of linear programming[END_REF], [START_REF] Sánchez-Torres | Predened-time stability of dynamical systems with sliding modes[END_REF] to overcome the shortcomings and the diculties of FxTS concept in tuning the control parameters to ensure a convergence within a nite time upper-bounded by a desired predened constant. These diculties are mainly due to the unclear relationship between the control parameters and the upper bound of the settling time. In order to overcome these diculties, the PdTS concept introduces explicitly the upper bound for the settling time as an adjustable control parameter. In 2015, Sánchez-Torres et al. [START_REF] Sánchez-Torres | Predened-time stability of dynamical systems with sliding modes[END_REF], [START_REF] Sánchez-Torres | A class of predened-time stable dynamical systems[END_REF] introduced a Lyapunovbased characterization of the PdTS concept. Another characterization was given by Aldana-Lopez et al. [START_REF] Aldana-López | Enhancing the settling time estimation of a class of xed-time stable systems[END_REF] in 2019 and a more generalized one was given later on by Jiménez-Rodríguez et al. [START_REF] Jiménez-Rodríguez | A Lyapunov-like characterization of predened-time stability[END_REF].

Prescribed-time stability

Note that the PdTS concept can be also improved in a sense where the settling time itself can be chosen a priori independently of initial conditions and the system's parameters. This type of stability is called prescribed-time stability (PTS) and was introduced by Song et al. [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF] in 2017 where a Lyapunov-based characterization of the PTS concept was given. The key idea behind this characterization is the use of time-varying tools, in particular, time-dependent gains that grow to innity as time tends toward the prescribed settling time. The resulting controllers that allow to achieve this Non-Asymptotic property are time-varying feedbacks (or what we call prescribed-time controllers). Because of the blow-up characteristic of the time-varying gains, the PTS concept is limited to processes that take eect within a nite time interval (e.g. missile guidance [99][101]).

For more details about this concept, we refer to the most recent survey on this topic: [START_REF] Song | Prescribed-Time Control and Its Latest Developments[END_REF].

One of the most striking features of time-varying prescribed-time controllers that use growing gains is not only to achieve convergence in desired time but to reduce state peaking during stabilization and to also reduce the control eort by distributing it more evenly over the time interval of convergence.

It is essential to say that Non-Asymptotic concepts (that, in this thesis, we will mostly refer to as either nite-time, xed-time, or prescribed-time stability) discussed above have been extensively studied for nite-dimensional systems. The question that arises now is, what about innitedimensional systems? Although there are few results in the literature (that we further detail in Chapter 2), too much research on this area is still needed. This motivates and shapes the core content of the thesis. However, as we will see, we limit the scope of the thesis to some Non-Asymptotic results for some classes of innite-dimensional systems that we discuss in the sequel.

Problems addressed in the thesis, structure and main contributions

This thesis provides Non-Asymptotic estimation and stabilization results for some classes of innite-dimensional systems, namely LTI systems subject to input/sensor delays and 1D reactiondiusion PDEs. Inspired by the comprehensive tools and results in the context of nite-dimensional systems, we extend state-dependent and time-varying feedbacks to innite-dimensional settings.

We start with the problem of input and sensor delay compensation in nite/xed time of LTI We start, in Chapter 2, by recalling some noteworthy results on nite/xed/prescribed-time stabilization for some classes of innite-dimensional systems. We introduce then the necessary background and main tools -including the stability concepts, some important Lyapunov-based stability results, and main techniques-that are instrumental to working out our contributions.

Finally, we position our contributions with respect to the most relevant works in the literature.

Our main contributions are summarized in order:

Part I C1) In Chapter 3, we present our rst contribution that deals with the problem of nite/xedtime estimation of LTI systems in the observable form with measurement delay:

ż(t) = Az(t) + BU (t), t ≥ t0 , Y (t) = Cz(t -D), t ≥ t0 , (1.2)
(1.3)
where the initial time t0 ≥ 0, the state z

(t) = [z 1 (t), ..., z n (t)] ∈ R n , n ∈ N * , the control U (t) ∈ R, the output Y (t) ∈ R, the delay D > 0 is a known constant.
To accomplish this, we reformulate the original system into a cascade ODE-PDE system where the PDE part is a transport equation that models the eect of the delay on the output. We construct suitable nonlinear (non-smooth) gains in a way that ensures the error system is nite-time stable. To prove this, we use an invertible backstepping transformation to convert the error system into a target system which is shown to be nite-time stable using Lyapunov-based analysis and homogeneity tools. We use the inverse transformation to transfer this property to the error system.

C2)

In Chapter 4, we present our second contribution that deals with the problem of nite/xedtime stabilization of a chain of integrators with pointwise input delay:

ż(t) = Az(t) + BU (t -D), t ≥ t0 , (1.4)
where the initial time t0 ≥ 0, the state z(t) = [z 1 (t), ..., z n (t)] ∈ R n , n ∈ N * , the control U (t) ∈ R, D > 0 is a known constant delay, the system matrix and input vector of appropriate dimensions and are given respectively by A :

= {A i,j } ∈ M n (R), where, for i, j ∈ {1, • • • , n}, A i,j = 1 if j = i + 1 and A i,j = 0 otherwise, B = e n := 0 • • • 0 1 .
The idea of this contribution is similar to our rst contribution. To be precise, we rst reformulate the chain of integrators with input delay as a cascade ODE-PDE system (i.e., a cascade of a linear transport PDE with the chain of integrators) where the transport equation models the eect of the delay on the input as we explained before. However, more challenges arise in this problem, as we use a nonlinear innite-dimensional backstepping transformation to convert the cascade system to a suitable target system that is chosen to be nite-time or xed-time stable. We perform the stability analysis on the target system by means of classical Non-Asymptotic concepts and tools such as linear homogeneity and "generalized KL" functions. Then, we use the inverse transformation to transfer back the stability property to the closed-loop system. Finally, we give some characterizations of nite/xed time predictor-based controllers followed by numerical simulations.

     z t (t, x) = z xx (t, x) + λz(t, x), t ≥ t0 , x ∈ [0, 1], z(t, 0) = 0, t ≥ t0 , z(t, 1) = U (t), t ≥ t0 , (1.5) (1.6) (1.7)
with the initial time t0 ≥ 0, the reaction term λ ∈ R, the state z(t, x) ∈ R, and the control U (t) ∈ R.

As aforementioned, we tackle this problem by using a Lyapunov-based approach. The idea of the approach is to use a "spatially weighted L 2 -norm" as a Lyapunov functional to design a nonlinear controller that ensures that the closed-loop system is stable with any desired convergence including (nite/xed-time convergences). As an application, we focus on the nite/xed-time notions for which we give some particular explicit controllers. We provide some hints on how we can extend the approach to Input-to-state stability (ISS) results as well as the problem of tracking. We point out limitations to observer design.

Part II C4)

In Chapter 6, we present our fourth contribution that deals with the problem of prescribedtime stabilization of a class of controllable linear systems with distributed input delay:

ż(t) = Az(t) + D 0 B(D -σ)U (t -σ)dσ, t ∈ [ t0 , t0 + T ), (1.8)
where the initial time t0 ≥ 0, the prescribed-time T > 0, the state z Then, we prove the bounded invertibility of the transformations and hence we show that the prescribed-time stability property is transferred to (1.8).

(t) = [z 1 (t), ..., z n (t)] ∈ R n , n ∈ N * , the control U (t) ∈ R, D > 0 is
C5) Finally, in Chapter 7, our fth contribution deals with the problem of boundary stabilization of one-dimensional (1D) reaction-diusion systems with boundary input delay of the form

           z t (t, x) = z xx (t, x) + λz(t, x), t ∈ [ t0 , t0 + T ), x ∈ [0, 1], z(t, 0) = 0, t ∈ [ t0 , t0 + T ), z(t, 1) = U (t -D), t ∈ [ t0 , t0 + T ), Y (t) = z x (t, 1), t ∈ [ t0 , t0 + T ), (1.9)
(1.10)

(1.11) (1.12)
with the initial time t0 ≥ 0, the prescribed-time T > 0, the reaction term λ ∈ R, the state z(t, x) ∈ R, the control U (t) ∈ R, with the following initial condition: U ( t0 + s) = 0 for all s ∈ [-D, 0], the collocated output Y (t) ∈ R, and D > 0 is a known constant delay,.

To deal with this problem, we rst reformulate the system as a PDE-PDE cascade system (i.e., a cascade of a linear transport PDE with a linear reaction-diusion PDE), where the transport equation represents the eect of the input delay. We then apply a time-varying innite-dimensional backstepping transformation, in conjunction with predictor-based concepts to convert the cascade system into a prescribed-time stable (PTS) target system. The stability analysis is conducted on the target system, and the desired stability property is transferred back to the closed-loop system using the inverse transformation. We pay special attention to the analysis of the growth-in-time of the gains relative to the decreasing-in-time of solutions while ensuring convergence of the closed-loop solutions and boundedness of the resulting output feedback controllers.

background and tools that are instrumental in the thesis by recalling rst the Non-Asymptotic stability concepts for nite-dimensional and innite-dimensional systems (Section 2.2 and 2.3), all of this accompanied by several examples and remarks. We nally position our contributions (summarized in Chapter 1) with respect to the most relevant works in the literature (Section 2.4).

Notice that additional helpful results related to the homogeneity theory are left to Appendix A) for the sake of brevity.

Introduction

In line with the discussion given in the previous chapter, we recall that Non-Asymptotic concepts have been extensively studied within the framework of stabilization of linear and nonlinear

ODEs. Numerous noteworthy contributions can be cited for nite/xed/predened/ prescribedtime stability based on: the backstepping approach for ODEs (see e.g. [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF], [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF][108]), the desingularization technique [START_REF] Novel | Small-time stabilization of homogeneous cascaded systems with application to the unicycle and the slider examples[END_REF], the control Lyapunov functions (CLF) (see e.g. [START_REF] Moulay | Lyapunov-based approach for nite time stability and stabilization[END_REF], [START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF]), [START_REF] Lasiecka | Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary[END_REF] Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings the weighted homogeneity concept (see e.g. [START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF], [START_REF] Bhat | Continuous nite-time stabilization of the translational and rotational double integrators[END_REF], [START_REF] Polyakov | Nonlinear feedback design for xed-time stabilization of linear control systems[END_REF], [START_REF] Jiménez-Rodríguez | A Lyapunov-like characterization of predened-time stability[END_REF], [112][122]), the generalized homogeneity concept (see e.g. [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF], [START_REF] Zimenko | Homogeneous observer design for linear mimo systems[END_REF]), the implicit Lyapunov functions (ILF) (see e.g. [START_REF] Polyakov | Finite-time and xed-time stabilization: Implicit lyapunov function approach[END_REF],

[125] [START_REF] Lopez-Ramirez | Finite-time and xedtime observer design: Implicit lyapunov function approach[END_REF]), the time-varying tools (see e.g. [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF], [START_REF] Song | Prescribed-Time Control and Its Latest Developments[END_REF], [129][133]), a mix of homogeneity tools and time-varying tools (see e.g. [134][138]), or periodic delay feedbacks (see e.g. [START_REF] Engel | A continuous-time observer which converges in nite time[END_REF], [START_REF] Zhou | Fixed-time stabilization of linear delay systems by smooth periodic delayed feedback[END_REF]), ...

Non-Asymptotic concepts of TDS

In the framework of nite/xed/prescribed-time stabilization of TDS, very few results can be found even for nite-dimensional systems with input delay. For nite-time stabilization, one may refer to some of the pioneering contributions starting with the work of Karafyllis [141] (2006) for triangular time-varying systems described by retarded functional dierential equations and nishing with the work of Moulay et al. [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF] (2008) for linear time-delay systems using the Artstein's transformation and weighted homogeneity. For xed-time stabilization, one can mention two key recent contributions [START_REF] Zuo | Fixed-time stabilization of general linear systems with input delay[END_REF] and [START_REF] Michiels | On the xed-time stabilization of input delay systems using act-and-wait control[END_REF] dealing with general LTI systems with input delay.

The former builds upon the ArtsteinKwonPearson reduction transformation and uses weighted homogeneity in a similar fashion to [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF]. The latter proposes a novel strategy based on act-andwait predictor-based control and opens new research avenues on xed-time control design.

Prescribed-time stabilization [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF] and predictor feedback for compensation of input delay [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF] are a perfect match because both techniques deal with nite-time dynamics. Intuitively, by applying predictor feedback to prescribed-time feedback (for either an ODE or PDE plant), the former feedback being time-varying and the latter innite-dimensional, one should be able to obtain convergence in a time that is the sum of the prescribed time and the input delay. And yet, already six years after [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF], one can still nd very few results that combine prescribed-time feedback with predictors. In an exception from this slow progress, in [START_REF] Espitia | Prescribed-time predictor control of LTI systems with input delay[END_REF], an ODE-PDE cascade representation of a class of LTI systems with input delay is provided and a time-varying backstepping-based approach is used to design a predictor feedback that ensures delay compensation in prescribed time.

An extension of the methodology was carried out in [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diusion PDEs[END_REF] (to a class of 1D reaction-diusion PDE with boundary input delay), and after that in [START_REF] Irscheid | Prescribed-time control for a class of semilinear hyperbolic PDE-ode systems[END_REF] (to a class of rst-order semi-linear hyperbolic PDE that is bidirectionally interconnected with nonlinear ODEs).

Finite/xed/prescribed-time stabilization of time-delay systems still remains sparse and constitutes challenging topics.

Non-Asymptotic concepts for PDEs

1D hyperbolic PDEs

The rst result related to this topic for hyperbolic PDEs has been achieved, to the best of our knowledge, by Coron et al. based on the backstepping approach in [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF] where they mainly proved exponential output-feedback stabilization, followed by the nite-time convergence for a class of 2×2

quasi-linear hyperbolic systems with single boundary control. In 2016, a similar approach [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF] was used to design output feedback laws for a system of n×n coupled rst-order hyperbolic linear PDEs achieving nite-time convergence to zero. Later on, this result was optimized in [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF] in the sense of the target system was slightly modied to minimize the settling time to t with the theoretical optimal nite-time of convergence (see e.g. [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF]). In 2017, this last result was revised by Coron et al. in [START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via fredholm backstepping transformation[END_REF] and the result was re-proved in a simplied way. The idea of the approach was to use an invertible Volterra transformation of the second kind, followed by a Fredholm backstepping transformation to convert the coupled system into a much simpler target system from which the nite-time stabilization with the theoretical optimal nite-time of convergence can be recovered.

In 2018, an interesting nite-time stabilization result was provided by Annsen et al. in [START_REF] Annsen | Control of a time-variant 1-d linear hyperbolic PDE using innite-dimensional backstepping[END_REF] for a class of 1D linear hyperbolic PDEs using some backstepping transformations that make use of time-dependent kernels. In 2021, an extension of all the previous results to time-dependent systems was presented by Coron et al. in [START_REF] Coron | Boundary stabilization in nite time of onedimensional linear hyperbolic balance laws with coecients depending on time and space[END_REF] in the same fashion as in [START_REF] Annsen | Control of a time-variant 1-d linear hyperbolic PDE using innite-dimensional backstepping[END_REF]. In addition to these previous references, some recent related noteworthy results can be also cited starting from [START_REF] Perrollaz | Finite-time stabilization of 2 × 2 hyperbolic systems on treeshaped networks[END_REF] that solved the problem of nite-time stabilization of a 2 × 2 hyperbolic systems (and in particular for the Saint-Venant equations) using some boundary control that satises a nite-time stable nonlinear ODE; [START_REF] Coron | Null-controllability of linear hyperbolic systems in one dimensional space[END_REF] in which Coron and Nguyen managed to solve the problem null-controllability of a general linear hyperbolic system in one space dimension using boundary controls on one side; [START_REF] Novel | Finite-time stabilization of an overhead crane with a exible cable[END_REF], [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a exible cable submitted to an ane tension[END_REF] which study the problem of nite-time stabilization of a hyperbolic wave PDE coupled with a second order ODE; and [START_REF] Coron | Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space[END_REF], [START_REF] Coron | Lyapunov functions and nite-time stabilization in optimal time for homogeneous linear and quasilinear hyperbolic systems[END_REF] that deals with nite-time stabilization in optimal time for homogeneous linear and quasi-linear hyperbolic systems.

1D parabolic PDEs

For 1D parabolic PDEs, the backstepping approach has continued to dominate in solving the problem of boundary stabilization in nite time. The rst result related to this topic was introduced in 2017 by Coron and Nguyen [START_REF] Coron | Null controllability and nite-time stabilization for the heat equations with variable coecients in space in one dimension via backstepping approach[END_REF] where they solved not only the problem of null controllability but also the problem of nite-time stabilization for a general class of 1D parabolic PDEs with variable coecients in space, by making use of some periodic time-varying feedback laws. In 2017, Polyakov et al. [162] used also the backstepping method to solve the problem of nite-time stabilization of the heat equation using state-dependent switching boundary control. Later on, with the emergence of the prescribed-time stability concept in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF], and inspired by [START_REF] Coron | Null controllability and nite-time stabilization for the heat equations with variable coecients in space in one dimension via backstepping approach[END_REF] and [162], Espitia et al were able to take the backstepping approach to another level while solving the problem of prescribed-time stabilization of a class of 1D reaction-diusion in [163][165]. The key idea is to use of a time-varying backstepping transformations and choose target system with the desired prescribed-stability property. This last result was later generalized to the case of output feedback stabilization of the same class of systems in [START_REF] Steeves | Prescribed-time stabilization of reaction-diusion equation by output feedback[END_REF], then for a pair of coupled reactiondiusion equations in [START_REF] Steeves | Boundary prescribedtime stabilization of a pair of coupled reactiondiusion equations[END_REF]; and since then has opened new research avenues for prescribed-time control for PDEs. More technical details are given in Section 2.3.2.

It is worth mentioning that as an alternative to the backstepping approach, as soon as one deals with in-domain control, the notion of generalized homogeneity (introduced in [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]) as in [START_REF] Polyakov | On homogeneous nite-time control for linear evolution equation in hilbert space[END_REF],

[170] and Lyapunov techniques have been instrumental for the design on controllers achieving nite-time stabilization for this class of PDE.

Other classes of PDEs

For other classes of PDEs, one can refer to some pioneering results, that are also based on the backstepping approach: [START_REF] Espitia | On local nite-time stabilization of the viscous burgers equation via boundary switched linear feedback[END_REF] for local nite-time stabilization of the Viscous Burgers equations; [START_REF] Steeves | Boundary prescribedtime stabilization of a pair of coupled reactiondiusion equations[END_REF], [START_REF] Steeves | Prescribedtime stabilization of the linearized schrödinger equation[END_REF] for prescribed-time stabilization of the linearized Schrödinger equation; [START_REF] Steeves | Prescribed-time stabilization of odes with diusive actuator dynamics[END_REF]

ż = K(z), z ∈ Ω, (2.1)
where K : Ω ⊂ R n → R n is a continuous function and K(0) = 0. Let us assume that K is such that (2.1) has the property of existence and uniqueness of solutions in forward time outside the origin.

Denition 2.1

The origin of the system (2.1) is said to be: stable if and only if for every ε > 0, there exists δ > 0 such that, for every initial time t0 ≥ 0 and any initial condition z 0 ∈ Ω,

( z 0 2 ≤ δ) =⇒ ( z(t) 2 ≤ ε) ,
for all t ≥ t0 ; asymptotically stable (AS) if and only if it stable and lim t→+∞ z (t) 2 = 0, for any z 0 ∈ Ω; exponentially stable (ES) if and only if it is AS and there exist M > 0 and λ > 0 such that z(t) 2 ≤ M z 0 2 e -λt , for all t ≥ t0 ≥ 0 and any z 0 ∈ Ω, (λ is referred to as the rate of the exponential convergence of the system); nite-time stable (FTS) [START_REF] Bhat | Lyapunov analysis of nite-time dierential equations[END_REF] if and only if it is stable and for any z 0 ∈ Ω there exists 0 < T z0 < +∞ such that z (t) 2 = 0 for all t ≥ t0 + T z0 . The functional T (z 0 ) = inf {T z0 ≥ 0 : z (t) 2 = 0, ∀t ≥ t0 + T z0 } denes the settling time of the system (2.1); nearly xed-time stable (nearly FxTS) if and only if it is stable and for any > 0 there exists 0 < T < +∞ such that z(t) 2 ≤ for all t ≥ t0 + T and all z 0 ∈ Ω; xed-time stable (FxTS) [START_REF] Polyakov | Nonlinear feedback design for xed-time stabilization of linear control systems[END_REF] if and only if it is FTS and sup z0∈Ω T (z 0 ) < +∞; predened-time stable (PdTS) [START_REF] Jiménez-Rodríguez | A Lyapunov-like characterization of predened-time stability[END_REF] if and only if it is FxTS and sup z0∈Ω T (z 0 ) ≤ T max where T max is a tuning parameter chosen, in advance, independently of z 0 ∈ Ω; prescribed-time stable (PTS) [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF] if and only if it is PdTS and T is a tuning parameter chosen, in advance, independently of z 0 ∈ Ω.

The set Ω is called the domain of stability/attraction. If Ω = R n , then the corresponding properties are global.

In the above denitions uniformly with respect to initial time t0 has been omitted for the sake of brevity.

Remark 2.1

Notice that when studying the prescribed-time stability notion later on, we will need to deal with a non-autonomous version of (2.1) of the form:

ż = K(t, z), (t, z) ∈ [ t0 , t0 + T ) × Ω. (2.2)
Here, T > 0 is a positive constant and K : [ t0 , t0 + T ) × Ω → R n is a continuous function such that K(t, 0) = 0 for all t ∈ [ t0 , t0 + T ) and such that (2.2) has the property of existence and uniqueness of solutions in forward time outside the origin. However, given that we are interested in uniform stability properties with respect to the initial time t0 , the previously introduced denitions remain unchanged (as seen in, for example, [86, Denitions 1 and 2] for uniform FTS/FxTS denitions and in [START_REF] Khalil | Nonlinear systems[END_REF]Dention 4.4] for both uniform stability and uniform asymptotic stability).

For this reason and to avoid unnecessarily complicating the notations, we will only invoke the non-autonomous system (2.2) when studying the PTS concept. It is worth noting that alternative non-uniform stability notions could be employed in this Thesis with some modications following the lines of [START_REF] Khalil | Nonlinear systems[END_REF]Denition 4.4] Let us consider the following scalar system:

ż(t) = -cz(t), c > 0, t ≥ t0 ≥ 0, z( t0 ) = z 0 , (2.3) (2.4)
whose solution is given by z(t) = z 0 e -c(t-t0) .

(2.5)

It is clear from (2.3) that the origin is the equilibrium point. In addition, from (2.5) we can see rst that the system is stable (|z(t)| ≤ |z 0 | : ∀t ≥ t0 ). Furthermore, from (2.5) we recover that when t tends to innity, the trajectories of (2.3) converge to the origin. Thus, the origin of (2.3) is ES (and thus also AS).

Example 2.2 (FTS)

Next, let us consider the following scalar system:

ż(t) = -c{z(t)} α , c > 0, α ∈ (0, 1), t ≥ t0 ≥ 0, z( t0 ) = z 0 , (2.6) (2.7)
whose solution is

z(t) =    sign(z 0 ) |z 0 | 1-α -c(1 -α)(t -t0 ) 1 1-α , t < t0 + |z0| 1-α c(1-α) , 0, t ≥ t0 + |z0| 1-α c(1-α) .
(2.8)

One can see that (2.6) has an equilibrium at the origin and from (2.8) we can see rst that the system is stable (always because |z(t)| ≤ |z 0 | : ∀t ≥ t0 ). Furthermore, from (2.8) we recover that the trajectories of (2.6) converge to the equilibrium in a nite time T (z 0 ) = t0 + |z0| The evolution of (2.8) (in logarithmic scale on the right) for t0 = 0, α = 0.8, c = 15, and 4 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, and in black using z 0 = 1000. We can observe that the larger the initial condition, the larger the settling time (i.e. the times of convergence depend on the initial condition).

Remark 2.2

Note that the oscillations of the solutions (in the neighborhood of 10 -20 ) shown on the right-hand side of Figure 2.1 are due to our ODE solver reaching the maximum precision of the machine, and thus should be discarded.

Example 2.3 (FxTS)

Now, let us consider the following scalar system:

ż(t) = -c{z(t)} α1 -c{z(t)} α2 , c > 0, α 1 ∈ (0, 1), α 2 > 1, t ≥ t0 ≥ 0, z( t0 ) = z 0 .
(2.9)

(2.10)

Notice that solutions of (2.9) can only be computed explicitly for some particular cases. For instance, when α 2 = 2 -α 1 , the solutions are given by

z(t) =    sign(z 0 ) tan arctan |z 0 | 1-α1 -c(1 -α 1 )(t -t0 ) 1 1-α 1 , t < t0 + arctan(|z0| 1-α 1 ) c(1-α1) , 0, t ≥ t0 + arctan(|z0| 1-α 1 ) c(1-α1)
.

(2.11)

As before, we can clearly see that (2.9) has an equilibrium at the origin and from (2.11) that the system is stable and the trajectories of (2.9) converge to the equilibrium in a nite time

T (z 0 ) = t0 + arctan(|z0| 1-α 1 ) c(1-α1)
upper-bounded by T max = t0 + π 2c(1-α1) which means that the system is FxTS (see Figure 2.2). Example 2.4 (PdTS) Now, let us consider the following scalar system: The evolution of (2.11) (in logarithmic scale on the right) for t0 = 0, α 1 = 0.25, α 2 = 1.75, c 1 = c 2 = 5, and 4 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, and in black using z 0 = 1000. We can observe that the settling time is upper bounded by a constant that does not depend on the initial conditions (i.e. the time of the convergence does not depend on the initial conditions).

ż(t) = - π 2(1-α1)Tc {z(t)} α1 + {z(t)} 2-α1 , T c > 0, α 1 ∈ (0, 1), t ≥ t0 ≥ 0, z( t0 ) = z 0 .
whose solutions of (2.12) are given by

z(t) =    sign(z 0 ) tan arctan |z 0 | 1-α1 -π 2Tc (t -t0 ) 1 1-α 1 , t < t0 + 2Tc arctan(|z0| 1-α 1 ) π , 0, t ≥ t0 + 2Tc arctan(|z0| 1-α 1 ) π . (2.14)
As before, we can clearly see that (2.12) has an equilibrium at the origin and from (2.14) that the system is stable and the trajectories of (2.12) converge to the equilibrium in a nite time

T (z 0 ) = t0 + 2Tc arctan(|z0| 1-α 1 )
π upper-bounded by T max = t0 + T c which means that the system is PdTS (see Figure 2.3). The evolution of (2.14) (in logarithmic scale on the right) for t0 = 0, α 1 = 0.5, T c = 0.3, and 5 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, in black using z 0 = 1000, and in orange using z 0 = 10000. We can observe that the settling time is upper bounded by a constant that is chosen a priori independently of the initial conditions. Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings Example 2.5 (PTS)

Let us now consider the following scalar system:

     ż(t) = - cT t0 + T -t 2 z(t), c > 0, t ∈ [ t0 , t0 + T ), t0 ≥ 0, T > 0, z( t0 ) = z 0 , (2.15) (2.16)
whose solution is given by

z(t) = z 0 e c 2 T e -c 2 T 2 t0 +T -t .
(2.17)

In this case, we can clearly see that (2.15) has an equilibrium at the origin and from (2.17) that the system is stable (when it is dened) and the trajectories of (2.15) converge to the equilibrium in a nite time t0 + T prescribed independently of the initial conditions as shown in Figure 2 The evolution of (2.17) (in logarithmic scale on the right) for t0 = 0, T = 1, c = 1, and 4 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, and in black using z 0 = 1000.

z(t) 2 ≤ β z 0 2 , µ(t -t0 ) , (2.21)
for every t ∈ [ t0 , t0 + T ), any z 0 ∈ Ω.

Remark 2.3

Following similar arguments to the ones given in Remark 2.1, the previously introduced alternative denitions of the stability concepts remain unchanged when dealing with non-autonomous systems of the form (2.2).

Remark 2.4

Note that a simple characterization of the stability concepts for the scalar case of (2.1) when K ∈ L 1 loc (Ω) (i.e. K : Ω ⊂ R → R) can be formulated using the Landau notation 1 . More precisely, the origin of (2.1) is asymptotically stable (AS) if and only if K ∈ S := {K ∈ L 1 loc (Ω) :

K(z) = 0 ⇔ z = 0, zK(z) > 0, ∀z ∈ Ω \ {0}} ; nite-time stable (FTS) if and only if K ∈ E k0,a0 = {K ∈ S : K(z) ∼ z→0 k 0 {z} a0 , lim |z|→∞ K(z) = 0}
, where {•} is the signed power function (see the Notation Section), a 0 ∈ [0, 1), and k 0 > 0;

xed-time stable (FxTS) if and only if K ∈ E k∞,a∞ k0,a0

= {K ∈ E k0,a0 : K(z) ∼ |z|→∞ k ∞ {z} a∞ }; where a 0 ∈ [0, 1), a ∞ > 1, k 0 > 0, and k ∞ > 0. Example 2.6 Let a 0 ∈ [0, 1), a ∞ > 1, k 0 > 0, k 1 > 0, k 2 > 0, k ∞ >
0 and ψ be any continuous positive function which is zero at 0 and at ∞

K 1 (z) = k 0 {z} a0 (1 + ψ(z)), K 2 (z) = (k 0 {z} a0 + k ∞ {z} a∞ )(1 + ψ(z)), K 3 (z) = k 0 {z} a0 1 + k 1 |z| a∞-a 0 k 2 k2 , (2.22)
(2.23)

(2.24) then K 1 ∈ E k0,a0 , K 2 ∈ E k∞,a∞ k0,a0 , K 3 ∈ E k∞,a∞ k0,a0
where in this last case

k ∞ = k 0 k k2 1 . 1 f (z) ∼ x→a g(z) if and only if f (z)-g(z) g(z)
-→ x→a 0. Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings

Lyapunov-based approach for ODEs

The use of Denitions 2.1 and 2.2 requires to know the explicit solution of (2.1) which can be hard to achieve for general classes of dierential equations (nonlinear ODEs for instance or even for PDEs). Therefore, Lyapunov analysis is a powerful tool that can provide sucient conditions to determine the stability of general nonlinear systems without the need of computing their solutions explicitly. The key idea behind this technique is to use and analyze the properties of a scalar function, known as a Lyapunov function, which maps the system's state to a real number (some kind of "distance" to the origin), to determine whether the system is stable or not.

In this subsection, we recall the main Lyapunov analysis results for autonomous ODEs starting with the well-known Lyapunov stability theorem.

We recall that Ω ⊂ R n is an open connected set containing the origin, with n ∈ N * . Let V : Ω → R be a continuously dierentiable function such that

1. V (0) = 0 and V (z) > 0 in Ω\{0}, (V is positive-denite); 2. V (z) ≤ 0 in Ω, ( V (z) is negative-semidenite).
Then, the origin of (2.1) is Lyapunov stable on Ω. Moreover, if we replace condition 2 by

3. V (z) < 0 in Ω\{0}, ( V (z) is negative-denite),
then, the origin of (2.1) is AS on Ω.

If in addition, Ω = R n , and V is radially unbounded (i.e. V (z) → +∞ as z 2 → +∞, z ∈ R n ) then the stability properties are global.

Remark 2.5

When the function V satises the rst two conditions, V is referred to as a Lyapunov function, whereas when V satises conditions 1 and 3, V is called a strict Lyapunov function.

Remark 2.6

Notice that the fact that V is continuously dierentiable on Ω, implies that the time derivative of V along the trajectories of (2.1) is given by

V (z) = ∂V ∂z K(z) = n i=1 ∂V ∂z i K i (z), z ∈ Ω\{0}, (2.25)
where we recall that K : Ω ⊂ R n → R n is a continuous function with K(0) = 0 given in (2.1).

To illustrate this result, let us look at the following example:

Example 2.7

Consider the following second-order ODE:

ż1 (t) = z 2 (t), t ≥ t0 ≥ 0, ż2 (t) = -c 1 z 1 (t) 1 5 -c 2 z 2 (t) 1 3 , t ≥ t0 ≥ 0, (2.26)
(2.27)

with z(t) = [z 1 (t), z 2 (t)] ∈ R 2 , and c 1 , c 2 > 0.

Let V be given by the following function:

V (z) = c 1 5 6 |z 1 | 6 5 + 1 2 z 2 2 , (2.28)
It easy to check that V satises Condition 1: V (0) = 0 and V (z) = c 1 Let V : Ω → R and W : Ω\{0} → R be two continuously dierentiable functions such that 1. V (0) = 0, V (z) > 0 and W (z) > 0 in Ω\{0};

V (z) = ∂V (z) ∂z 1 ż1 + ∂V (z) ∂z 2 ż2 , = ∂V (z) ∂z 1 z 2 + ∂V (z) ∂z 2 -c 1 z 1 5 1 -c 2 z 1 3 2 = c 1 z 1 5 1 z 2 + z 2 -c 1 z 1 5 1 -c 2 z 1 3 2 , = -c 2 |z 2 | 4 3 , = -c 2 3 |z 2 | 4 ≤ 0, (2.29)
2. V (z) ≤ 0 in Ω.
Then, the origin of (2.1) is Lyapunov stable on Ω. Moreover, if we replace condition 2 by

3. V (z) ≤ -W (z) in Ω\{0},
then, the origin of (2.1) is AS on Ω.

If in addition, Ω = R n , and V is radially unbounded then the origin is globally AS. Remark 2.7 Note that if in addition, W (z) = -cV (z) for c > 0, then the origin of (2.1) is (globally) ES. Remark 2.8

It is worth mentioning that the Lyapunov-based characterization of stability and asymptotic stability, previously introduced, can be seamlessly adapted to t the framework of non-autonomous ODEs of the form (2.2). To do that, we start rst by making the function of interest V depend explicitly on the time variable t in a continuously dierentiable way (i.e. V : R + × Ω → R is a continuously dierentiable function on [ t0 , +∞) × Ω); Then we replace conditions 1. and 3.

given in Theorem 2.2, respectively, by the following two alternative conditions as elaborated in [68, Theorems 4.84.10]:

1 . W 1 (z) ≤ V (t, z) ≤ W 2 (z) in [ t0 , +∞) × Ω, Chapter 2.
Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings

3 . V (t, z) = ∂V (t,z) ∂t + ∂V (t,z) ∂z K(t, z) ≤ -W 3 (z), in [ t0 , +∞) × Ω\{0},
where W 1 , W 2 , and W 3 are continuous positive denite functions on Ω. Let V : Ω → R be a continuously dierentiable function such that 1. V (0) = 0 and V (z) > 0 in Ω\{0};

2. V (z) ≤ -cV (z) α in Ω for some real numbers c > 0 and α ∈ (0, 1).

Then, the origin of (2.1) is FTS and the settling time T is a continuous function on Ω that satises

T (z) ≤ 1 c(1 -α) V (z) 1-α , ∀z ∈ Ω.
(2.34)

If in addition, Ω = R n , and V is radially unbounded, then the origin of (2.1) is a globally FTS.

To illustrate this, let us give an example.

Example 2.8 (FTS)

Consider the following second-order ODE:

     ż1 (t) = z 2 (t) - c 2 {z 1 (t)} 2α-1 , t ≥ t0 ≥ 0, ż2 (t) = -z 1 (t) - c 2 {z 2 (t)} 2α-1 , t ≥ t0 ≥ 0, (2.35) (2.36) with z(t) = [z 1 (t), z 2 (t)] ∈ R 2 , c > 0, α ∈ 1 2 , 1 .
The conditions of Theorem 2.3 are satised for the following choice of function V :

V (z) = |z 1 | 2 + |z 2 | 2 .
(2.37) Indeed, V satises Condition 1 (i.e. V (0) = 0 and V (z) > 0 for all z = [0, 0]) and condition 2. Let V : Ω → R be a continuously dierentiable function and r : R + → R + be a continuous positive denite function such that 1. V (0) = 0 and V (z) > 0 in Ω\{0};

V (z) = ∂V (z) ∂z 1 ż1 + ∂V (z) ∂z 2 ż2 , = ∂V (z) ∂z 1 z 2 - c 2 {z 1 } 2α-1 + ∂V (z) ∂z 2 -z 1 - c 2 {z 2 } 2α-1 , = 2z 1 z 2 - c 2 {z 1 } 2α-1 + 2z 2 -z 1 - c 2 {z 2 } 2α-1 , = -c z 2α 1 + z 2α 2 , ≤ -c |z 1 | 2 + |z 2 | 2 α , = -cV (z) α . ( 2
2. ε 0 ds r(s) < +∞ for some ε > 0;

3. V (z) ≤ -r(V (z)) in Ω.
Then, the origin of (2.1) is FTS and the settling-time function T is continuous on Ω and satises the inequality:

T (z) ≤ V (z) 0 ds r(s) , ∀z ∈ Ω.
(2.44)

If in addition, Ω = R n , and V is radially unbounded then the origin is globally FTS.

The following two results extend Theorems 2.3 and 2.4 to the case of xed-time stability. Let V : Ω → R be a continuously dierentiable function such that 1. V (0) = 0 and V (z) > 0 in Ω\{0};

2. V (z) ≤ -c 1 V (z) α1 -c 2 V (z) α2
in Ω for some real numbers c 1 > 0, c 2 > 0, α 1 ∈ (0, 1), and α 2 > 1.

Then, the origin of (2.1) is FxTS and the settling time T is a continuous function on Ω that satises

T (z) ≤ 1 c 1 (1 -α 1 ) + 1 c 2 (α 2 -1)
, ∀z ∈ Ω.

(2.45)

If in addition, Ω = R n , and V is radially unbounded, then the origin of ( Let V : Ω → R be a continuously dierentiable function and r : R + → R + be a continuous positive denite function such that 1. V (0) = 0 and V (z) > 0 in Ω\{0};

2.

sup z∈Ω V (z) 0 ds r(s) < +∞ for some ε > 0 3. V (z) ≤ -r(V (z)) in Ω.
Then, the origin of (2.1) is FxTS and the settling-time function T is continuous on Ω and satises the inequality:

T (z) ≤ sup z∈Ω V (z) 0 ds r(s) , ∀z ∈ Ω. (2.46)
If in addition, Ω = R n , and V is radially unbounded then the origin is globally FxTS.

Consider the following example:

Example 2.9 (FxTS)

Consider the following second-order ODE:

     ż1 (t) = z 2 (t) - c 1 2 {z 1 (t)} 2α1-1 - c 2 2 {z 1 (t)} 2α2-1 , t ≥ t0 ≥ 0, ż2 (t) = -z 1 (t) - c 1 2 {z 2 (t)} 2α1-1 - c 2 2 {z 2 (t)} 2α2-1 , t ≥ t0 ≥ 0, (2.47)
(2.48) Let T max > 0 be chosen a priori and assume that (2.1) depends on some tunable parameters ρ ∈ R l , l ∈ N. Let κ ∈ K 1 (a class-K 1 function) be dierentiable in R\{0} and depending on ρ, and let V : Ω → R be a continuously dierentiable function such that 1. V (0) = 0 and V (z) > 0 in Ω\{0};

with z(t) = [z 1 (t), z 2 (t)] ∈ R 2 , c 1 , c 2 > 0, α 1 ∈ 1 2 ,
2. V (z) ≤ - 1 (1-p)Tmax κ(V (z)) p κ (V (z)) in Ω\{0} for some 0 ≤ p < 1.
Then, the origin of is PdTS in a time t0 + T (z 0 ) where sup z0∈Ω T (z 0 ) ≤ T max . Moreover, if 2) is replaced by an equality, then sup z0∈Ω T (z 0 ) = T max .

If in addition, Ω = R n , and V is radially unbounded, then the origin of (2.1) is a globally PdTS. Let T max > 0 be chosen a priori and assume that (2.1) depends on some tunable parameters ρ = [ρ 1 , . . . , ρ 6 ] ∈ R × R 5 + . Let V : Ω → R be a continuously dierentiable function such that 1. V (0) = 0 and V (z) > 0 in Ω\{0};

2. V (z) ≤ -ρ1 Tmax (ρ 2 V (z) ρ3 + ρ 4 V (z) ρ5 ) ρ6 in Ω\{0} for ρ is chosen such that ρ 6 ρ 3 < 1, ρ 6 ρ 5 > 1 and ρ 1 = Γ 1-ρ6ρ3 ρ5-ρ3 Γ ρ6ρ5-1 ρ5-ρ3 α ρ6 Γ(ρ 6 )(ρ 5 -ρ 3 ) ρ 2 ρ 4 1-ρ6ρ3 ρ5-ρ3 (2.49)
where Γ is the Gamma Function (see Notation section).

Then, the origin of (2.1) is PdTS in a time t0 + T (z 0 ) where sup z0∈Ω T (z 0 ) ≤ T max . Moreover, if 2) is replaced by an equality, then sup z0∈Ω T (z 0 ) = T max .

If in addition, Ω = R n , and V is radially unbounded, then the origin of (2.1) is a globally PdTS.

Proof. The proof is a direct application of Proposition 2.1 with the particular selections of κ(r

) = I 1-ρ6ρ3 ρ5-ρ3 , ρ6ρ5-1 ρ5-ρ3 , ρ4r ρ 5 -ρ 3 βr ρ 5 -ρ 3 +ρ2
, with p = 0 (see [ V : R + × Ω → R is a continuously dierentiable function on [ t0 , +∞) × Ω); Then we replace conditions 1. given in each result by the following alternative condition as shown for example in [START_REF] Song | Prescribed-Time Control and Its Latest Developments[END_REF], [START_REF] Moulay | Finite time stability conditions for non-autonomous continuous systems[END_REF]:

1 . W 1 (z) ≤ V (t, z) ≤ W 2 (z) in [ t0 , +∞) × Ω,
where W 1 and W 2 are continuous positive denite functions on Ω.

As highlighted in Remarks 2.1-2.9, dealing with prescribed-time stability (PTS), we will need to deal with a non-autonomous ODEs of the form (2.2). For this reason, we nish this subsection, by a Lyapunov-based characterization of the prescribed-time stability (PTS) concept for this class of systems.

Proposition 2.2

Let T > 0 be chosen a priori and let V :

[ t0 , t0 + T ) × Ω → R be a continuously dierentiable function such that 1. V (t, 0) = 0 and V (t, z) > 0 in [ t0 , t0 + T ) × Ω\{0}; 2. V (t, z) ≤ -c(t -t0 )V (t, z) in Ω for some continuous function c : [ t0 , t0 + T ) → R + such that t t0 c(s -t0
)ds is nite for all t ∈ [ t0 , t0 + T ) and becomes unbounded as t approaches t0 + T . Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings

Then, the origin of (2.1) is PTS in a time t0 + T .

If in addition, Ω = R n , and V is radially unbounded, then the origin of (2.1) is a globally PTS.

Proof. The proof follows directly from Theorem 2.1 and from Grönwall's lemma on condition 2).

Indeed, applying Grönwall's lemma, we get

V (t, z(t)) ≤ e -t t0 c(s-t0)ds V ( t0 , z( t0 )).
(2.50) from which we can deduce that V (t, z(t)) → 0 as t → t0 + T and in turn z(t) 2 → 0 as t → t0 + T .

Example 2.10 (PTS)

Based on [START_REF] Espitia | Prescribed-time predictor control of LTI systems with input delay[END_REF], [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF], consider the following second-order ODE:

       ż1 (t) = z 2 (t), t ∈ [ t0 , t0 + T ), ż2 (t) = -(r 1 + r 2 )γ 2 (t -t0 ) -2 γ2,0T γ 2 (t -t0 ) z 2 (t) -r 1 r 2 γ 2 (t -t0 ) 2 z 1 (t), t ∈ [ t0 , t0 + T ), (2.51) (2.52)
with

z(t) = [z 1 (t), z 2 (t)] ∈ R 2 , t0 ≥ 0, r 1 , r 2 > 0 such r 1 = r 2 ,
T > 0 is a priori xed, and the function γ 2 (•) is chosen as follows:

γ 2 (s) := γ 2 2,0 T 2 (T -s) 2 , s ∈ [0, T ), (2.53) for γ 2,0 > 0 where t t0 γ 2 (s -t0 )ds = γ 2,0 T γ 2 (s -t0 ) t t0 , (2.54) = γ 2,0 T ( γ 2 (t -t0 ) -γ 2,0 ) -→ +∞, (2.55)
as t → t0 + T . Let V be given by the following function:

V (t, z) = V 1 (t, z) + V 2 (t, z), (2.56)
where

V i (t, z) = r i z 1 + γ 2 (t -t0 ) -1 z 2 2 , i ∈ {1, 2}.
(2.57)

It easy to check that V satises Condition 1: V (t, 0) = 0 and V (t, z) > 0 for all t ∈ [ t0 , t0 + T ] and all z = [0, 0]. Moreover, from the time derivative of V i along the trajectories of (2.51)-(2.52), we get

Vi (t, z) = ∂V i (t, z) ∂z 1 ż1 + ∂V i (t, z) ∂z 2 ż2 + ∂V i (t, z) ∂t , = ∂V i (t, z) ∂z 2 -r 1 r 2 γ 2 (t -t0 ) 2 z 1 -(r 1 + r 2 )γ 2 (t -t0 ) -2 γ2,0T γ 2 (t -t0 ) z 2 + ∂V i (t, z) ∂z 1 z 2 + ∂V i (t, z) ∂t , = 2 r i z 1 + γ 2 (t -t0 ) -1 z 2 r i z 2 + γ 2 (t -t0 ) -1 -r 1 r 2 γ 2 (t -t0 ) 2 z 1 -(r 1 + r 2 )γ 2 (t -t0 ) -2 γ2,0T γ 2 (t -t0 ) z 2 -2 γ2,0T γ 2 (t -t0 ) -1 2 z 2 , = 2 r i z 1 + γ 2 (t -t0 ) -1 z 2 [-r 2-i z 2 -r i r 2-i γ 2 (t -t0 )z 1 ] , = -2r 2-i γ 2 (t -t0 ) r i z 1 + γ 2 (t -t0 ) -1 z 2 2 , = -2r 2-i γ 2 (t -t0 )V i (t, z), ≤ -2r min γ 2 (t -t0 )V i (t, z), (2.58)
(2.59)

(2.60) (2.61) (2.62) (2.63) (2.64)
which yields

V (t, z) ≤ -2r min γ 2 (t -t0 )V (t, z). (2.65)
Thus, V satises Condition 2. and by Proposition 2.2 the system is (2.51)-(2.52) is PTS.

Notice that using Grönwall's lemma, we obtain

V (t, z(t)) ≤ e -2rmin t t0
γ2(s-t0)ds V ( t0 , z( t0 )).

(2.66)

Moreover, since V can be rewritten as follows:

V (t, z) = z 1 z 2 r 1 r 2 γ 2 (t -t0 ) -1 γ 2 (t -t0 ) -1 r 1 γ 2 (t -t0 ) -1 r 2 γ 2 (t -t0 ) -1 z 1 z 2 , (2.67) = z 1 z 2 r 2 1 + r 2 2 (r 1 + r 2 )γ 2 (t -t0 ) -1 (r 1 + r 2 )γ 2 (t -t0 ) -1 2γ 2 (t -t0 ) -2 z 1 z 2 , (2.68)
:= z W (γ 2 (t -t0 ))z, (2.69)
we get the following coercivity property:

λ min ( W (γ 2 (t -t0 ))) z 2 2 ≤ V (t, z) ≤ λ max ( W (γ 2 (t -t0 ))) z 2 2 .
(2.70)

By applying this property on (2.66), we obtain

z(t) 2 2 ≤ λ max ( W (γ 2,0 ))λ min ( W (γ 2 (t -t0 ))) -1 e -2rmin t t0 γ2(s-t0)ds z( t0 ) 2 2 , (2.71)
from this, we can clearly see that z(t) 2 2 → 0 as t → t0 + T .

Non-smooth tools in nite-dimensional setting

In this section, we will recall some results of nite-time stabilization of nonlinear systems starting by [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF] (2002) where Hong managed to use a constructive iterative method to solve the problem Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings of nite-time stabilization of a power chain of power-integrators of the form:

                   ż1 (t) = z 2 (t) m1 , t ≥ t0 ≥ 0, . . . żn-1 (t) = z n (t) mn-1 , t ≥ t0 ≥ 0, żn (t) = U (t), t ≥ t0 ≥ 0, z( t0 ) = z 0 , (2.72)
(2.73)

(2.74) (2.75)
where

z(t) = [z 1 (t), . . . , z n (t)]
T ∈ R n and U (t) ∈ R are the system's state and control, and

m i > 0, i ∈ {1, . . . , n} are odd integers, n ∈ N * .
This recursive approach was based on the paper of Coron and Praly [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF] for desingularization, which basically consists in using, for each step j of the induction, a homogeneous C 1 -Lyapunov function V j and a virtual controller U j dened for the chain of power-integrators of the size j while depending recursively on the previous induction steps. More precisely, for each step j of the induction, the virtual controller U j is constructed in terms of previous virtual controllers, to make the closed-loop chain of power integrators of size j homogeneous with a negative degree.

After that, the closed-loop chain of power integrators of size j is proved to be AS on some given compact set (i.e. Vj is denite negative on some compact set). This last property is then extended to all the domain using the homogeneity property of Vj , and the closed-loop system (2.72)-(2.75), with the control

U (t) = U n (z 1 (t), • • • , z n (t)), is deduced to be FTS, in the light of Theorem A.3,
where the last virtual controller U n (z

1 (t), • • • , z n (t)) is given in terms of all the previous virtual controllers U i (z 1 (t), • • • , z i (t)), i ∈ {1, .
. . , n -1} which are dened as follows:

   U 1 (z 1 (t)) = 0, i ∈ {1, . . . , n -1} U i+1 (z 1 (t), . . . , z i+1 (t)) = -l i+1 {{z i+1 (t)} miαi -{U i (z 1 (t), . . . , z i (t))} αi } ri+1 +k ri+1 m i α i .
(2.76)

(2.77)

with ri , α i-1 , i ∈ {1, . . . , n} and k be constants satisfying the following given inequalities:

r1 = 1, . . . , ri = ri-1 + k m i-1 , ri > -k > 0, i ∈ {1, . . . , n}, α 0 = r2 , (α i m i + 1) ri+1 (α i-1 m i-1 + 1) ri > 0, i ∈ {1, . . . , n -2}, α n-1 > 0,
for m 0 = 1 and for some positive constants l i , i ∈ {1, . . . , n}.

Remark 2.10

Note that for small n, the recursive approach is easy to handle and the resulting controller is simple. However, when n is large (e.g. n ≥ 4), the recursive approach becomes hard to use and the controller becomes complicated.

Remark 2.11

It is possible to replace the odd integer exponents m i , i ∈ {1, . . . , n}, by some positive real exponents mi ∈ R + , i ∈ {1, . . . , n}. However, for that, we need to replace the terms z i (t) mi-1 by {z

i (t)} mi-1 = sign(z i (t))|z i (t)| mi-1 .

A homogeneity-based method

It is worth mentioning that if all the exponents m i , i ∈ {1, . . . , n}, are equal to 1, then the previous result still holds. Note that even though this system looks easy to stabilize using the Lyapunov analysis, sadly, if we take

U (t) = n j=1 k j {z j (t)} αj , (2.78)
for k j < 0, α j ∈ (0, 1), j ∈ {1, ..., n}, we can not nd an explicit Lyapunov function, when n > 2 without using the recursive method. For the particular case of n = 2 with α 1 = α2

2-α2 , one can use the following Lyapunov function

V (z 1 , z 2 ) = -k 1 (2 -α 2 ) 2 |z 1 | 2 2-α 2 + z 2 2 2 . (2.79)
which is continuously dierentiable, r-homogeneous of degree 2 (see Appendix A) and

V (z) = k 2 |z 2 | 1+α . Since k 1 < 0 and k 2 < 0, the function V is positive-denite, and V is negative- semidenite.
A direct application of the LaSalle invariance principle shows that the origin of the double chain of integrators is globally AS. Being homogeneous of negative degree implies that the system is globally FTS (see Appendix A). If we want to prove the asymptotic stability without relying on the LaSalle principle, we have to change the Lyapunov function as Bernuau et al.

showed in [START_REF] Bernuau | Finite-time output stabilization of the double integrator[END_REF]. More precisely, if the control parameters k 1 and k 2 are taken such that:

k 2 < M M 2 (1 -α 2 ) -1 , (2.80)
and

k 1 ∈ 1 2 + (2 -α 2 )k 2 2M (3 -α 2 ) - √ ∆ 2(3 -α 2 ) , M (1 -α 2 )k 2 , (2.81)
with

∆ = (3 -α 2 ) + (2 -α 2 )k 2 M 2 + 4(1 -α 2 )(3 -α 2 )k 2 2 , (2.82)
and

M = (2 -α 2 ) 1-α2 2 α α2 2 2 2 , (2.83)
then, one can ensure stabilization of the double chain of integrators in nite time. Moreover, the proof of stability property can be checked directly using a dierent Lyapunov function.

Time-Varying tools in nite-dimensional setting

In this section, we recall the rst key contribution of prescribed-time stabilization for nonlinear systems which was achieved by Song et al. [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF]

                   ż1 (t) = z 2 (t), t ∈ [ t0 , t0 + T ) , . . . żn-1 (t) = z n (t), t ∈ [ t0 , t0 + T ) , żn (t) = f (z(t), t) + b(z(t), t)U (t), t ∈ [ t0 , t0 + T ) , z( t0 ) = z 0 , (2.84)
(2.85)

(2.86) (2.87)
where the state z(t) = [z 1 (t), ..., z n (t)] ∈ R n , the control U (t) ∈ R, the initial condition z 0 ∈ R n , the initial time t0 , the prescribed-time T ≥ 0, and the functions b, f are possible uncertain and non-vanishing such that: Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings

b ≤ |b(z, t)| < ∞ for some known b > 0, and for all z ∈ R n , t ∈ R + . |f (z, t)| ≤ d(t)ψ(z), where d(t) is a disturbance with an unknown bound d [ t0,t] := sup τ ∈[ t0,t] |d(τ )|
and ψ(z) ≥ 0 is a known scalar-valued continuous function.

It is worth mentioning that when b(z, t) and f (z, t) are known, Song et al., provided a similar proof in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF].

The main idea of the proof is the use of a scaling transformation involving a blow-up function that grows to innity as the time gets closer to the settling time t0 + T . This function is given explicitly as follows :

µ 1 (s) = T T -s , s ∈ [0, T ) .
(2.88)

The following step is to design a controller that stabilizes the system in the scaled state representation. To conclude, they used the inverse transformation to transport the stability property from the scaled state representation to the original one.

To better illustrate the idea of Song et al. [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF], let us focus on the scalar case,

ż(t) = f (z(t), t) + b(z(t), t)U (t), t ∈ [ t0 , t0 + T ) , z ( t0 ) = z 0 , , (2.89) (2.90)
where z(t) ∈ R and U (t) ∈ R. In this case, the scaling transformation is given by

ω(t) = µ (t -t0 ) z(t), (2.91)
with

µ (t -t0 ) = T 1+m ( t0 + T -t) 1+m = µ 1 (t -t0 ) 1+m = ν 1 (t -t0 ) -1-m = ν(t -t0 ) -1 . (2.92)
where m > 0 is a positive constant and ν is a monotonically decreasing function with the properties that ν(0) = 1 and ν(T ) = 0.

Using this transformation, the original scalar system (2.89)-(2.90) is converted into

ω(t) = μ (t -t0 ) ν(t -t0 )ω(t) + µ (t -t0 ) f (ν(t -t0 )ω(t), t) + b(ν(t -t0 )ω(t), t)U (t) . (2.93)
This last system is proven, using the Lyapunov function

V (ω(t)) = |ω(t)| 2 2
, to be prescribed-time input-to-state stable in time t0 + T and converge to zero under the following control (see [98, Proof of Theorem 4]):

U (t) = - 1 b k + λψ(ν(t -t0 )ω(t)) 2 + 1 + m T ω(t), (2.94)
for some positive constants k > 0 and λ > 0. Using the inverse transformation the control input is then expressed in terms of the original state z(t) as follows:

U (t) = - 1 b k + λψ(z(t)) 2 + 1 + m T µ(t -t0 )z(t), (2.95)
and the prescribed-time input-to-state stability and the convergence to zero properties are transferred to (2.89)-(2.90). Moreover, the following inequality is recovered

|z(t)| ≤ ν (t -t0 ) e -kT m+n-1 (µ1(t-t0) m+n-1 -1) |z ( t0 ) | + d [ t0,t] 2 √ kλ , (2.96)
for all t ∈ [ t0 , t0 + T ).

Non-Asymptotic concepts in an innite-dimensional setting 35 2.3 Non-Asymptotic concepts in an innite-dimensional setting

In the framework of innite-dimensional systems, all the denitions mentioned earlier in Subsection 2.2 can be adapted by substituting the Euclidean norm with an appropriate spatial norm, as we will explain below. In this section, we will focus on one-dimensional partial dierential equations of the rst order with respect to time. This class of PDEs can be described using an evolution equation of the form:

z t (t, •) = Az(t, •), (2.97)
with t ≥ t0 ≥ 0, where A :

D(A) ⊂ L 2 (a, b) → L 2 (a, b) (a, b ∈ R, a ≤ b) is a (possibly unbounded)
linear operator, t0 is the initial time, and z 0 will denote the initial condition.

Example 2.11

Consider the following heat equation with homogeneous Dirichlet-type boundary conditions

z t (t, x) = z xx (t, x), t ≥ t0 , x ∈ [0, 1], z(t, 0) = z(t, 1) = 0, t ≥ t0 , (2.98)
(2.99)

with t0 ≥ 0. For this example, we can clearly rewrite (2.98)-(2.99) as in (2.97) by taking the linear operator A as A = ∂ xx , which corresponds to the second-order partial derivative with respect to the variable x, dened on the domain

D(A) = {z ∈ H 1 (0, 1) : z xx ∈ L 2 (0, 1), z(0) = z(1) = 0} := H 2 (0, 1) ∩ H 1 0 (0, 1).
Now, let us give the following global stability denition:

Denition 2.3

The origin of system (2.97) is said to be stable [START_REF] Coron | Boundary stabilization in nite time of onedimensional linear hyperbolic balance laws with coecients depending on time and space[END_REF]Denition 1.1.] if and only if for every ε > 0, there exists δ > 0 such that, for every t0 ≥ 0 and z 0 ∈ L 2 (a, b),

z 0 L 2 (a,b) ≤ δ =⇒ z(t, •) L 2 (a,b) ≤ ε , for any t ≥ t0 ;
asymptotically stable (AS) 2 if and only if it is stable and

lim t→+∞ z (t, •) L 2 (a,b) = 0, for any z 0 ∈ L 2 (a, b);
exponentially stable (ES) if and only if it is AS and there exist M > 0 and β > 0 such that

z (t, •) L 2 (a,b) ≤ M z 0 L 2 (a,b) e -βt ,
for any t ≥ t0 and any z 0 ∈ L 2 (a, b);

nite-time stable (FTS) if and only if it is stable and for any z 0 ∈ L 2 (a, b) there ex-

ists 0 < T z0 < +∞ such that z (t, •) L 2 (a,b) = 0 for all t ≥ t0 + T z0 . The functional T (z 0 ) = inf T z0 ≥ 0 : z (t, •) L 2 (a,b) = 0, ∀t ≥ t0 + T z0 denes
T (z 0 ) < +∞;
prescribed-time stable (PTS) if and only if it is FxTS and the settling time T is chosen in advance independently of the initial condition z 0 ∈ L 2 (a, b).

In the above denitions uniformly with respect to initial time t0 and global with respect to initial conditions z 0 have been omitted for the sake of brevity.

Remark 2.12

Using the comparison functions, it is possible to give equivalent denitions to Denition 2.3.

Remark 2.13

Similarly to the nite-dimensional case (see Remark 2.1), when studying the prescribed-time stability (PTS) notion, we will have to deal with a non-autonomous version of (2.97) of the form (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Dierential Equations[END_REF]Chapter 5]):

z t (t, •) = A(t)z(t, •), (2.100)
where in this case A(t) :

D(A(t)) ⊂ L 2 (a, b) → L 2 (a, b) (a, b ∈ R, a ≤ b) is a time-depending (possibly unbounded) linear operator (e.g. A(t) = ∂ xx + a(t) Id where a : [ t0 , +∞) → R is a continuous function).
As mentioned in Remark 2.1, we are interested, in this Thesis, in uniform stability properties with respect to t0 . In this situation, the previously introduced stability denitions remain unchanged as presented for example in [154, Denition 1.1]). To avoid unnecessarily complicating the notations, we will only switch to non-autonomous systems when dealing with the PTS concept.

Lyapunov-based approach for PDEs

In this section, we present some sucient conditions for the stability notions in the framework of innite-dimensional systems.

Let Ω ⊂ D(A) is an open connected set containing the origin.

Denition 2.4

A continuous function V : Ω ⊂ D(A) → R + is said to be coercive if and only if there exist two class-K ∞ functions σ 1 and σ 2 such that

σ 1 ( z L 2 (a,b) ) ≤ V (z) ≤ σ 2 ( z L 2 (a,b) ), (2.101)
for any z ∈ Ω.

Based on Denition 2.3, the following results generalize the Lyapunov stability Theorem 2.1 for innite-dimensional systems.

Proposition 2.3 ([178, Proposition 3.2] )

Let V : Ω ⊂ D(A) → R + be a continuous function, continuously dierentiable on Ω\{0}, satisfying the coercivity condition, such that

1. V (z) = ∂V (z) ∂z , Az L 2 (a,b) ≤ 0 in Ω.
Then, the origin of (2.97) is Lyapunov stable. Moreover, if there exists a class-

K ∞ function σ 3 such that 2. V (z) ≤ -σ 3 ( z L 2 (a,b) ) in Ω,
then, the origin of (2.97) is asymptotically stable (AS).

Corollary 2.2

Let V : Ω ⊂ D(A) → R + be a continuous function, continuously dierentiable on Ω\{0}, satisfying the coercivity condition, such that

1. V (z) ≤ -cV (z)
in Ω, with some real positive constant c > 0.

Then, the origin of (2.97) is exponentially stable (ES).

Proof. The proof is a direct application of Proposition 2.3 and the coercivity property of V .

Example 2.12

Consider the following parabolic reaction-diusion PDE with a homogeneous Dirichlet-type boundary

z t (t, x) = z xx (t, x) -(λ 0 -λ)z(t, x), t ≥ t0 , x ∈ [0, 1], z(t, 0) = z(t, 1) = 0, t ≥ t0 , (2.102)
(2.103)

where the initial time t0 ≥ 0, the reaction term (λ 0 -λ) > 0 with λ, λ 0 ∈ R.

Let V be given by

V (z(t, •)) = z(t, •) 2 L 2 (0,1) = 1 0 |z(t, x)| 2 dx. (2.104)
Clearly, V is a continuous function, continuously dierentiable, coercive, and positive-denite. Moreover, by computing the time derivative of V along the solution of (2.102)-(2.103), we get

d dt V (z(t, •)) = 1 0 z(t, x) (z xx (t, x) -(λ 0 -λ)z(t, x)) dx, ≤ -(λ 0 -λ)V (z(t, •)), (2.105)
(2.106)

where the term 1 0 z(t, x)z xx (t, x)dx can be proven to be less or equal to 0 by double integration by parts. Thus, V satises the conditions of Corollary

z t (t, x) = z x (t, x), t ≥ t0 , x ∈ [0, 1], z(t, 1) = 0, t ≥ t0 , (2.107)
(2.108)

with t0 ≥ 0. Let V be given by (see [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]):

V (z(t, •)) = 1 0 e σx |z(t, x)| 2 dx, σ > 0.
(2.109) Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings V is a continuous function, continuously dierentiable, coercive, positive-denite, and the time derivative of V along the solution of (2.107)-(2.108) satises,

d dt V (z(t, •)) = 2 1 0 e σx z(t, x)z x (t, x)dx, = 1 0 e σx ∂|z(t, x)| 2 ∂x dx, = -|z(t, 0)| 2 -σ 1 0 e σx |z(t, x)| 2 dx, ≤ -σV (z(t, •)), ≤ -σ z(t, •) 2 L 2 (0,1) , (2.110)
(2.111)

(2.112) (2.113) (2.114)
where we integrated by parts and then used the coercivity of V . Then, from Proposition 2. Proposition 2.4

Let V : Ω ⊂ D(A) → R + be a continuous function, continuously dierentiable on Ω\{0}, satisfying the coercivity condition, such that

1. V (z) ≤ 0 in Ω. 2. there exists 0 ≤ T V (z0) < +∞ such that V (z(t, •)) = 0 for all t ≥ T V (z0) .
Then, the origin of (2.97) is FTS and the settling time T (V (z 0 )) is dened similarly as in Denition 2.3. Moreover, if

3. sup z0∈Ω T (V (z 0 )) < +∞.
Then, the origin of (2.97) is FxTS.

Proof. The proof of Proposition 2.4 is a direct application of Denition 2.4.

Remark 2.14

Note that if one can nd a suitable coercive Lyapunov function then, using the Comparison principle, one may reduce the complexity of the stability analysis to the study of a simple scalar ODE

(2.1) (i.e. ż = -K(z), z ∈ R where K : Ω ⊂ R n → R n is a continuous function with K(0) = 0 given in (2.1)).
In light of Remarks 2.4 and 2.14, we present the following result:

Proposition 2.5

Under the conditions of Proposition 2.4, if there exists a continuous function K :

R + → R + such that d dt V (z(t, •)) ≤ -K(V (z(t, •))), (2.115)
and K ∈ E k0,a0 (resp. K ∈ E k∞,a∞ k0,a0 ), given in Remark 2.4, with a 0 ∈ [0, 1) (resp. a 0 ∈ [0, 1), a ∞ > 1), then the origin of (2.97) is FTS (resp. FxTS).

Example 2.14 (FTS)

Inspired from [START_REF] Polyakov | On homogeneous nite-time control for linear evolution equation in hilbert space[END_REF], [181][183], consider the following reaction-diusion PDE with a homogeneous Dirichlet-type boundary

     z t (t, x) = z xx (t, x) - c 2 z(t, x) z(t, •) 2-2α L 2 (0,1) , t ≥ t0 , x ∈ [0, 1], z(t, 0) = z(t, 1) = 0, t ≥ t0 , (2.116)
(2.117)

with t0 ≥ 0, c > 0 and α ∈ (0, 1). Let V be given by

V (z(t, •)) = z(t, •) 2 L 2 (0,1) = 1 0 |z(t, x)| 2 dx. (2.118)
Clearly, V is a continuous function, continuously dierentiable, coercive, and positive-denite. Moreover, by computing the time derivative of V along the solution of (2.116)-(2.117), we get

d dt V (z(t, •)) = 2 1 0 z(t, x) z xx (t, x) - c 2 z(t, x) z(t, •) 2-2α L 2 (0,1) dx, = 2 1 0 z(t, x)z xx (t, x)dx -c 1 z(t, •) 2-2α L 2 (0,1) 1 0 |z(t, x)| 2 dx, = 2 1 0 z(t, x)z xx (t, x)dx -c z(t, •) 2α L 2 (0,1) , ≤ -cV (z(t, •)) α , (2.119) (2.120) (2.121) (2.122)
where the 1 0 z(t, x)z xx (t, x)dx can be proven to be less or equal to 0 by double integration by parts. Then, V satises the conditions of Proposition 2.5 for K ∈ E k0,a0 : s → cs α which means that (2.116)-(2.117) is FTS.

To nish this section, let us give a Lyapunov characterization of prescribed-time stability for non-autonomous PDEs of the form (2.100), Proposition 2.6 Let T > 0 be chosen a priori and let V :

[ t0 , t0 + T ) × Ω → R be a continuously dierentiable function such that 1. V (t, z) ≤ -c(t-t0 )V (t, z) in [ t0 , t0 +T )×Ω for some continuous function c : [ t0 , t0 +T ) → R + such that t t0 c(s-t0 )ds is nite for all t ∈ [ t0 , t0
+T ) and becomes unbounded as t approaches t0 + T .

Then, the origin of (2.97) is PTS in a time t0 + T .

Proof. The proof is a direct consequence of Grönwall's lemma applied to Condition 2, Inspired from [START_REF] Espitia | Boundary time-varying feedbacks for xed-time stabilization of constant-parameter reactiondiusion systems[END_REF], consider the following reaction-diusion PDE with a homogeneous Dirichlettype boundary and time-varying damping term:

V (t, z) ≤ V (t 0 , z 0 )e -t t0 c(s-t0)ds . (2.
z t (t, x) = z xx (t, x) -(γ m (t -t0 ) -λ)z(t, x), t ∈ [ t0 , t0 + T ), x ∈ [0, 1], z(t, 0) = z(t, 1) = 0, t ∈ [ t0 , t0 + T ), (2.124)
(2.125)

where t0 ≥ 0, T > 0 is a priori xed and the function γ m (•) is chosen as follows:

γ m (t -t0 ) := γ m m,0 T m ( t0 + T -t) m , m ∈ N * , m = 1, (2.126) for γ m,0 > 0. Let V be given by V (z(t, •)) = z(t, •) 2 L 2 (0,1) = 1 0 |z(t, x)| 2 dx. (2.127)
Clearly, V is a continuous function, continuously dierentiable, coercive, and positive-denite. Moreover, by computing the time derivative of V along the solution of (2.124)-(2.125), we get

d dt V (z(t, •)) = 2 1 0 z(t, x) (z xx (t, x) -(γ m (t -t0 ) -λ)z(t, x)) dx, ≤ -2(γ m (t -t0 ) -λ)V (z(t, •)), (2.128)
(2.129)

where the term 1 0 z(t, x)z xx (t, x)dx can be proven to be less or equal to 0 by double integration by parts. Moreover, we have

t t0 (γ m (s -t0 ) -λ)ds = γm,0T m-1 γ m (s -t0 ) m-1 m -λs t t0 , (2.130)
= γm,0T m-1 γ m (t -t0 ) m-1 m -γ m-1 m,0 -λ(t -t0 ) -→ +∞, (2.131)
as t → t0 + T . Thus, V satises the conditions of Proposition 2.6 and (2.124)-(2.125) is PTS.

Finite/prescribed-time boundary stabilization key results for 1D reaction-diusion PDEs

In this subsection, we present key results on nite-time and prescribed-time boundary stabilization for 1D reaction-diusion PDEs. The main employed method is the backstepping approach. Before we enter into details, it is important to remember that the backstepping approach for PDEs has gained a lot of popularity in recent years due to its eectiveness in addressing the challenging problem of stabilizing PDEs. This technique provides a systematic and structured approach to designing feedback control laws that stabilize PDE systems by transforming the original PDEs into a suitable target system that satises the chosen stability property. To better illustrate this approach, let us briey study rst the problem of exponential boundary stabilization (inspired from [START_REF] Smyshlyaev | Explicit state and output feedback boundary controllers for partial dierential equations[END_REF]) of 1D parabolic reaction-diusion PDEs with boundary control:

           z t (t, x) = z xx (t, x) + λz(t, x), t ≥ t0 , x ∈ [0, 1], z(t, 0) = 0, t ≥ t0 , z(t, 1) = U (t), t ≥ t0 , z( t0 , x) = z 0 (x), (2.132)
(2.133)

(2.134) (2.135) with initial time t0 ≥ 0, reaction term λ ∈ R, state z(t, •) ∈ D(A) := {z ∈ H 1 (0, 1) : zxx ∈ L 2 (0, 1), z(0) = 0, z(1) = U (t)} where the operator A = ∂ 2 ∂x 2 + λ Id with ∂ 2
∂x 2 is the second-order partial derivative with respect to the space variable x, Id is the identity operator, the control U (t) ∈ R, and the initial condition z 0 ∈ D(A).

Notice that the open-loop system (2.132)-(2.135) (i.e. z(t, 1) = 0) is unstable for any λ > π 2 . Let us assume that λ > π 2 , otherwise, there is no reason to study the stabilization problem.

We recall that the key steps of the Backstepping approach are as follows:

The rst and the most important step of this approach is to choose a suitable target system of the same type of (2.132)- (2.135). For instance, one can choose the following exponentially stable system (see Example 2.12):

     ω t (t, x) = ω xx (t, x) -λ 0 ω(t, x), t ≥ t0 , x ∈ [0, 1], ω(t, 0) = 0, t ≥ t0 , ω(t, 1) = 0, t ≥ t0 , (2.136) (2.137) (2.138)
for λ 0 > 0.

z(t, x) = ω(t, x) + x 0 L exp (x, y)ω(t, y)dy.
(2.140)

The nal step is to recover the expression of the control either from the direct transformation (2.139), for x = 1, as follows:

U exp (t) = 1 0 K exp (1, y)z(t, y)dy, (2.141)
or from the inverse transformation, for x = 1, as follows: As we discussed earlier on, for parabolic PDEs, the backstepping approach has contributed signicantly to solving the problem of prescribed-time stabilization. The rst result related to this topic was introduced in 2017 by Coron and Nguyen [START_REF] Coron | Null controllability and nite-time stabilization for the heat equations with variable coecients in space in one dimension via backstepping approach[END_REF] where they solved the problems of null controllability followed by the problem of nite-time stabilization for a general class of parabolic PDEs with variable coecients in space, by making use of some periodic time-varying feedback laws.

U exp (t) = 1 0 L exp (1, y)ω(t,
To illustrate the key points of their contribution, let us focus on the reaction-diusion equation (2.132)-(2.135) dened on [ t0 , t0 + T ) where t0 + T will be considered as the settling time. For the null controllability problem, the idea is to consider a strictly increasing sequence of real numbers (t j ) j∈N dened such as lim j→+∞ t j = t0 + T (e.g.

t 0 = t0 , t j = ( t0 + T ) 1 -1 2j 2 for j ∈ N * )
and to apply the backstepping approach (whose steps we recalled above), on each time interval [t j , t j+1 ). To be precise, we choose the target system to be the same as (2.136)-(2.138). The only dierence is the term λ 0 is replaced by λ 0,j on each time interval where the sequence (λ 0,j ) j∈N is chosen to be an increasing sequence of positive numbers converging to innity (e.g. λ 0,0 = λ 0 , λ 0,j = (j + 1) 8 for j ∈ N * ). After that, one uses backstepping transformations (2.139) and (2.140) were we replace, respectively, the kernels K exp (x, y) and L exp (x, y) by K FTS j (x, y) and L FTS j (x, y)

on each time interval. Making use of these kernels, the boundary control is dened as a sequence of controllers

U FTS (t) = 1 0 K FTS j (1, y)z(t, y)dy, (2.143)
for every t ∈ [t j , t j+1 ), j ∈ N. Under this boundary control, the authors were able to prove that the solution of the closed-loop system (2.132)-(2.135) satises the following property:

z(t, •) L 2 (0,1) → 0, (2.144)
as t → T , for any initial condition z 0 ∈ L 2 (0, 1) with z 0 L 2 (0,1) ≤ M for some M > 0 (see [161, Theorem 1]). In particular, they proved that

|U FTS (t)| → 0, (2.145)
as t → T .

Furthermore, the authors proved that using the following modied version of (2.143):

U FTS (t) = 1 0 K FTS j (1, y)z(t, y)dy, t ∈ [t j , t j+1 ), j ∈ N, j ≤ N -1, φ j ( z(t, •) L 2 (0,1) ) 1 0 K FTS j (1, y)z(t, y)dy, t ∈ [t j , t j+1 ), j ∈ N, j ≥ N.
(2.146)

the original system (2.132)-(2.135) is FTS (see [161, Theorem 2]
), where N is a xed chosen positive large integer, and (φ j ) j∈N is a sequence of class C 1 (R)-functions such that 0 ≤ φ j ≤ 1, φ j (s) = 1 for s ≤ µ j and φ j (s) = 0 if s ≥ 2µ j with (µ j ) j∈N be dened by µ n := e -j α , ∀j ∈ N for some real number α ∈ (4, 5).

Inspired by this result and with the emergence of the prescribed-time stability concept in [START_REF] Song | Time-varying feedback for regulation of normal-form nonlinear systems in prescribed nite time[END_REF], Espitia et al. [START_REF] Espitia | Boundary time-varying feedbacks for xed-time stabilization of constant-parameter reactiondiusion systems[END_REF] solved the problem of prescribed-time stabilization of (2.132)-(2.135)

(restricted to [ t0 , t0 + T ) × [0, 1])
U (t) = 1 0 K PTS 2 (1, s, t -t0 )z(t, s)ds, (2.147)
or equivalently

U (t) = 1 0 L PTS 2 (1, s, t -t0 )ω(t, s)ds. (2.148)
is ensured to be PTS (see [START_REF] Espitia | Boundary time-varying feedbacks for xed-time stabilization of constant-parameter reactiondiusion systems[END_REF]Theorem 3]) provided that 2γ 2,0 T > 1.

(2.149)

In particular, z(t, •) L 2 (0,1) → 0, and |U (t)| → 0 when t → t0 + T .

Positioning of our contributions with respect to results

for systems which are of the same class as those addressed in this thesis

Having completed a literature review on the most relevant works on Non-Asymptotic concepts for some classes of innite-dimensional systems, and having presented the background and the main tools that we rely on, we end this chapter by positioning our contributions (summarized in Chapter 1, Section 1.3) with respect to up-to-now-known results of nite/xed/prescribed-time stabilization/estimation for 3 dierent classes of dierential equations including LTI systems with input delay or output delay, 1D reaction-diusion PDE with boundary control, and 1D reactiondiusion PDE with boundary input delay. In the table below, we refer to our contributions using `Ci)', and to extensions that can be possibly achieved using our contributions using the `' symbol.

The open problems are left `blank'. In this chapter, we design a nonlinear observer for linear time-invariant (LTI) systems in the observable form with measurement delay. The design guarantees the convergence of the error state to the origin within a nite time that depends on the initial conditions. To accomplish this, we reformulate the original system into a cascade ODE-PDE system where the PDE part is a transport equation that models the eect of the delay on the output. We construct the nonlinear gains in a way that ensures the error system is nite-time stable (FTS). To prove this, we use an invertible backstepping transformation to convert the error system into a target system which is shown to be nite-time stable using Lyapunov-based analysis and homogeneity tools. We use the inverse transformation to transfer this property to the error system.

Introduction

State estimation is one of the most important topics in control theory. Indeed, usually, the full state is not available and/or sensors costs are prohibitive. Moreover, in engineering applications, delays are often in dierent parts of the system. In particular, the delay can be caused by transmitting the state's measurements via a communication network. Furthermore, nite-time observation becomes very desired in several applications that require the transient process to nish in a nite amount of time.

For instance, in Teleoperation, the surgery is performed using a robotic hand controlled by the doctor using a remote control. The robotic hand consists of a number of rigid links connected with joints. Using the Euler-Lagrange equation the dynamic model of the robotic hand is given by M (q)q + C(q, q) + G(q) = U (t -D),

(3.1)
where U is the applied torque generated by the actuators, q ∈ R n , q ∈ R n and q ∈ R n are respectively vectors of joint positions, velocities, and accelerations, M (q) is the inertia matrix, C(q, q) the vector of Coriolis and centrifugal forces, G(q) the vector of gravitational forces, and D is the delay. In this situation, transmitting the measurements through a communication channel causes a time delay in reception and response. In addition, using a nite-time observer in this situation is critical, as the surgery requires as much precision in the control objective, and also it ends in a nite period.

In this chapter, we design a robust nonlinear observer which compensates for the delay and that is able to estimate the sensor dynamics (innite-dimensional), as well. Indeed, we study the problem of nite-time observation for LTI systems in the observable form with a delay in the output measurement. To solve this problem, we rst rewrite the original system into an ODE-PDE form, where the PDE part models the eect of the delay on the output. Next, we propose an observer with nonlinear injection terms, that reconstruct the system's states in a nite time.

To achieve this, we use an invertible backstepping transformation to transform the error system into a nite-time stable target system. Finally, we use the inverse transformation to transfer this property to the error system.

This chapter is organized as follows: In Section 3.2, we introduce the main class of systems that we are interested in (LTI systems with output delay), and we rewrite it into an ODE-PDE setting.

Next, we present our nonlinear nite-time observer from which we recover the error system. Then, we use a suitable choice of transformation to link this error system to a well-chosen target system.

In Section 3.3, we make use of the transformation and the fact that the target system is FTS to prove the main result of this chapter. In Section 3.4 we consider a numerical example to illustrate the results.

Problem Statement

Let us consider an nth-order linear time-invariant system with delayed output. We assume that delay-free system is observable. Then one can rewrite the system in the form

ż(t) = Az(t) + BU (t), t ≥ t 0 , Y (t) = Cz(t -D), t ≥ t 0 , (3.2) (3.3)
where

A =          a n-1 1 0 • • • 0 a n-2 0 . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . 1 a 0 0 • • • • • • 0          , B =      0 . . . 0 1      , and C = [ 1 0 • • • 0 ] , are the system's matrices ([a 0 , . . . , a n-1 ] ∈ R n ), z(t) = [z 1 (t), . . . , z n (t)] ∈ R n (n ∈ N * ) is the state with initial condition z(θ) = z 0 (θ) = [z 1,0 (θ), . . . , z n,0 (θ)] for all θ ∈ [t 0 -D, t 0 ], where t 0 ≥ 0 is the initialization time, U (t) ∈ R
           ż(t) = Az(t) + BU (t), t ≥ t 0 , u t (t, x) = u x (t, x), t ≥ t 0 , x ∈ [0, D], u(t, D) = Cz(t), t ≥ t 0 , Y (t) = u(t, 0), t ≥ t 0 , (3.4)
(3.5)

(3.6) (3.7)
where the initial condition of (3.5)-(3.6) is taken as u (t 0 , x) = Cz (t 0 -D + x) for all t 0 0.

Next, we propose the following observer system for (3.4)-(3.7):

       ż(t) = Aẑ(t) + BU (t) -e AD L (Y (t) -û(t, 0)) , t ≥ t 0 , ût (t, x) = ûx (t, x) -Ce Ax L(Y (t) -û(t, 0)), t ≥ t 0 , x ∈ [0, D], û(t, D) = C ẑ(t), t ≥ t 0 , (3.8)
(3.9)

(3.10)
where the initial condition is given by ẑ(θ) = ẑ0 (θ) = [ẑ 1,0 (θ), . . . , ẑn,0 (θ)] for all θ ∈ [t 0 -D, t 0 ] and the observer gain function

L(•) = [L 1 (•), • • • , L n (•)
] will be designed later. Note that usually in practice, we take ẑ(θ) = 0.

Remark 3.1

ż(t) = Aẑ(t) + BU (t) -e AD L Y (t) -Ce -AD ẑ(t) + C t t-D e A(t-D-θ) BU (θ) dθ , (3.11)
where the nonlinear function L = [ L1 (•), 2. predictor-based observers do not have a straightforward extension to systems with timevarying delays, state-dependent delays, or distributed delays, whereas our formulation opens a path for extending the obtained results to these kinds of systems and more general ones.

3. lastly, predictor-based observers do not provide a construction of the Lyapunov Krassovki functional, which allows us to asses directly the desired FTS property of the error system and to estimate the time of convergence (the settling time), whereas our observer helps in establishing an explicit Lyapunov-based stability for the error system.

Our goal is to prove that, for a well-designed vector function L, [ẑ(t), û(t, •)] solution of the observer (3.8)-(3.10) converges in a nite-time to [z(t), u(t, •)] solution of (3.4)-(3.7). To achieve this, we consider the following error variables: Next, we consider the following transformation:

z(t) = e -AD (z(t) -ẑ(t)), ũ(t, x) = u(t, x) -û(t, x),
       ż(t) = Az(t) + L(ũ(t, 0)), t ≥ t 0 , ũt (t, x) = ũx (t, x) + Ce Ax L(ũ(t, 0)), t ≥ t 0 , x ∈ [0, D], ũ(t, D) = Ce AD z(t), t ≥ t 0 , (3.14)
ω(t, x) = ũ(t, x) -Ce Ax z(t), (3.17
     ż(t) = Az(t) + L(ω(t, 0) + C z(t)), t ≥ t 0 , ωt (t, x) = ωx (t, x), t ≥ t 0 , x ∈ [0, D], ω(t, D) = 0, t ≥ t 0 , (3.18)
(3.19) (3.20)
which is equivalent to the following system: In order to characterize transformation (3.17), let us compute its time and spatial derivatives and the value of ω(t, D).

                         ż1 (t) = a n-1 z1 (t) + z2 (t) + L 1 (ω(t, 0) + z1 (t)) , t ≥ t 0 , . . . żn-1 (t) = a 1 z1 (t) + zn (t) + L n-1 (ω(t, 0) + z1 (t)) , t ≥ t 0 , żn (t) = a 0 z1 (t) + L n (ω (t, 0) + z1 (t)) , t ≥ t 0 , ωt (t, x) = ωx (t, x), t ≥ t 0 , x ∈ [0, D], ω(t, D) = 0, t ≥ t 0 . (3.21)
ωt (t, x) = ũt (t, x) -Ce Ax ż(t), (3.26)
= ũx (t, x)

+ Ce Ax L(ũ(t, 0)) -Ce Ax Az(t) -Ce Ax L(ũ(t, 0)), (3.27)
= ũx (t, x) -Ce Ax Az(t).

(3.28)

(3.29)
On the other hand,

ωx (t, x) = ũx (t, x) -CAe Ax z(t), (3.30)
= ωt (t, x), (3.31) where we have used the fact that Ae Ax = e Ax A. Using ũ(t, D) = Ce AD z(t) in (3.17) at x = D, we obtain

ω(t, D) = ũ(t, D) -Ce AD z(t), (3.32)
= Ce AD z(t) -Ce AD z(t), Choosing L 1 and L 2 respectively as follows:

L 1 (ω(t, 0) + z1 (t)) = -k 1 {ω (t, 0) + z1 (t)} α1 -a 1 (ω (t, 0) + z1 (t)) , (3.35)
and

L 2 (ω(t, 0) + z1 (t)) = -k 2 {ω(t, 0) + z1 (t)} α2 -a 0 (ω(t, 0) + z1 (t)) , (3.36)
gives us

           ż1 (t) = z2 (t) -k 1 {ω(t, 0) + z1 (t)} α1 -a 1 ω(t, 0), t ≥ t 0 , ż2 (t) = -k 2 {ω(t, 0) + z1 (t)} α2 -a 0 ω(t, 0), t ≥ t 0 , ωt (t, x) = ωx (t, x), t ≥ t 0 , x ∈ [0, D], ω(t, D) = 0, t ≥ t 0 , (3.37)
(3.38)

(3.39) (3.40)
where

k 1 > 0, k 2 > 0, α 1 ∈ 1 2 ,
L 1 (ω(t, 0) + z1 (t)) = -k 1,0 {ω (t, 0) + z1 (t)} α1,0 -k 1,∞ {ω (t, 0) + z1 (t)} α1,∞ -a 1 (ω (t, 0) + z1 (t)) , (3.41)
and

L 2 (ω(t, 0) + z1 (t)) = -k 2,0 {ω (t, 0) + z1 (t)} α2,0 -k 2,∞ {ω (t, 0) + z1 (t)} α2,∞ -a 0 (ω(t, 0) + z1 (t)) , (3.42)
for some

k 1,0 > 0, k 2,0 > 0, k 1,∞ > 0, k 2,∞ > 0, α 1,0 ∈ 1 2 , 1 , α 2,0 ∈ (0, 1), α 1,∞ > 1, and α 2,∞ > 1.

Characterisation of the observer-gain functions for nth-order LTI systems

In the general case, we can choose the observer-gain functions in a similar fashion to the secondorder case:

L i (ω(t, 0) + z1 (t)) = -k i {ω (t, 0) + z1 (t)} αi -a n-i (ω (t, 0) + z1 (t)) , i ∈ {1, . . . , n}. (3.43) for some positive coecients k i > 0, i ∈ {1, • • • , n} and some constants α i ∈ (0, 1), i ∈ {1, • • • , n}.
However, in this case, the nite-time stability analysis of the target system when ω ≡ 0 will be hard. To avoid this, we can propose instead (see [START_REF] Zimenko | Homogeneity based nite/xedtime observers for linear mimo systems[END_REF]Theorem 6]):

L (ω(t, 0) + z1 (t)) = (p |ω(t, 0) + z1 (t)|) ν- d ln p |ω(t, 0) + z1 (t)| 1 L F T (ω(t, 0) + z1 (t)) + L 0 (ω(t, 0) + z1 (t)) , (3.44
G d (see Appendix A), v ∈ [-1, 0) and L 0 = H + Īn -1 X 0 ∈
R n are chosen such that the system of matrix equations and inequalities

     AH -HA -A -X 0 C = 0, CH = 0, H + H T + 2 Īn > 0, (3.45)
(3.46) (3.47) is feasible with H ∈ M n,n (R), X 0 ∈ R n , and p = X 1 2 1 ∈ R, L F T = P -1 X 2 ∈ R n , ∈ R + , G d = vH + Īn are parameters chosen such that for some ξ > τ > 0, θ ∈ R + the system of matrix inequalities                                    P A + A T P + P L 0 C + C T L T 0 P + X 2 C + C T X T 2 + ξP P P -X1 ≤ 0, P > 0, X1 > 0, X 1 > 0, τ X 1 X T 2 X 2 θP ≥ 0 P -C T X 1 C ≥ 0, vP H + vH T P + 2 P > 0, ∀λ ∈ [0, 1] : λ 2 e (vH+v Īn) ln λ -Īn T X1 e (vH+v Īn) ln λ -Īn ≤ 1 θ P, (3.48)
(3.49)

(3.50) (3.51) (3.52) (3.53) is feasible with P ∈ M n,n (R), X1 ∈ M n,n (R), X 1 ∈ R, X 2 ∈ R n .
Moreover, in this case, if, in addition, the following inequality is feasible:

-

λ
L F xT (ω(t, 0) + z1 (t)) = 1 2 p v- 1 |ω(t, 0) + z1 (t)| v- d ln p 1 1 |ω(t, 0) + z1 (t)| 1 +p -ν 2 |ω(t, 0) + z1 (t)| -ν d -ln p 1 2 |ω(t, 0) + z1 (t)| 1 L F xT (ω(t, 0) + z1 (t)) + L 0 (ω(t, 0) + z1 (t)), (3.55)
where d is a dilation with the generator

G d = vH + Īn , L 0 = H + Īn -1 X 0 ∈ R n , and v ∈ [-1, 0)
are chosen such that the system of matrix equations is feasible with H ∈ M n,n (R), X 0 ∈ R n , and ∈ R + . In addition, the parameters pi = δ i X is feasible with

           AH -HA -A -X 0 C = 0, CH = 0, H + H T + 2 Īn > 0, 2 Īn -vH -vH T > 0,
1 2 1 ∈ R, i ∈ {1, 2}, L F xT = P -1 X 2 ∈ R n , ∈ R + are chosen such that for some ξ i > τ > 0, δ i ∈ R + , 3.3. Stability analysis 53 i ∈ {1, 2}, θ ∈ R + , the system of matrix inequalities                                                          τ X 1 X T 2 X 2 θP ≥ 0, P -C T X 1 C ≥ 0, vP H + vH T P + 2 P > 0, P A + A T P + P L 0 C + C T L T 0 P + X 2 C + C T X T 1 + ξ i P P P -Xi ≤ 0, P > 0, Xi > 0, X 1 > 0, ∀[λ 1 , λ 2 ] ∈ [0, 1] 2 : 1 4 λ 2 1 δ 1 λ 1 λ 1-i 2 v-d ln δ 1 λ 1 λ 1-i 2 + δ 2 λ 1 λ 2-i 2 ε-v d -ln δ 2 λ 1 λ 2-i 2 -2 Īn × Xi δ 1 λ 1 λ 1-i 2 v-d ln δ 1 λ 1 λ 1-i 2 + δ 2 λ 1 λ 2-i 2 ε-v d -ln δ 2 λ 1 λ 2-i 2 -2 Īn ≤ 1 θ P, (3.60)
P ∈ M n,n (R), Xi ∈ M n,n (R), i ∈ {1, 2}, X 1 ∈ R, and X 2 ∈ R n .

Stability analysis

In this section, we present and prove the main result of this chapter. To better illustrate, we focus on the case of second-order LTI systems for which we prove that the error state of (3.14)- (3.16) goes to zero within a nite time. To prove this result we need to establish some intermediate results. First, we prove that the solutions of the target system are bounded for all t ∈ [t 0 , t 0 + D].

Next, we estimate, in two dierent ways, the solution of the z-dynamics of the target system at time t 0 + D by a function that depends on the initial condition of the target system. Then, we prove that these solutions converge to zero within a nite time that can be estimated.

Lemma 3.1

Let t 0 ≥ 0, ω0 ∈ L 2 and bounded, and z0 = [z 1,0 , z2,0 ] ∈ R 2 . Then for any t ∈ [t 0 , t 0 + D] and any x ∈ [0, D], (z 1 , z2 , ω) the solution of the target system (3.37)-(3.40) is bounded. Moreover, the solution of the z-dynamics satises the following inequalities:

sup t∈[t0,t0+D] |z 1 (t)| 2 ≤ M 1 ω0 2 L 2 (0,D) + M 2 ( z0 2 ), (3.66)
sup

t∈[t0,t0+D] |z 2 (t)| 2 ≤ N 1 ω0 2 L 2 (0,D) + N 2 sup t∈[t0,t0+D] |z 1 (t)| 2 + N 3 ( z0 2 ), (3.67)
where

M 1 = 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 1 -4D 2 ε 2 (4D 2 k 2 2 + k 2 1 ) , M 2 ( z0 2 ) = 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 1 -4D 2 ε 2 (4D 2 k 2 2 + k 2 1 )
,

N 1 = 2D(k 2 + |a 0 |) 2 , N 2 = 4D 2 k 2 2 ε 2 , N 3 ( z0 2 ) = 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + z0
ε ∈ (0, 1) is chosen small enough so that 1 -4D 2 ε 2 4D 2 k 2 2 + k 2 1 is positive.
Proof. Using the initial condition ω(t 0 , x) = ω0 (x) and by the method of characteristics, we have that ω(t, x) = ω0 (x + t -t 0 ) for any t ≤ t 0 + D -x and ω(t, x) = 0 for any t ≥ t 0 + D -x. Thus ω is bounded for all t ∈ [t 0 , t 0 + D].

Next, integrating (3.38) from t 0 to t, gives us

z2 (t) = -k 2 t t0 {ω(s, 0) + z1 (s)} α2 ds -a 0 t t0
ω(s, 0)ds + z2,0 .

(3.69)

Using the fact that t 0 ≤ t ≤ t 0 + D, we get

|z 2 (t)| ≤ k 2 t0+D t0 |ω(s, 0) + z1 (s)| α2 ds + |a 0 | t0+D t0 |ω(s, 0)|ds + |z 2,0 |. (3.70) Using the fact that ∀(a, b) ∈ R 2 + , ∀β ∈ (0, 1) : (a + b) β ≤ a β + b β and since α 2 ∈ (0, 1), we obtain |z 2 (t)| ≤ k 2 t0+D t0 |ω(s, 0)| α2 ds + k 2 t0+D t0 |z 1 (s)| α2 ds + |a 0 | t0+D t0 |ω(s, 0)|ds + |z 2,0 |. (3.71)
Now, using the fact that ∀a ∈ R + , ∀β ∈ (0, 1) : a β ≤ βa + (1 -β) on the rst term and the fact that ∀a ∈ R + , ∀ε ∈ (0, 1), ∀β ∈ (0, 1) :

a β ≤ εa + (1 -β) β ε β 1-β on the second term, we get |z 2 (t)| ≤ α 2 k 2 t0+D t0 |ω(s, 0)|ds + (1 -α 2 )Dk 2 + |a 0 | t0+D t0 |ω(s, 0)|ds + k 2 ε t0+D t0 |z 1 (s)|ds + k 2 (1 -α 2 )D α 2 ε α 2 1-α 2 + |z 2,0 |, (3.72
+ k 2 ε t0+D t0 |z 1 (s)|ds + k 2 Dε -α 2 1-α 2 + |z 2,0 |, (3.73)
≤ (k 2 + |a 0 |) t0+D t0 |ω(s, 0)|ds + k 2 ε t0+D t0 |z 1 (s)|ds + Dk 2 1 + ε -α 2 1-α 2 + |z 2,0 | . (3.74)
By squaring this last inequality, we nd

|z 2 (t)| 2 ≤ 2(k 2 + |a 0 |) 2 t0+D t0 |ω(s, 0)|ds 2 + 4k 2 2 ε 2 t0+D t0 |z 1 (s)|ds 2 + 4 Dk 2 1 + ε -α 2 1-α 2 + |z 2,0 | 2 . (3.75) (3.76)
Finally, using the Jensen inequality on each integral and Young's inequality on the last term, we get

|z 2 (t)| 2 ≤ 2D(k 2 + |a 0 |) 2 t0+D t0 |ω(s, 0)| 2 ds + 4Dk 2 2 ε 2 t0+D t0 |z 1 (s)| 2 ds + 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + |z 2,0 | 2 .
(3.77)

Stability analysis 55

Using the value of ω(t, 0) in the previous inequality, gives us

|z 2 (t)| 2 ≤ 2D(k 2 + |a 0 |) 2 t0+D t0 |ω 0 (s -t 0 )| 2 ds + 4Dk 2 2 ε 2 t0+D t0 |z 1 (s)| 2 ds + 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + |z 2,0 | 2 , (3.78) ≤ 2D(k 2 + |a 0 |) 2 D 0 |ω 0 (x)| 2 dx + 4Dk 2 2 ε 2 t0+D t0 |z 1 (s)| 2 ds + 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + |z 2,0 | 2 , (3.79) ≤ 2D(k 2 + |a 0 |) 2 ω0 2 L 2 (0,D) + 4Dk 2 2 ε 2 t0+D t0 |z 1 (s)| 2 ds + 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + |z 2,0 | 2 , (3.80)
which implies that

|z 2 (t)| 2 ≤ 2D(k 2 + |a 0 |) 2 ω0 2 L 2 (0,D) + 4D 2 k 2 2 ε 2 sup t∈[t0,t0+D] |z 1 (t)| 2 + 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + z0 2 2 .
(

Then,

sup t∈[t0,t0+D] |z 2 (t)| 2 ≤ N 1 ω0 2 L 2 (0,D) + N 2 sup t∈[t0,t0+D] |z 1 (t)| 2 + N 3 ( z0 2 ), (3.82)
where

N 1 = 2D(k 2 + |a 0 |) 2 , (3.83
)

N 2 = 4D 2 k 2 2 ε 2 , (3.84) N 3 ( z0 2 ) = 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + z0 2 2 .
(3.85)

Similarly, integrating (3.37) from t 0 to t, gives us

z1 (t) = t t0 z2 (s)ds -k 1 t t0 {ω(s, 0) + z1 (s)} α1 ds -a 1 t t0
ω(s, 0)ds + z1,0 . Now, using the fact that ∀a ∈ R + , ∀β ∈ (0, 1) : a β ≤ βa + (1 -β) on the second term and fact that ∀a ∈ R + , ∀ε ∈ (0, 1), ∀β ∈ (0, 1) :

a β ≤ εa + (1 -β) β ε β 1-β
on the third term of the previous inequality, we get

|z 1 (t)| ≤ t0+D t0 |z 2 (s)|ds + α 1 k 1 t0+D t0 |ω(s, 0)|ds + Dk 1 (1 -α 1 ) + k 1 ε t0+D t0 |z 1 (s)|ds + Dk 1 (1 -α 1 ) α 1 ε α 1 1-α 1 + |a 1 | t0+D t0 |ω(s, 0)|ds + |z 1,0 |, (3.89)
≤ t0+D t0 |z 2 (s)|ds + k 1 t0+D t0 |ω(s, 0)|ds + Dk 1 + k 1 ε t0+D t0 |z 1 (s)|ds + Dk 1 ε -α 1 1-α 1 + |a 1 | t0+D t0 |ω(s, 0)|ds + |z 1,0 |, (3.90)
≤ t0+D t0 |z 2 (s)|ds + (k 1 + |a 1 |) t0+D t0 |ω(s, 0)|ds + k 1 ε t0+D t0 |z 1 (s)|ds + Dk 1 1 + ε -α 1 1-α 1 + |z 1,0 | . (3.91)
Squaring this last inequality gives us

|z 1 (t)| 2 ≤ 4 t0+D t0 |z 2 (s)|ds 2 + 4(k 1 + |a 1 |) 2 t0+D t0 |ω(s, 0)|ds 2 + 4k 2 1 ε 2 t0+D t0 |z 1 (s)|ds 2 + 4 Dk 1 1 + ε -α 1 1-α 1 + |z 1,0 | 2 . (3.92)
Applying the Jensen's inequality on each integral and the Young's inequality on the last term, we obtain

|z 1 (t)| 2 ≤ 4D t0+D t0 |z 2 (s)| 2 ds + 4D(k 1 + |a 1 |) 2 t0+D t0 |ω(s, 0)| 2 ds + 4Dk 2 1 ε 2 t0+D t0 |z 1 (s)| 2 ds + 8 D 2 k 2 1 1 + ε -α 1 1-α 1 2 + |z 1,0 | 2 . (3.93)
Using the inequality (3.77) on the last inequality, gives us

|z 1 (t)| 2 ≤ 4D 2 2D(k 2 + |a 0 |) 2 t0+D t0 |ω(s, 0)| 2 ds + 4Dk 2 2 ε 2 t0+D t0 |z 1 (s)| 2 ds + 8 D 2 k 2 2 1 + ε -α 2 1-α 2 2 + z0 2 2 + 4D(k 1 + |a 1 |) 2 × t0+D t0 |ω(s, 0)| 2 ds + 4Dk 2 1 ε 2 t0+D t0 |z 1 (s)| 2 ds + 8 D 2 k 2 1 1 + ε -α 1 1-α 1 2 + z0 2 2 .
(3.94)

Stability analysis 57

By developing this last inequality and using the value of ω(t, 0), we get for all t ∈ [t 0 , t 0 + D]

|z 1 (t)| 2 ≤ 4Dε 2 4D 2 k 2 2 + k 2 1 t0+D t0 |z 1 (s)| 2 ds + 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 × t0+D t0 |ω(s, 0)| 2 ds + 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 , (3.95)
≤ 4Dε 2 4D 2 k 2 2 + k 2 1 t0+D t0 |z 1 (s)| 2 ds + 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 × t0+D t0 |ω 0 (s -t 0 )| 2 ds + 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 , (3.96)
≤ 4Dε 2 4D 2 k 2 2 + |k 1 | 2 t0+D t0 |z 1 (s)| 2 ds + 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 × D 0 |ω 0 (x)| 2 dx + 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 , (3.97)
≤ 4D 2 ε 2 4D 2 k 2 2 + k 2 1 sup t∈[t0,t0+D] |z 1 (t)| 2 + 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 × ω0 2 L 2 (0,D) + 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 , (3.98)
which implies that

sup t∈[t0,t0+D] |z 1 (t)| 2 ≤ 4D 2 ε 2 4D 2 k 2 2 + k 2 1 sup t∈[t0,t0+D] |z 1 (t)| 2 + 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 × ω0 2 L 2 (0,D) + 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 .
(3.99)

Then :

1 -4D 2 ε 2 4D 2 k 2 2 + k 2 1 sup t∈[t0,t0+D] |z 1 (t)| 2 ≤ 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 ω0 2 L 2 (0,D) + 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 .
(3.100)

Finally, choosing ε small enough so that 1

-4D 2 ε 2 4D 2 k 2 2 + k 2 1 is positive, (for instance, we can take ε = 8D 2 (4D 2 k 2 2 + k 2 1 ) - 1
2 ), gives us

sup t∈[t0,t0+D] |z 1 (t)| 2 ≤ M 1 sup t∈[t0,t0+D] |ω(t, 0)| 2 + M 2 ( z0 2 ), (3.101)
where

M 1 = 4D 2D 2 (k 2 + |a 0 |) 2 + (k 1 + |a 1 |) 2 1 -4D 2 ε 2 (4D 2 k 2 2 + k 2 1 )
,

(3.102) M 2 ( z0 2 ) = 8D 2 4D 2 k 2 2 1 + ε -α 2 1-α 2 2 + k 2 1 1 + ε -α 1 1-α 1 2 + 8 4D 2 + 1 z0 2 2 1 -4D 2 ε 2 (4D 2 k 2 2 + k 2 1 )
.

Lemma 3.2

Let ω0 ∈ L 2 and bounded, and ẑ0 , z 0 ∈ R 2 . For any t 0 , D > 0, we have

z(t 0 + D) 2 2 ≤ G 1 ( ω0 L 2 (0,D) , z 0 -ẑ0 2 ), (3.104)
where G 1 is the function given by

G 1 (s 1 , s 2 ) = (N 1 + N 2 M 1 )s 2 1 + N 2 M 2 ( e -AD 2 s 2 ) + N 3 ( e -AD 2 s 2 ) , (3.105)
where M 1 , M 2 , N 1 , N 2 , and N 3 are given in (3.68).

Proof. Using Lemma 3.1, we can clearly see that

z(t) 2 2 ≤ (N 1 + N 2 M 1 ) ω0 2 L 2 (0,D) + [N 2 M 2 ( z0 2 ) + N 3 ( z0 2 )] for all t ∈ [t 0 , t 0 + D], (3.106)
where M 1 , M 2 , N 1 , N 2 , and N 3 are given in (3.68).

Using (3.12)-(3.13), we obtain

z(t) 2 2 ≤ (N 1 + N 2 M 1 ) ω0 2 L 2 (0,D) + N 2 M 2 ( e -AD 2 z 0 -ẑ0 2 ) + N 3 ( e -AD 2 z 0 -ẑ0 2 ) . (3.107)
In particular for t = t 0 + D, we have

z(t 0 + D) 2 2 ≤ G 1 ω0 L 2 (0,D) , z 0 -ẑ0 2 . (3.108)
where G 1 is the function given by (3.105).

Lemma 3.3

For any t 0 , D > 0 and any ẑ0 in R 2 , we have

z(t 0 + D) 2 2 ≤ G 2 ( z 0 -ẑ0 2 ) , (3.109)
where G 2 is a function dened for any s ≥ 0 by

G 2 (s) = 4e 2D(|a0|+|a1|+1) e -AD 2 2 s 2 + D 2 (|a 0 | + |a 1 |) 2 M 2 + 2D 2 (k 1 + k 2 ) 2 ( M 2α1 + M 2α2 ) , (3.110)
with M is such that sup

t∈[t0,t0+D] |ũ(t, 0)| ≤ M . ( 3
|z 1 (t)| + |z 2 (t)| ≤ (|a 0 | + |a 1 | + 1) t t0 |z 1 (s)| + |z 2 (s)| ds + |z 1,0 | + |z 2,0 | + (k 1 + k 2 ) × t0+D t0 |ũ(s, 0)| α2 + |ũ(s, 0)| α1 ds + (|a 0 | + |a 1 |) t0+D t0 |ũ(s, 0)|ds. (3.116)
Using the Grönwall Lemma, we nd

|z 1 (t)| + |z 2 (t)| ≤ e (|a0|+|a1|+1)(t-t0) |z 1,0 | + |z 2,0 | + (k 1 + k 2 ) t0+D t0 |ũ(s, 0)| α2 + |ũ(s, 0)| α1 ds +(|a 0 | + |a 1 |) t0+D t0 |ũ(s, 0)|ds , (3.117
)

≤ e (|a0|+|a1|+1)D |z 1,0 | + |z 2,0 | + (k 1 + k 2 ) t0+D t0 |ũ(s, 0)| α2 + |ũ(s, 0)| α1 ds +(|a 0 | + |a 1 |) t0+D t0
|ũ(s, 0)|ds .

(3.118)

Squaring this inequality gives us

z(t) 2 2 ≤ (|z 1 (t)| + |z 2 (t)|) 2 ≤ e 2D(|a0|+|a1|+1) 2(|z 1,0 | + |z 2,0 |) 2 + 4(|a 0 | + |a 1 |) 2 × t0+D t0 |ũ(s, 0)|ds 2 + 4(k 1 + k 2 ) 2 × t0+D t0 |ũ(s, 0)| α2 + |ũ(s, 0)| α1 ds 2   . ( 3
z(t) 2 2 ≤ e 2D(|a0|+|a1|+1) 4 z0 2 2 + 4D(|a 0 | + |a 1 |) 2 t0+D t0 |ũ(s, 0)| 2 ds +8D(k 1 + k 2 ) 2 t0+D t0 |ũ(s, 0)| 2α2 + |ũ(s, 0)| 2α1 ds , (3.120)
≤ e 2D(|a0|+|a1|+1) 4 z0

2 2 + 4D 2 (|a 0 | + |a 1 |) 2 sup t∈[t0,t0+D] |ũ(t, 0)| 2 + 8D 2 (k 1 + k 2 ) 2 sup t∈[t0,t0+D] |ũ(t, 0)| 2α2 + 8D 2 (k 1 + k 2 ) 2 × sup t∈[t0,t0+D] |ũ(t, 0)| 2α1 . ( 3
z(t) 2 2 ≤ 4e 2D(|a0|+|a1|+1) z0 2 2 + D 2 (|a 0 | + |a 1 |) 2 M 2 + 2D 2 (k 1 + k 2 ) 2 ( M 2α1 + M 2α2 ) . (3.123)
This inequality is true for all t ∈ [t 0 , t 0 + D], in particular for t = t 0 + D, we have :

z(t 0 + D) 2 2 ≤ 4e 2D(|a0|+|a1|+1) z0 2 2 + D 2 (|a 0 | + |a 1 |) 2 M 2 + 2D 2 (k 1 + k 2 ) 2 ( M 2α1 + M 2α2 ) .
2 ≤ F (t -t 0 -D, z(t 0 + D) 2 ) + F (t -t 0 -D, z(t 0 + D) 2 ) 2 1+α , (3.126)
where F is an increasing function with respect to the second variable dened for any s ≥ 0 by

F (t -t 0 -D, s) = 1 P min -δ 1 -ν ν (t -t 0 -D) + (P max ) 1-ν ν s α+1 + s 2 1-ν ν ν ν -ν , (3.127) 3.3. Stability analysis 61 with α ∈ (0, 1), ν = α + 1, ν = 3α+1 2 .
In particular, when t tends to the settling time t 0 + D + T max ( z(t 0 + D) 2 ), the function F and the norm z 2 2 go to zero, where T max is given for any s ≥ 0 by

T max (s) = ν δ(ν -ν ) (P max ) 1-ν ν s α+1 + s 2 1-ν ν . ( 3
ż1 (t) = z2 (t) -k 1 {z 1 (t)} α1 , t ≥ t 0 , ż2 (t) = -k 2 {z 1 (t)} α2 , t ≥ t 0 . (3.129) (3.130)
To prove that this system is FTS, we choose α 1 and α 2 so that the system (3.129)-(3.130) is r-homogeneous of degree ν, where r = [r 1 , r 2 ] ∈ R 2 and ν < 0 satisfy

r 1 + ν = r 2 = r 1 α 1 , r 2 + ν = r 1 α 2 . (3.131) (3.132)
Let α 2 = α ∈ (0, 1) and r 1 = 1. Then we nd

ν = (α -1) 2 < 0, (3.133
)

α 1 = r 2 = (α + 1) 2 ∈ 1 2 , 1 .
(3.134)

Next, we will prove that the system is asymptotically stable. In order to do that, let ϕ be a function dened by

ϕ(z 1 , z2 ) = {z 1 } α+1 2 , z2 . (3.135)
We can clearly see that ϕ is 1, α+1

2

-homogeneous of degree µ = α+1

2 .

Now, let V be a Lyapunov candidate function dened by (as in [START_REF] Bernuau | Finite-time output stabilization of the double integrator[END_REF])

V (z 1 , z2 ) = 1 2 ϕ(z 1 , z2 ) P ϕ(z 1 , z2 ) , (3.136)
with P = a c c b , where for any k 1 , k 2 > 0, the parameters a, b, and c are given as follows:

a = (-2ck 2 + 1) (α + 1)k 1 > 0, b = (α + 1)(a -ck 1 ) 2k 2 > 0, c = -1 < 0.
ab -c 2 = (a -ck 1 )(-2ck 2 + 1) -2k 1 k 2 c 2 2k 1 k 2 , (3.139)
= (a + k 1 )(1 + 2k 2 ) -2k 1 k 2 2k 1 k 2 , (3.140)
= a + k 1 + 2ak 2 + 2k 1 k 2 -2k 1 k 2 2k 1 k 2 , (3.141)
= a + k 1 + 2ak 2 2k 1 k 2 > 0. (3.142)
On the other hand, the time derivative of V is given for all t ≥ t 0 + D as follows:

V (z 1 , z2 ) = {z 1 } α+1 2 , z2 a c c b α+1 2 ż1 |z 1 | α-1 2 ż2 , (3.143)
= a {z 1 }

α+1 2 + cz 2 , c {z 1 } α+1 2 + bz 2 α+1 2 |z 1 | α-1 2 z2 -k 1 {z 1 } α+1 2 -k 2 {z 1 } α , (3.144) = 1 2 [-2bk 2 + (α + 1)(a -ck 1 )] {z 1 } α z2 + 1 2 [-ak 1 (α + 1) -2ck 2 ] |z 1 | 3α+1 2 + 1 2 c(α + 1)|z 1 | α-1 2 z2 2 , (3.145)
= 1 2 |z 1 | α-1 2 z2 2 [-2bk 2 + (α + 1)(a -ck 1 )] {z 1 } α+1 2 z2 + [-ak 1 (α + 1) -2ck 2 ] × {z 1 } α+1 2 z2 2 + c(α + 1)   , (3.146) = 1 2 |z 1 | α-1 2 z2 2 [-2bk 2 + (α + 1)(a -ck 1 )] E + [-ak 1 (α + 1) -2ck 2 ] E 2 + c(α + 1) , (3.147) = |z 1 | 3α+1 2 2E 2 [-2bk 2 + (α + 1)(a -ck 1 )] E + [-ak 1 (α + 1) -2ck 2 ] E 2 + c(α + 1) , (3.148)
where

E = {z1} α+1 2 z2 . We can clearly see that V is (1, α+1 2 )-homogeneous of degree ν = 3α+1 2 .
Let Q be the polynomial function given by

Q(E) = āE 2 + bE + c = ā E + b 2ā 2 - b2 -4āc 4ā 2 , (3.149)
where ā, b, and c are given as follows:

ā = [-ak 1 (α + 1) -2ck 2 ] , (3.150) b = [-2bk 2 + (α + 1)(a -ck 1 )] , (3.151)
c = c(α + 1).

(3.152)

Our goal is to prove that V is denite negative, which is equivalent to proving that Q is negative, or more precisely it is equivalent to proving that ā and b2 -4āc are negative. In fact using the

= -2k 2 -1 k 1 (α + 1) k 1 (α + 1) + 2k 2 , (3.154) = -2k 2 -1+2k 2 , (3.155)
= -1 < 0,

(3.156) and b2 -4āc = [-2bk 2 + (α + 1)(a -ck 1 )] 2 -4c(α + 1) [-ak 1 (α + 1) -2ck 2 ] , (3.157)
= -2k 2 (α + 1)(a -ck 1 ) 2k 2 + (α + 1)(a -ck 1 ) 2 -4(α + 1), (3.158)
= -4(α + 1) < 0.

(3.159)

Thus, we proved that V is denite negative and that the system (3.

W := {(z 1 , z2 ) ∈ R 2 : V (z 1 , z2 ) = 1}, (3.160)
and set the constant δ as follows:

δ = -max (z1,z2)∈W { V (z 1 , z2 )} = min (z1,z2)∈W {-V (z 1 , z2 )} > 0. (3.161)
Using the homogeneity of V for λ = V (z 1 , z2 )

-1 ν , we have V (λ r1 z1 , λ r2 z2 ) = V (z 1 , z2 ) V (z 1 , z2 ) ν ν . (3.162)
On the other hand, we have

(λ r1 z1 , λ r2 z2 ) : (z 1 , z2 ) ∈ R 2 \{[0, 0]} = W, (3.163)
because

V (λ r1 z1 , λ r2 z2 ) = V (z 1 , z2 ) -1 ν ν V (z 1 , z2 ) = 1. (3.164) Therefore, sup (z1,z2) =[0,0] V (z 1 , z2 ) [V (z 1 , z2 )] ν ν = sup (z1,z2) =[0,0] V (λ r1 z1 , λ r2 z2 ) , (3.165)
= sup

(z1,z2)∈W V (z 1 , z2 ) , (3.166)
= -δ.

(3.167)

Hence it follows that:

V (z 1 , z2 ) ≤ -δ V (z 1 , z2 ) ν ν . ( 3
V (z 1 (t), z2 (t)) 1-ν ν ≤ -δ 1 -ν ν (t -t 0 -D) + V (z 1 (t 0 + D), z2 (t 0 + D)) 1-ν ν . (3.169)
Then, we get

V (z 1 (t), z2 (t)) ≤ -δ 1 -ν ν (t -t 0 -D) + V (z 1 (t 0 + D), z2 (t 0 + D)) 1-ν ν ν ν -ν . (3.170)
Moreover, we obtain

|z 1 (t)| α+1 + |z 2 (t)| 2 ≤ 1 P min -δ 1 -ν ν (t -t 0 -D) + (P max ) 1- ν ν |z 1 (t 0 + D)| α+1 +|z 2 (t 0 + D)| 2 1-ν ν ν ν -ν , (3.171) ≤ 1 P min -δ 1 - ν ν (t -t 0 -D) + (P max ) 1-ν ν z(t 0 + D) α+1 2 + z(t 0 + D) 2 2 1-ν ν ν ν -ν , (3.172)
where we have used the following property:

P min |z 1 | α+1 + |z 2 | 2 ≤ V (z 1 , z2 ) ≤ P max |z 1 | α+1 + |z 2 | 2 , (3.173)
with P min and P max are the eigenvalues of P with 0 < P min ≤ P max .

Finally, using the fact that

|z 1 (t)| 2 ≤ 1 P min -δ 1 -ν ν (t -t 0 -D) + (P max ) 1-ν ν z(t 0 + D) α+1 2 + z(t 0 + D) 2 2 1-ν ν ν ν -ν    2 1+α , (3.174) we get z(t) 2 2 ≤ 1 P min -δ 1 -ν ν (t -t 0 -D) + (P max ) 1-ν ν z(t 0 + D) α+1 2 + z(t 0 + D) 2 2 1-ν ν ν ν -ν + 1 P min -δ 1 - ν ν (t -t 0 -D) + (P max ) 1- ν ν z(t 0 + D) α+1 2 + z(t 0 + D) 2 2 1- ν ν   ν ν -ν     2 1+α . (3.175) Then, z(t) 2 2 ≤ F (t -t 0 -D, z(t 0 + D) 2 ) + F (t -t 0 -D, z(t 0 + D) 2 ) 2 1+α , (3.176)

Stability analysis 65

where F is the increasing function dened by (3.127). In particular, when t tend to the settling time t 0 + D + T max ( z(t 0 + D) 2 ), the function F and the norm z(t) 2 goes to zero, where T max is given by (3.128). Remark 3.6 For the general case, we can recover a similar inequality to (3.126) from [START_REF] Zimenko | Homogeneity based nite/xedtime observers for linear mimo systems[END_REF]Theorem 6] to get FTS or from [START_REF] Zimenko | Homogeneity based nite/xedtime observers for linear mimo systems[END_REF]Theorem 7] to get FxTS.

Remark 3.7

Note that the problem of nite-time stability of the target system (3.129)- (3.130) is studied in the presence of some disturbances in [START_REF] Polyakov | Robust nite-time stabilization and observation of a planar system revisited[END_REF] using an implicit Lyapunov-based approach and in [START_REF] Cruz-Zavala | Strict lyapunov functions for homogeneous nite-time second-order systems[END_REF] using a strict explicit Lyapunov-based approach. In the absence of the disturbance in [START_REF] Polyakov | Robust nite-time stabilization and observation of a planar system revisited[END_REF], the settling-time is then estimated by the following formulas:

T max (s 1 , s 2 ) = 3 + α δ(1 -α) V 0 (s 1 , s 2 ) 1-α 3+α , for any s 1 , s 2 ∈ R 2 , (3.177)
where δ = min

(z1,z2)∈W1 l 1 |z 1 | 2 2-α + l 2 z2 2 , W 1 = {(s 1 , s 2 ) ∈ R 2 : V 0 (s 1 , s 2 ) = 1}, l 1 = C (3+α)k1 2(1+α) 1-α 2+2α - √ k 2 - k 2 1 √ k2 , l 2 = 1 2 √ k2
, and such that l 1 > 0 and V 0 (s 1 , s 2 ) > 0. This estimation is quite similar to our estimation (3.128), but may be less accurate than our estimation or even far from the real settling-time as we will see in the simulation later. Now, let us state the main result of this chapter.

V 0 (s 1 , s 2 ) = C |s1| α+1 α+1 + s 2 2 2k2 3+α 2+2α -s1s2 √ k2 , with α = α 2 , k 1 ,

Theorem 3.1

Let the matrices A, B, and C be as in (3.2)-(3.3), the vector function L(•) = [(L 1 (•), L 2 (•))] be as in (3.35) and (3.36), the function F be as in (3.127), the function G 1 be as in (3.105) and G 2 as in (3.110), δ be given as in (3.161), P min and P max be as in (3.173). Let t 0 , D > 0. Then, for any initial conditions z 0 , ẑ0 the observer system guarantees that [ẑ, û] converges to [z, u] within a nite time. Moreover the quantity u(t,

•) -û(t, •) 2 L 2 (0,D) + z(t) -ẑ(t) 2
2 is bounded for all t ∈ [t 0 , t 0 + D], and for all t

≥ t 0 + D u(t, •) -û(t, •) 2 L 2 (0,D) + z(t) -ẑ(t) 2 2 ≤ M F (t -t 0 -D, G ( z 0 -ẑ0 2 )) + F (t -t 0 -D, G ( z 0 -ẑ0 2 )) 2 1+α , (3.178) with M = 2 Ce A• 2 L 2 (0,D) + 1 and G = G 1 (or G = G 2 ).
In particular, the norm u(t,

•) -û(t, •) 2 L 2 (0,D) + z(t) -ẑ(t) 2 2 → 0, as t → t 0 + D + Tmax ( z 0 -ẑ0 2 )
, where Tmax is given for any s ≥ 0 by

Tmax (s) = ν δ(ν -ν ) (P max ) 1-ν ν G (s) α+1 + G (s) 2 1-ν ν , (3.179)
with α ∈ (0, 1),

ν = α + 1, ν = 3α+1 2 .
Proof. Using the transformation (3.17 Squaring this inequality and integrating with respect to the variable x between 0 and D, gives us

ũ(t, •) 2 L 2 (0,D) ≤ 2 ω(t, •) 2 L 2 (0,D) + 2 Ce A• 2 L 2 (0,D) z(t) 2 2 .
(3.181)

Adding the norm z(t) -ẑ(t) 2 2 in both sides and using (3.12)-(3.13), we get

u(t, •) -û(t, •) 2 L 2 (0,D) + z(t) -ẑ(t) 2 2 ≤ 2 ω(t, •) 2 L 2 (0,D) + 2 Ce A• 2 L 2 (0,D) z(t) 2 2 + z(t) -ẑ(t) 2 2 , (3.182) ≤ 2 ω(t, •) 2 L 2 (0,D) + 2 Ce A• 2 L 2 (0,D) + e AD 2 2 × z(t) 2 2 .
(3.183)

Using Lemma 3.1 and the last inequality, we see that ũ(t,

•) 2 L 2 (0,D) + z(t) 2 2 is bounded for all t ∈ [t 0 , t 0 + D].
On the other hand, for all t ≥ t 0 + D, employing Lemma 3.4 and the fact that ω(t, •) ≡ 0 for all t ≥ t 0 + D, we get

u(t, •) -û(t, •) 2 L 2 (0,D) + z(t) -ẑ(t) 2 2 ≤ M F (t -t 0 -D, z(t 0 + D) 2 ) + F (t -t 0 -D, z(t 0 + D) 2 ) 2 1+α , (3.184) where M = 2 Ce A• 2 L 2 (0,D) + e AD 2
2 and F is given for any s ≥ 0 in (3.127) as follows:

F (t -t 0 -D, s) = 1 P min -δ 1 - ν ν (t -t 0 -D) + (P max ) 1- ν ν s α+1 + s 2 1-ν ν ν ν -ν . (3.185)
Employing Lemma 3.2 and the fact that F is increasing with respect to the second variable, we obtain

u(t, •) -û(t, •) 2 L 2 (0,D) + z(t) -ẑ(t) 2 2 ≤ M F (t -t 0 -D, G ( z 0 -ẑ0 2 )) + F (t -t 0 -D, G ( z 0 -ẑ0 2 )) 2 1+α
, (3.186) where G = G 1 and G 1 is the function dened by (3.105). In addition, note that using Lemma 3.3 instead of Lemma 3.2 we nd the same inequality as (3.186) with G = G 2 and G 2 is dened by (3.110).

Thus, we nally conclude that the norm u(t,

•) -û(t, •) 2 L 2 (0,D) + z(t) -ẑ(t) 2 2 → 0, as t → t 0 + D + Tmax ( z 0 -ẑ0 2 )
, where Tmax is given for any s ≥ 0 by

Tmax (s) = T max (G(s)) = ν δ(ν -ν ) (P max ) 1-ν ν G(s) α+1 + G(s) 2 1-ν ν . (3.187) Remark 3.8
Notice that it is possible to use some change of variables (i.e. z(t) = z(t -D)) to pass the delay (given in (3.3)) to the input (appearing in (3.2)). In this situation, the observer design is simplied as we are only required to design a nite-dimensional observer. This task can be achieved using classical techniques based, for instance, on the Homogeneity theory (see e.g. [START_REF] Zimenko | Homogeneity based nite/xedtime observers for linear mimo systems[END_REF]). However, passing the delay to the input will yield diculties for the control design. To deal with it, we can use similar techniques based on the backstepping approach. Solving this problem will be the main objective of our next chapter.

3.4. Simulation 67

Simulation

In this section, we illustrate our results using some numerical examples.

An academic example

Let us start with a simple academic example. Consider (3.4)-(3.7) with a 0 = a 1 = 1. For our simulation, we consider 5 dierent initial conditions: z(t 0 ) := z 0 = [START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF][START_REF] Liu | Networked control systems: A time-delay approach[END_REF] , 10z 0 , 100z 0 , 1000z 0 and 10000z 0 . We take the initial time t 0 = 0, the control U (t) = -7z

k 1 = k 3 = 1.5, k 2 = k 4 = 2.5, α 2 = 0.7, α 1 = 1+α2 2 , β 2 = 2, β 1 = 1+β2 2
and the delay D = 1s. ) in solid blue lines and the estimated states ẑ1 and ẑ2 of (3.8)-(3.10) in dashed red lines, with feedback U (t) = -7z 1 (t)-5z 2 (t), delay D = 1s, and initial condition z(0) = [START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF][START_REF] Liu | Networked control systems: A time-delay approach[END_REF] . Figure 3.2, we have on the left the evolution of the error states of the ODE part of the error system (3.14) multiplied by e AD with feedback U (t) = -7z 1 (t) -5z 2 (t) and delay D = 1s. On the right hand, the evolution of the error state of the PDE part of the error system (3.15)-(3.16). Finally, Figure 3.3 presents on the left, the evolution of the norm z(t) -ẑ(t) 2 2 + u(t, •) -û(t, •) 2 L 2 (0,D) of the error system, shown in a logarithmic scale, with feedback U (t) = -7z 1 (t) -5z 2 (t) and delay D = 1s, where we can observe that the solutions converge to the origin in a nite-time upper bounded by t 0 + D + T max = 10.6s

(numerically estimated) for all the initial conditions. On the right, we observe the evolution of the outputs Y (t) = z 1 (t -D) = u(t, 0) in a solid line and the estimated outputs û(t, 0) in a dashed line with feedback U (t) = -7z 1 (t) -5z 2 (t) and delay D = 1s

Application to a simplied robot manipulator

Now, let us focus on a simplied robot manipulator described in Figure 3.4, where for i ∈ {1, 2}, q i and m i are, respectively, the angle and the mass of the i th joint of the robot. Let us assume that

(t) -ẑ(t) 2 2 + u(t, •) -û(t, •) 2
L 2 (0,D) of the error system, shown in a logarithmic scale, with feedback U (t) = -7z 1 (t) -5z 2 (t) and delay D = 1s, for 5 dierent initial conditions z 0 = [START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF][START_REF] Liu | Networked control systems: A time-delay approach[END_REF] , 10z 0 , 100z 0 , 1000z 0 , and 10000z 0 . On the right, the evolution of the outputs Y (t) = z 1 (t -D) = u(t, 0) in a solid line and the estimated outputs û(t, 0) in a dashed line with feedback U (t) = -7z 1 (t) -5z 2 (t) and delay D = 1s.

the manipulator is fully actuated, i.e., the number of actuators is equal to 2 (degree of freedom), and that the masses of the links are neglected with respect to the masses of the motors and payload (m 2 ). Then, the model is (3.1) ( i.e., M (q)q + C(q, q) + G(q) = τ ), with

C(q, q) = m 2 l 2 l 1 sin (q 2 ) -q2 2 -2 q1 q2 q2 1 , G(q) = g m 2 l 2 sin (q 1 + q 2 ) + (m 1 + m 2 ) l 1 sin (q 1 ) m 2 l 2 sin (q 1 + q 2 ) , M (q) = (m 1 + m 2 )l 2 1 + 2m 2 l 1 l 2 cos (q 2 ) + m 2 l 2 2 m 2 l 2 2 + m 2 l 1 l 2 cos (q 2 ) m 2 l 2 2 + m 2 l 1 l 2 cos (q 2 ) m 2 l 2 2 ,
τ = M (q)W + C(q, q) + G(q), (3.188)
where W = [w 1 , w 2 ] ∈ R 2 is the new input, gives the following closed-loop system:

qi (t) = w i (t), t ≥ t 0 , i ∈ {1, 2}, (3.189)
(where we have used the fact that M (q) is a symmetric positive-denite matrix). However, since q and q are not directly available by measurements we can not use feedback (3.188) but instead, we can use feedback τ = M (q)W + C(q, q) + G(q).

(3.190) Thus, the system (3.189) should be replaced by a similar perturbed model, qi (t) = w i (t) + δ i W (t), q(t), q(t), q(t), q(t) , t ≥ t 0 , i ∈ {1, 2},

where, for i ∈ {1, 2}, δ i is a disturbance reecting mismatched models given by δ 1 W, q, q, q, q , δ 2 W, q, q, q, q = M (q) -1 [τ (q, q) -M (q)W -C(q, q) -G(q)]. (3.192) This fact explains why such a nite-time robust state estimation has to be performed as seen in the simulations below. Using the following change of coordinates:

z 1 = q 1 , z 2 = q1 , z 3 = q 2 , z 4 = q2 , (3.193)
the system (3.191) is transformed into the following two perturbed chains of double integrators:

           ż1 (t) = z 2 (t), t ≥ t 0 , ż2 (t) = w 1 (t) + δ 1 W (t), z(t), z(t), ẑ(t), ẑ(t) , t ≥ t 0 , ż3 (t) = z 4 (t), t ≥ t 0 , ż4 (t) = w 2 (t) + δ 2 W (t), z(t), z(t), ẑ(t), ẑ(t) , t ≥ t 0 , (3.194)
δ 1 W, z, z, ẑ, ẑ , δ 2 W, z, z, ẑ, ẑ = M (z 3 ) -1 [τ (ẑ, ẑ) -M (z 3 )W -C(z 2 , z 3 , z 4 ) -G(z 1 , z 3 )], (3.198)
where

ẑ = [ẑ 1 , ẑ2 ] , ẑ = [ẑ 3 , ẑ4 ] .
Then, we take the initialization time t 0 = 0s, the initial positions [z 1 (θ), z 3 (θ)] = [0, 0] rad for all θ ∈ [t 0 -D, t 0 ) and [z 1 (t 0 ), z 3 (t 0 )] = [START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF]2] rad, and the initial velocities [z 2 (θ), z 4 (θ)] = [0, 0] rad/s for all θ ∈ [t 0 -D, t 0 ]. In addition, we assume that z 1 and z 3 are available by delayed measurements, i.e.,

Y 1 (t) = z 1 (t -D), t ≥ t 0 , Y 2 (t) = z 3 (t -D), t ≥ t 0 , (3.199) (3.200)
with a delay of D units of time.

Next, by following the same steps as in Section 3.3, we rewrite the two subsystems into an ODE-PDE cascade system,

           ż(t) = Az(t) + B w 1 (t) + δ 1 W (t), z(t), z(t), ẑ(t), ẑ(t) , t ≥ t 0 , u 1,t (t, x) = u 1,x (t, x), t ≥ t 0 , x ∈ [0, D], u 1 (t, D) = Cz(t), t ≥ t 0 , Y 1 (t) = u 1 (t, 0), t ≥ t 0 , (3.201)
(3.202)

(3.203) (3.204) and            ż(t) = Az(t) + B w 2 (t) + δ 2 W (t), z(t), z(t), ẑ(t), ẑ(t) , t ≥ t 0 , u 2,t (t, x) = u 2,x (t, x), t ≥ t 0 , x ∈ [0, D], u 2 (t, D) = C z(t), t ≥ t 0 , Y 2 (t) = u 2 (t, 0), t ≥ t 0 , (3.205) (3.206)
       ż(t) = Aẑ(t) + Bw 1 (t) -e AD L(Y 1 (t) -û1 (t, 0)), t ≥ t 0 , û1,t (t, x) = û1,x (t, x) -Ce Ax L(Y 1 (t) -û1 (t, 0)), t ≥ t 0 , x ∈ [0, D], û1 (t, D) = C ẑ(t), t ≥ t 0 , (3.209) (3.210) (3.211)
and

       ż(t) = A ẑ(t) + Bw 2 (t) -e AD L(Y 2 (t) -û2 (t, 0)), t ≥ t 0 , û2,t (t, x) = û2,x (t, x) -Ce Ax L(Y 2 (t) -û2 (t, 0)), t ≥ t 0 , x ∈ [0, D], û2 (t, D) = C ẑ(t), t ≥ t 0 , (3.212)
(3.213)

(3. 214
)
where L is given in (3.35) and (3.36).

In this case, since we do not design a predictor feedback, one can only ensure the robustness of the controller to small delays (although the nite-time convergence of the observer is guaranteed for larger delays). we rst discretize them using the two-step variant of the Lax-Friedrichs numerical method introduced in [START_REF] Shampine | Two-step laxfriedrichs method[END_REF], then we use its corresponding solver in Matlab. For simplicity, we chose the parameters of L as follows: k 1 = k 2 = 4, α 2 = 0.8 and α 1 = 0.9 for both observers, and the delay D = 0.04s.

+ D) -ẑ(t 0 + D) 2 ) = 3.5380s.
Next, in Figure 3.5 we give the evolution of the states z 1 , z 2 , z 3 , and z 4 of the system (3.194)- (3.197) in solid blue lines and the estimated states ẑ1 , ẑ2 , ẑ3 , and ẑ4 in dashed red lines with the delay D = 0.04s, the feedback τ in (3.190) instead of τ and the new inputs w 1 (t) = 2 sin(10t) and w 2 (t) = 1.4 sin(20t). Then, in Figure 3.6 we show the evolution of the error states [z 1 , z2 ] of the ODE part of the error system linked to the subsystem (3.201)- (3.204) where we have used the feedback τ (3.188), multiplied by e AD . In addition, we can observe that the solutions converge to the origin in a nite-time less than t 0 + D + T 1,max = 3.3396s (numerical estimation using (3.128)): the log plot (not shown) conrmed that t 0 + D + T 1,max ≈ 3.31s. On the right-hand side, we show the evolution of the error state of the PDE part of the error system. Similarly, Figure 3.7 shows the evolution of the error states [z 3 , z4 ] of the ODE part of the error system linked to the subsystem (3.205)-(3.208) with the feedback τ (3.188), multiplied by e AD . In addition, we can observe that the solutions converge to the origin in a nite-time less than t 0 + D + T 2,max = 3.5780s (numerical estimation using (3.128)): the log plot (not shown) conrmed that t 0 + D + T 2,max ≈ 3.5s. On the right hand, we show the evolution of the error state of the PDE part of the error system. Finally, Figure 3.8 present on the left, the evolution of the norm z(t)-ẑ(t) 2 2 + z(t)-ẑ(t

) 2 2 + 2 i=1 u i (t, •)- ûi (t, •) 2
L 2 (0,D) of the error systems (in a logarithmic scale), where is shown to converge to zero in nite-time less than t 0 +D +T max ≈ 3.5s (compare to 3.578s obtained using (3.128)). On the right, we observe the evolution of the outputs Y

1 (t) = z 1 (t-D) = u 1 (t, 0) and Y 2 (t) = z 3 (t-D) = u 2 (t, 0)
in solid lines and the estimated outputs û1 (t, 0) and û2 (t, 0) in dashed lines.

In Figures 3.9, 3.10, 3.11, and 3.12, we add noise measurement of power 0.0001 and we give the same simulations as before and verify the impact of noise to the convergence of the closed-loop.

As anticipated, the observer is robust with respect to the modeled uncertainties and the noise measurement. However, we only ensure a nite-time convergence to a neighborhood of the origin characterized by the magnitude of the noise.

Remark 3.9

Now, following Remark 3.7 with C = 1.7, α, k 1 , and k 2 as chosen in the above simulations (i.e., k 1 = k 2 = 4, α = α 2 = 0.8), one can numerically estimate (3.177) as follows: T 1,max = 16.9744s and T 2,max = 18.4054s. Indeed, to get these estimates, we have l 1 = 2.5049, l 2 = 0.25,

V 0,1 (z(t 0 + D) -ẑ(t 0 + D)) = 0.6883, V 0,2 (z(t 0 + D) -ẑ(t 0 + D)) = 3.

Conclusion

In this chapter, we dealt with the problem of nite-time estimation of linear time-invariant (LTI) systems, in the observable form, with delayed output. The main ideas relied on rewriting the system into an ODE-PDE cascade setting, where the PDE part modeled the eect of the delay on the output. The nonlinear gains were designed such that the error observer system is either FTS or FxTS. To achieve this, we used the backstepping approach where we chose a suitable nonlinear target system satisfying a chosen nite/xed-time convergence property. Finally, we used the invertibility of the backstepping transformation to pass this property to the error system.

(t) -ẑ(t) 2 2 + z(t) -ẑ(t) 2 2 + 2 i=1 u i (t, •) - ûi (t, •) 2 L 2 (0,D)
of the two error systems (plotted in logarithmic scale) with feedback τ in (3.190) and without noise measurement, with the delay D = 0.04, and T max = 3.5380s. On the right, the evolution of the outputs Y 1 (t) = z 1 (t -D) = u 1 (t, 0) and Y 2 (t) = z 3 (t -D) = u 2 (t, 0) in solid lines and the estimated outputs û1 (t, 0) and û2 (t, 0) in dashed with feedback τ in (3.190) and lines without noise measurement.

(t) -ẑ(t) 2 2 + z(t) -ẑ(t) 2 2 + 2 i=1 u i (t, •) - ûi (t, •) 2 L 2 (0,D)
of the two error systems (plotted in logarithmic scale) with feedback τ in (3.190) and noise of power 0.0001, and the delay D = 0.04, and T max = 3.5381s. On the right, the evolution of the outputs Y 1 (t) = z 1 (t -D) = u 1 (t, 0) and Y 2 (t) = z 3 (t -D) = u 2 (t, 0) in solid lines and the estimated outputs û1 (t, 0) and û2 (t, 0) in dashed lines with feedback τ in (3.190) In this chapter, we present a general approach to studying the problem of nite-time and xedtime stabilization of a chain of integrators with input delay. To accomplish this, we rst reformulate the chain of integrators with input delay as a cascade ODE-PDE system (i.e., a cascade of a linear transport partial dierential equation (PDE) with the chain of integrators) where the transport equation models the eect of the delay on the input. Next, we use a nonlinear innite-dimensional backstepping transformation to convert the cascade system into a suitable target system that is chosen to be FTS or FxTS. We perform the stability analysis on the target system by means of classical Non-Asymptotic concepts and tools such as the linear homogeneity and "generalized KL"

functions. Then, we use the inverse transformation to transfer back the stability property to the closed-loop system. Finally, we give some characterizations of nite/xed time predictor-based controllers followed by numerical simulations.

Introduction

In this chapter, we revisit the problem of nite/xed stabilization of a chain of integrators with input delay and propose a more general approach for the design of nite/xed-time state-dependent predictor-based controllers. We use a cascade ODE-PDE system (i.e., a cascade of a linear transport partial dierential equation (PDE) with the chain of integrators) where the transport equation 80 Chapter 4. Finite/xed-time stabilization of LTI systems with pointwise input delay models the eect of the delay on the input and builds on a nonlinear innite-dimensional backstepping transformation inspired by [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]. Compared to [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF]-which uses a linear transformation and time-varying tools to ensure the prescribed-time stability property for an LTI system with input delay-our approach uses a nonlinear transformation and nonsmooth tools to ensure a dierent (i.e. nite/xed) stability property. Both methods bring dierent challenges and have specic issues. The approach in this chapter allows to perform the stability analysis on a suitable target system (chosen to exhibit the desired stability properties, i.e., either nite time or xed time) while employing classical notions and tools such as Lyapunov-based characterization of nite/xed-time stability property of ODEs, linear homogeneity [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] (as those discussed in Chapter 2) as well as the "generalized KL" (in short "GKL") functions [START_REF] Hong | Finite-time input-to-state stability and applications to nite-time control design[END_REF]. Hence, we can provide some characterizations of the resulting nite/xed-time predictor-based controllers.

It is worth mentioning that [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF], [START_REF] Zuo | Fixed-time stabilization of general linear systems with input delay[END_REF] achieve similar results (nite/xed-time stabilization of LTI systems with input delay) to ours using Artstein's model reduction. Nevertheless, no state estimates of the closed-loop solution are provided. The actuator dynamic is not identied throughout the analysis, either. Moreover, extensions of [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF], [START_REF] Zuo | Fixed-time stabilization of general linear systems with input delay[END_REF] to complex innite-dimensional systems (including cascaded systems) with constant/time-varying/state-dependent delays, distributed delays are not straightforward. In contrast, our approach does account for the innite dimensionality of the input, and may allow possible extensions to more complex innite-dimensional systems (e.g., 1D reaction-diusion PDEs with delayed boundary) or when just cascading nite/xed-time ISS subsystems.

This chapter is organized as follows. In Section 4.2, we give the problem statement in which we present the chain of integrators with input delay and its reformulation within an ODE-PDE setting. In Section 4.3, we give a general approach to stabilize the chain of integrators in a nite time or xed time. We present the nonlinear backstepping transformation to transform the ODE-PDE setting to a suitable target system and to come up with a nite/xed-time predictor-based control. Next, in Section 4.4 we apply our approach to dierent target systems to attain nitetime stability or xed-time stability. Then, we give in Section 4.5 some numerical simulations to illustrate the results.

Problem statement

We consider the following chain of integrators with input delay:

żj (t) = z j+1 (t), j = 1, . . . , n -1, t ≥ t 0 , żn (t) = U (t -D), t ≥ t 0 , (4.1) (4.2)
where z(t) = [z 1 (t), . . . , z n (t)] ∈ R n (n ∈ N\{0}) is the instantaneous state of the system, U (t) ∈ R is the control input, and D > 0 is a known constant delay.

Our goal is to design a nonlinear predictor-based controller for the system (4.1)-(4.2) to achieve FTS or FxTS. To this end, the methodology developed in this chapter relies on representing the actuator delay as a linear transport PDE and builds upon the cascade ODE-PDE setting (i.e., a cascade of linear hyperbolic PDE with an LTI system) of [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF].

Remark 4.1

Note that in [START_REF] Bekiaris-Liberis | Stabilization of linear strict-feedback systems with delayed integrators[END_REF], the problem of exponential stabilization of the following class of strict-feedback 4.3. Finite/xed-time predictor-based controller via PDE-based backstepping approach 81

system with delayed integrators and delayed input

             żj (t) = j i=1 a ji z i (t) + z j+1 (t -D j ), j = 1, . . . , n -1, t ≥ t 0 , żn (t) = n i=1 a ni z i (t) + U (t -D n ), t ≥ t 0 , (4.3)
           żj (t) = z j+1 (t), j = 1, . . . , n -1, t ≥ t 0 , żn (t) = u(t, 0), t ≥ t 0 , u t (t, x) = u x (t, x), t ≥ t 0 , x ∈ [0, D] u(t, D) = U (t), t ≥ t 0 , (4.5) (4.6)
u(t, x) = u 0 (t + x -t 0 ), t 0 ≤ t + x ≤ t 0 + D, U (t + x -D), t + x ≥ t 0 + D, (4.9)
where u 0 is a bounded function in L 2 (0, D).

The objective of the rst part of this chapter is to give a general approach to design a controller (predictor-type) for the system (4.1)-(4.2), to attain FTS and/or FxTS. We employ a nonlinear innite-dimensional backstepping transformation. The key idea is to transform the original system into a suitable target system that is chosen to exhibit the FTS or FxTS properties.

Finite/xed-time predictor-based controller via PDE-based backstepping approach 4.3.1 Nonlinear innite-dimensional backstepping transformation

Inspired by [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF]

           ϕ j,x (t, x) = ϕ j+1 (t, x), j = 1, . . . , n -1, ϕ j (t, 0) = z j (t), j = 1, . . . , n -1, ϕ n,x (t, x) = u(t, x),
ϕ n (t, 0) = z n (t). Notice that ϕ i (t, x) = z i (t + x) for all i ∈ {1, • • • , n}, all t ≥ t 0 and all x ∈ [0, D]. Then, by the variation of the constant formula, we obtain:

ϕ i (t, x) = x 0 (x -y) n-i (n -i)! u(t, y)dy + n j=i x j-i (j -i)! z j (t). (4.15)
The proof of (4.15) is as follows:

z(t + x) = e Ax z(t) + t+x t e A(t+x-s) Bu(s, 0)ds, (4.16
)

with z = [z 1 , • • • , z n ] , B = e n := [0, • • • , 0, 1] , and A := {A ij } ∈ M n,n (R), where A ij = 1 if j = i + 1 and A ij = 0 otherwise.
Next, using the change of variables y = s -t, we get z(t + x) = e Ax z(t) +

x 0 e A(x-y) Bu(t + y, 0)dy,

= e Ax z(t) +

x 0 e A(x-y) Bu(t, y)dy. x k k! A k , we recover

z(t + x) = n-1 k=0 x k k! A k z(t) + n-1 k=0 x 0 (x -y) k k! A k Bu(t, y)dy. (4.19)
Then, using the fact

A k z(t) = [z k+1 (t), • • • , z n (t), 0, • • • , 0] and A k B = e n-k , we obtain z i (t + x) = n-i k=0 x k k! z k+i (t) + x 0 (x -y) n-i (n -i)! u(t, y)dy, (4.20)
= n j=i Note that using the fact that ϕ(t, x) = z(t+x) for all t ≥ t 0 and al x ∈ [0, D], it is clear that the nonlinear transformation (4.10) satises the PDE part of (4.22)-(4.25) (i.e. ω t (t, x) = ω x (t, x)).

x j-i (j -i)! z j (t) + x 0 (x -y) n-i (n -i)! u(t, y)dy.
           żj (t) = z j+1 (t), j = 1, . . . , n -1, t ≥ t 0 , żn (t) = F(z 1 (t), . . . , z n (t)) + ω(t, 0), t ≥ t 0 , ω t (t, x) = ω x (t, x), t ≥ t 0 , x ∈ [0, D] ω(t, D) = 0, t ≥ t 0 , (4.22)

Inverse transformation

The inverse transformation is given by, u(t, x) = ω(t, x) + F(ψ 1 (t, x), . . . , ψ n (t, x)), (4.26) where ψ 1 , . . . , ψ n are the solutions of: Similarly to the direct transformation, we recover from the inverse transformation: u t (t, x) = u x (t, x).

           ψ j,x (t, x) = ψ j+1 (t, x), j = 1, . . . , n -1, ψ j (t, 0) = z j (t), j = 1, . . . , n -1, ψ n,x (t, x) = F(ψ 1 (t, x), . . . , ψ n (t, x)) + ω(t, x), ψ n (t, 0) = z n (t).

On the selection of the nite/xed-time predictor-based controller

In this section, we give an important assumption on the nonlinear function F given in (4.22)- (4.25) to ensure FTS or FxTS properties. Then, we give the expression of our predictor-based controller U (t) using the transformation (4.10) or (4.26).

An assumption on the nonlinear function F

In order to ensure that the target system (4.

żj (t) = z j+1 (t), j = 1, . . . , n -1, t ≥ t 0 , żn (t) = U (t), t ≥ t 0 , (4.31)
(4.32) delay is FTS (resp. FxTS) i.e., there exists a class GKL function β such that the solution of the previous closed-loop system with a well chosen feedback control U (t), satises:

z(t) 2 ≤ β( z0 2 , t -t0 ), ∀t ≥ t0 , (4.33)
where z0 = [z 1,0 , . . . , zn,0 ] is the initial condition at time t0 . Moreover, there exists an increasing function T (•) such that z(t) 2 = 0 when t ≥ t0 + T ( z( t0 ) 2 ), (resp. a positive real constant T max such that z(t) 2 = 0 when t ≥ t0 + T max ).

Remark 4.2

Note that since

F is non-dierentiable on S = n i=1 {s = [s 1 , • • • , s n ] ∈ R n , s i = 0}, equations (4. 24
)
(i.e. ω t (t, x) = ω x (t, x)) and (4.7) (i.e. u t (t, x) = u x (t, x)) will not be dened everywhere (i.e. for all (t, x) ∈ [0, +∞) × [0, 1]). However, they can be veried almost everywhere (except in a set of measure zero). This last fact is sucient for us to dene u and ω, using the characteristic method, almost everywhere and then by using the continuity of F, and the transformations (4.10) and (4.26), we recover u and ω everywhere. The question now is how to prove that, for instance, (4.24) is satised almost everywhere. To do that, we need to rst prove that (t, x) → ω(t, x) is dierentiable almost everywhere. From (4.10), this is equivalent to proving that (t, x) → F • ϕ(t, x) is dierentiable almost everywhere (where ϕ(t, x) is given by ϕ

(t, x) = [ϕ 1 (t, x), • • • , ϕ n (t, x)] ∈ R n ) and since [ϕ 1 (t, x), • • • , ϕ n (t, x)] = [z 1 (t + x), • • • , z n (t + x)], it is equivalent to proving that t → F • ϕ(t, 0
) is dierentiable almost everywhere (except on a countable set η of measure zero) and that the set

S η := {(t, x) ∈ [0, +∞) × [0, 1] : t + x ∈ η} is of measure zero (S η is the set where (t, x) → F • ϕ is not dierentiable).
To better understand this, let us focus on the case n = 2. Let us recall t → ϕ(t, 0) = z(t) is absolutely continuous since it satises continuously an ODE with respect to t (see (4.5)-(4.6)). In addition, the mapping [s 1 , s 2 ] → F(s 1 , s 2 ) is absolutely continuous w.r.t. each variable, continuously dierentiable except on the set S = 2 i=1 s = [s 1 , s 2 ] ∈ R 2 , s i = 0 (of Lebesgue measure equal to zero). Notice, however, that the mapping t → F •ϕ(t, 0) may not be assured to be absolutely continuous due to the lack of Lipschitzness of F on S. Nevertheless, we can use the arguments of [START_REF] Moreno | A lyapunov approach to second-order sliding mode controllers and observers[END_REF]Remark 1] to state that since the trajectories z(t) of (4.5)-(4.6) may eventually cross the surface S but cannot stay on it (due to the trajectories' oscillatory nature), unless we reach the equilibrium, that the t → F • ϕ(t, 0) is dierentiable for almost every t except on a (countable) set η of measure zero. Moreover, the set S η := {(t, x) ∈ [0, +∞) × [0, 1] : t + x ∈ η} is of measure zero, since S η can be represented as a countable union of the lines {(t, x) ∈ [0, +∞)×[0, 1] : t+x = n i , n i ∈ η, i ∈ N} of measure zero. Finally, by the chain rule on (4.10) the following holds: = ω t (t, x), (4.38) almost everywhere. In particular, using (4.10) at x = D alongside (4.8), we get ω(t, D) = 0. This concludes the proof.

ω x (t, x) = u x (t, x) - ∂F (ϕ 1 (t, x), . . . , ϕ n (t, x)) ∂x , (4.34) = u x (t, x) - n i=1 z i,x (t + x) ∂F (ϕ 1 (t,
V (z(t), w(t, •)) ≤ -c 1 V α (z(t), w(t, •)) -c 2 V β (z(t), w(t, •)), c 1 , c 2 > 0, α ∈ (0,
U (t) = u(t, D) = F(ϕ 1 (t, D), . . . , ϕ n (t, D)), (4.40)
where where [ψ 1 (t, x), . . . , ψ n (t, x)] is solution of (4.27)-(4.30).

ϕ i (t, D) = D 0 (D -y) n-i (n -i)! u(t, y)dy + n j=i D j-i (j -i)! z j (t).

Stability analysis

In this subsection, we rst perform the stability analysis on the target system (4. Proof. By the method of the characteristics, the solution of the ω-dynamics of the target system (4.22)-(4.25) for any t ≥ t 0 + D is zero (i.e. ω(t, x) = 0 for all x ∈ [0, D] and t ≥ t 0 + D). Then, delay we can conclude using Assumption 4.1 (replacing "t" by "x") that the solution of the system (4.27)-(4.30) satises for t ≥ t

ψ(t, •) L 2 (0,D) ≤ β 1 ( z(t 0 + D) 2 , t -t 0 -D). (4.43) Moreover, ψ(t, •) L 2 (0,D) → 0 when t → t 0 + D + T ( z(t 0 + D) 2 ) (resp. t → t 0 + D + T max ), where ψ(t, •) 2 L 2 (0,D) = n j=1 ψ j (t, •) 2 L 2 (0,D) , T (
0 + D ψ(t, x) 2 ≤ β( z(t) 2 , x), ∀x ∈ [0, D], (4.44)
where β is a class GKL function. Moreover, there exists an increasing function T (•) such that ψ(t, x) 2 → 0 when x → T ( z(t) 2 ) (resp. x → T max ). Furthermore, when t

→ t 0 + D + T ( z(t 0 + D) 2 ) (resp. t → t 0 + D + T max ) ψ(t, x) 2 → 0.
Now, using the fact that β is decreasing with respect to the second variable x, we get,

ψ(t, x) 2 ≤ β( z(t) 2 , 0), ∀x ∈ [0, D], ∀t ≥ t 0 + D. (4.45)
By squaring and integrating with respect to x from 0 to D, then passing to the square roots, we nd,

ψ(t, •) L 2 (0,D) ≤ √ Dβ( z(t) 2 , 0), ∀t ≥ t 0 + D.
ψ(t, •) L 2 (0,D) ≤ √ Dβ(β( z(t 0 + D) 2 , t -t 0 -D), 0), ∀t ≥ t 0 + D. (4.47)
Then,

ψ(t, •) L 2 (0,D) ≤ β 1 ( z(t 0 + D) 2 , t -t 0 -D), ∀t ≥ t 0 + D, (4.48)
where for any s, t ∈ R + , β 1 is a class GKL function given by, β 1 (s, t) = √ Dβ(β(s, t), 0). Furthermore, when t → t 0 + D + T (s) (resp. t → t 0 + D + T max ), β(s, t) → 0, and by continuity β 1 (s, t) → 0.

Proposition 4.2

There exists a class GKL function β 2 such that for any x ∈ [0, D], F(ψ 1 (t, x), . . . , ψ n (t, x)) satises for t ≥ t 0 + D,

|F(ψ 1 (t, x), . . . , ψ n (t, x))| ≤ β 2 ( z(t 0 + D) 2 , t -t 0 -D), (4.49)
and

F(ψ 1 (t, •), . . . , ψ n (t, •)) L 2 (0,D) ≤ √ Dβ 2 ( z(t 0 + D) 2 , t -t 0 -D). (4.50)
Moreover, for all x ∈ [0, D], |F(ψ 1 (t, x), . . . , ψ n (t, x))| → 0 and F(ψ

1 (t, •), . . . , ψ n (t, •)) L 2 (0,D) → 0 when t → t 0 + D + T ( z(t 0 + D) 2 ) (resp. t → t 0 + D + T max ). Proof. Let x ∈ [0, D]. We can see from Proposition 4.1 that ψ(t, x) 2 → 0 when t → t 0 + D + T ( z(t 0 + D) 2 ) (resp. t → t 0 + D + T max ), and ψ(t, x) 2 = 0 when t ≥ t 0 + D + T ( z(t 0 + D) 2 ) (resp. t ≥ t 0 + D + T max ). Next, by continuity of F, we also have F(ψ 1 (t, x), . . . , ψ n (t, x)) → 0 when t → t 0 + D + T ( z(t 0 + D) 2 ) (resp. t → t 0 + D + T max ). Then, there exists a class GKL function β 2 such that F(ψ 1 (t, x), . . . , ψ n (t, x)) satises, |F(ψ 1 (t, x), . . . , ψ n (t, x))| ≤ β 2 ( z(t 0 + D) 2 , t -t 0 -D), (4.51)
for all x ∈ [0, D] and t ≥ t 0 + D.

Next, by squaring and integrating from 0 to D with respect to x and passing to the square roots, we nd,

F(ψ 1 (t, •), . . . , ψ n (t, •)) L 2 (0,D) ≤ √ Dβ 2 ( z(t 0 + D) 2 , t -t 0 -D), (4.52)
for all t ≥ t 0 + D. In addition, for all x ∈ [0, D], we have, |F(ψ 1 (t, x), . . . , ψ n (t, x))| → 0 and F(ψ Finally, by integrating from 0 to D with respect to the space variable x and passing to the square roots, we get

1 (t, •), . . . , ψ n (t, •)) L 2 (0,D) → 0 when t → t 0 + D + T ( z(t 0 + D) 2 ) (resp. t → t 0 + D + T max ).
u(t, •) L 2 (0,D) ≤ √ 2 F(ψ 1 (t, •), . . . , ψ n (t, •)) L 2 (0,D) , ∀t ≥ t 0 + D.
I(t) = z(t) 2 2 + u(t, •) 2 L 2 (0,D)
remains bounded for t ∈ [t 0 , t 0 + D], and for all t

∈ [t 0 + D, t 0 + D + T (B D ( z 0 2 , U t0 ∞ ))) (resp. t ∈ [t 0 + D, t 0 + D + T max ))
, there exists a class GKL function β 3 such that,

I(t) ≤ β 3 (B D ( z 0 2 , U t0 ∞ ), t -t 0 -D) , (4.59)
with B D (s 1 , s 2 ) = e D s 1 + De 2D s 2 for any s 1 , s 2 ≥ 0.

In particular, I(t) → 0 and |U (t

)| → 0, as t → t 0 + D + T (B D ( z 0 2 , U t0 ∞ )) (resp. t → t 0 + D + T max ).
Proof. Let us start by proving the boundedness of z(t) 2 for all t ∈ [t 0 , t 0 + D]. By the variation of the constant formula on (4.1)-(4.2) we recover, z(t) = e A(t-t0) z 0 + t t0 e A(t-y) BU (y -D)dy,

I(t) ≤ β( z(t 0 + D) 2 , t -t 0 -D) 2 + 2Dβ 2 ( z(t 0 + D) 2 , t -t 0 -D) 2 , (4.68)
which leads to

I(t) ≤ β 3 ( z(t 0 + D) 2 , t -t 0 -D), (4.69)
with

β 3 = β 2 + 2Dβ 2 2 is a class GKL function.
Then, using inequality (4.66), we obtain,

I(t) ≤ β 3 (B D ( z 0 2 , U t0 ∞ ), t -t 0 -D) .
(4.70)

In particular, we recover that I(t) → 0 when t → t 0 + D + T (B D ( z 0 2 , U t0 ∞ )) (resp. when t → t 0 + D + T max ) and that z(t) 2 is bounded for all t ≥ t 0 . Now, let us prove that u(t, •) L 2 (0,D) is bounded for all t ∈ [t 0 , t 0 +D]. Notice that the solution u is given by

u(t, x) = u 0 (t + x -t 0 ), t ∈ [t 0 , t 0 + D -x], U (t + x -D), t ∈ [t 0 + D -x, t 0 + D]. (4.71)
From this last equation, it is easy to deduce the boundedness of u(t, •) L 2 (0,D) using the transformation (4.26), the fact that |U (t + x -D)| ≤ |F (z 1 (t + x), . . . , z n (t + x)) | and the boundedness of z(t + x) 2 for all t + x ≥ t 0 . As a result, I(t) is bounded for all t ∈ [t 0 , t 0 + D]. Then, using inequality (4.66), we get

Finally, let us prove that |U (t)| → 0 as t → t 0 + D + T (B D ( z 0 2 , U t0 ∞ )) (resp. t → t 0 + D + T max ).
|U (t)| ≤ β 2 ( z(t 0 + D) 2 , t -t 0 -D), ∀t ≥ t 0 + D.
|U (t)| ≤ β 2 (B D ( z 0 2 , U t0 ∞ ), t -t 0 -D) . (4.74)
From where we deduce the desired property.

4.4 Some characterizations of F for the design of nite/xedtime predictor-based controllers

The previous section provides a general setting in which, as soon as one chooses F satisfying Assumption 4.1, one can design a nonlinear predictor-based controller to stabilize the system (4.5)-(4.8) in nite time or in xed time. In this section, let us give some characterizations of F.

For simplicity let us take t 0 = 0. inspired by [START_REF] Bernuau | Robust nite-time output feedback stabilization of the double integrator[END_REF]:

Explicit controllers for double chain of integrators

F(z 1 (t), z 2 (t)) = -k 1 {z 1 (t)} α1 -k 2 {z 2 (t)} α2 , (4.75)
which satises Assumption 4.1 as soon as k 1 , k 2 are any positive reals numbers and α 1 , α 2 are selected so that weighted homogeneity of negative degree κ is obtained for (4.

F(z 1 (t), z 2 (t)) = -k 1,0 {z 1 (t)} α1,0 -k 2,0 {z 2 (t)} α2,0 -k 1,∞ {z 1 (t)} α1,∞ -k 2,∞ {z 2 (t)} α2,∞ , (4.76)
where k 1,0 , k 2,0 , k 1,∞ , and k 2,∞ are any positive real numbers, and

α 1,0 = r0+2κ0 r0 , α 1,∞ = r∞+2κ∞ r∞ , α 2,0 = r0+2κ0 r0+κ0 ,α 2,∞ = r∞+2κ∞ r∞+κ∞ with κ 0 < 0, κ ∞ > 0, r 0 > -2κ 0 , r ∞ > 0.
Proof. Consider (4.22)-(4.25) with ω ≡ 0. Using LaSalle invariance principle with

V (z) = z1 0 k 1,0 {s} α1,0 + k 1,∞ {s} α1,∞ ds + z 2 2 2 , ( 4

Implicit Controllers for chain of integrators

For the z-subsystem of the target system (4.22)-(4.25),we can use the results from [START_REF] Zimenko | On simple scheme of nite/xedtime control design[END_REF], [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] to characterize a new F from which we can subsequently design a nonlinear predictor-based controller U (t) achieving FTS or nearly FxTS. However, for such a chain of integrators, it appears that G d has to be of the form G d = diag [r 1 , . . . , r n ], r i = r + (i -1)κ, r > max(0, -nκ), κ ∈ R (κ is the degree of homogeneity) then Note that AG d -G d A = κA (the driftless part "Az" is homogeneous) [88] can be rephrased as: Proposition 4.5 (see [START_REF] Polyakov | Finite-time and xed-time stabilization: Implicit lyapunov function approach[END_REF] for details)

Let a, b be chosen positive real numbers. For the z-subsystem of the target system where d is the dilation dened by (4.78) with r i = r + (i -1)κ, r > max(0, -nκ), z d is its associated homogeneous norm and gain k = yP is derived from the solution X ∈ R n×n (X = P -1 ), y ∈ R 1×n of the LMIs:

XA + AX + y B + By + aX ≤ 0, X > 0, bX ≥ G d X + XG d > 0, (4.80) (4.81)
where A = 0 (n-1)×1 , Īn-1 , 0 n×1 , B = e n := [0, . . . , 0, 1] . Then, the z-subsystem of

T (z 0 ) ≤ b a(-κ) z 0 -κ d ,
globally uniformly ES for κ = 0, globally nearly FxTS for κ > 0.

Similarly, we get: Proposition 4.6 (see [START_REF] Polyakov | Finite-time and xed-time stabilization: Implicit lyapunov function approach[END_REF], [START_REF] Zimenko | On simple scheme of nite/xedtime control design[END_REF], [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] for an equivalent formulation)

Select κ 0 < 0, κ ∞ > 0 and r 0 > -nκ 0 , r ∞ > 0. Let us dene r i,0 = r 0 + (i -1)κ 0 , r i,∞ = r ∞ + (i - 1)κ ∞ . Set d 0 (s) = e G d 0 s = diag [e r1,
F(z) = z r0+nκ0 d0 k 0 d 0 (-ln z d0 ) z for z < 1 z r∞+nκ∞ d∞ k ∞ d ∞ (-ln z d∞ ) z for z ≥ 1 (4.82)
where the gains k 0 and k ∞ are such that the LMIs

             X 0 A + AX 0 + y 0 B + By 0 + a 0 X 0 ≤ 0, X 0 > 0, b 0 X 0 ≥ G d0 X 0 + XG d0 > 0, X ∞ A + AX ∞ + y ∞ B + By ∞ + a ∞ X ∞ ≤ 0, X ∞ > 0, b ∞ X ∞ ≥ G d∞ X ∞ + XG d∞ > 0,
, P 0 = X -1 0 , k ∞ = y ∞ P ∞ , P ∞ = X -1 ∞ ).
Then, the z-subsystem of (4.22)-(4.25) with w ≡ 0 is globally FxTS.

Simulations

In this section, we focus on (4.3)-(4.4) for n = 2, i.e. By combining the state transformations ( 52)-( 53) introduced in [START_REF] Bekiaris-Liberis | Stabilization of linear strict-feedback systems with delayed integrators[END_REF] with the change of variables z1 (t) = z 1 (t), z2 (t) = z 2 (t -D 1 ), we recover the following transformations:

ż1 (t) = a 11 z 1 (t) + z 2 (t -D 1 ), t ≥ t 0 , ż2 (t) = a 21 z 1 (t) + a 22 z 2 (t) + U (t -D 2 ), t ≥ t 0 , (4.87)
z1 (t) = z 1 (t), z2 (t) = z 2 (t -D 1 ) + a 11 z 1 (t).
     ż1 (t) = z2 (t), t ≥ t 0 , ż2 (t) = -a 11 a 22 z1 (t) + (a 11 + a 22 )z 2 (t) + a 21 z1 (t -D 1 ) + U (t -D 1 -D 2 ), t ≥ t 0 , (4.91)
(4.92)
which is rewritten into

               ż1 (t) = z2 (t), t ≥ t 0 , ż2 (t) = -a 11 a 22 z1 (t) + (a 11 + a 22 )z 2 (t) + a 21 z1 (t -D 1 ) + u(t, 0), t ≥ t 0 , u t (t, x) = u x (t, x), t ≥ t 0 , x ∈ [0, D 1 + D 2 ], u(t, D 1 + D 2 ) = U (t), t ≥ t 0 . ( 4
U (t) = F(ϕ 1 (t, D 1 + D 2 ), ϕ 2 (t, D 1 + D 2 )) -a 21 ϕ 1 (t, D 2 ) -(a 11 + a 22 )ϕ 2 (t, D 1 + D 2 ) + a 11 a 22 ϕ 1 (t, D 1 + D 2 ), (4.97)
where ϕ 1 and ϕ 2 are solutions of Then, using F given in (4.76) to attain FxTS where we take the delays D 1 = 0.5s, D 2 = 0.75s, and the parameters as follows:

ϕ 1,x (t, x) = ϕ 2 (t, x), ϕ 2,x (t, x) = -
κ 0 = -0.5, r 0 = 2, k 1,0 = 10, k 2,0 = 11, κ ∞ = -0.2, r ∞ = 3, k 1,∞ = 11, k 2,∞ = 10
. Finally, we take the initial time t 0 = 0, the coecients a 11 = a 21 = a 22 = 1 and we give the simulations for three dierent initial conditions: z 0 = [START_REF] Castaños | Observer-based predictor for a susceptible-infectious-recovered model with delays: An optimal-control case study[END_REF][START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF] , 10z 0 and 100z 0 . to get FTS (logarithmic scale) for the initial condition z(t 0 ) = [START_REF] Castaños | Observer-based predictor for a susceptible-infectious-recovered model with delays: An optimal-control case study[END_REF][START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF] with dierent values for the delays D 1 and D 2 . On the right, the evolution of the applied predictor-based controller U (t)

given in (4.97) using the expression of F in (4.75) to get FTS in a blue solid line for the initial condition z(t 0 ) = [START_REF] Castaños | Observer-based predictor for a susceptible-infectious-recovered model with delays: An optimal-control case study[END_REF][START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF] , in a red dashed line for z(t 0 ) = [START_REF] Bresch-Pietri | Prediction-based stabilization of linear systems subject to input-dependent input delay of integral-type[END_REF][START_REF] Alexanderian | An age-structured model for the spread of epidemic cholera: Analysis and simulation[END_REF] , and in a black dotted line for z(t 0 ) = [500, 300] , with the delays D 1 = 0.75s and D 2 = 1s. In this chapter, we treat the problem of Lyapunov-based nonlinear boundary stabilization of a class of 1D reaction-diusion systems with any predened convergence (asymptotic or Non-Asymptotic). As an application, we focus on the Non-Asymptotic notions (nite-time and xedtime) for which we give some particular explicit control designs followed by some numerical simulations. The key idea of our approach is to use a spatially weighted L 2 -norm as a Lyapunov functional to design a nonlinear controller and to ensure stability with any desired convergence.

Introduction

In this chapter, we revisit the problem of boundary control design for a class of linear 1D reactiondiusion equations. In Section 5.2, we introduce the 1D reaction-diusion system with Dirichlet actuation. In Section 5.3, we introduce and give some properties of the spatially weighted L 2norm which is chosen as a Lyapunov functional (inspired from results in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]Chapter 11,page 178] for hyperbolic systems). Next, we establish the Lyapunov stability analysis where we design a nonlinear controller that will ensure stability with a predened convergence (asymptotic or Non-Asymptotic). Our approach is similar to the Control Lyapunov Function (CLF) approach, which has been investigated for parabolic PDEs in [START_REF] Karafyllis | Lyapunov-based boundary feedback design for parabolic PDEs[END_REF], [197], in the sense of using the Lyapunov function directly to design the boundary control. In Section 5.4, we provide some explicit control designs ensuring Non-Asymptotic stability (nite-time and xed-time). In Section 5.5, we give some numerical simulations to illustrate the results of both the nite-time and xed-time stabilization cases. Finally, in sections 5.6 and 5.7, we give some possible extensions of our approach and some limitations.

95

Proof. The proof of Theorem 5.1 is a direct application of the quadratic formula on the inequality (5.7), where we chose U (•) to be the solution of the following second-degree equation: Notice that by well choosing K in inequality (2.115), one may be able to recover dierent types of stability including asymptotic (exponential, hyper-exponential,...) or Non-Asymptotic (nite-time, xed-time, predened-time, prescribed-time,...).

σ 2 U (t) 2 -z x (t, 1)U (t) -B(V (z(t, •))) = 0.

Application to nite/xed-time stabilization

In this section, we use Theorem 5.1 to establish the second main result of our chapter which proves the FTS (resp. FxTS) of the closed-loop system (5.1)-(5.4) with the nonlinear control (5.20) for some well-chosen function K.

Theorem 5.2

Let t 0 ≥ 0, σ > 0. Let K : R + → R + be a continuous function such that K(0) = 0. Let B(•) be given as in (5.8). Then, if K is in the set E k0,a0 (resp. in the set E k∞,a∞ k0,a0 ), then the closed-loop system (5.1)-(5.4) with the nonlinear control (5.20) is nite-time stable (resp. xed-time stable).

Moreover, there exists a settling time T max (V (z 0 )) (upper bounded by a constant when FxTS) such that V (z(t, •)) = 0 when t ≥ t 0 + T max (V (z 0 )). By the coercivity condition z(t, •) 2 L 2 (0,1) = 0 when t ≥ t 0 + T max (V (z 0 )). In particular, if the control (5.20) is replaced by

U (t) = z x (t, 1) -sign(z x (t, 1)) z x (t, 1) 2 + 2σB(V (z(t, •))) σ , (5.23)
we have in addition, |U (t)| → 0 when t → t 0 + T max (V (z 0 )) and |U (t)| = 0 for any t ≥ t 0 + T max (V (z 0 )).

Proof. The proof of Theorem 5.2 is a straightforward application of Proposition 5.1 and Theorem 5.1. In fact, from Theorem 5.1, we have that (2.115) is satised for all t ≥ t 0 and any continuous function K : R + → R + such that K(0) = 0. In particular, for K ∈ E k0,a0 (resp. K ∈ E k∞,a∞ k0,a0 ).

Then from Proposition 2.5, we conclude that the closed-loop system (5.1)-(5.4) with (5.20) (or

(5.21)) is FTS (resp. FxTS).
Furthermore, if (5.20) is replaced by (5.23), we can prove that:

|U (t)| 2 = 1 σ 2 z x (t, 1) 2 -z x (t, 1) 2 + 2σB(V (z(t, •))) 2 , (5.24)
≤ 1 σ 2 z x (t, 1) 2 + 2σB(V (z(t, •))) -z x (t, 1) 2 2 , (5.25)
= 2 σ B(V (z(t, •))), (5.26)
where we have used the fact that |

√ a 1 - √ a 2 | ≤ |a 1 -a 2 |, for any a 1 , a 2 ≥ 0. Using in addition the fact that V (z(t, •)) = 0 =⇒ K(V (z(t, •))) = 0 =⇒ B(V (z(t, •))) = 0, we conclude that |U (t)| → 0 for any t → t 0 + T max (V (z 0 )
) and |U (t)| = 0 for any t ≥ t 0 + T max (V (z 0 )).

Simulations

In this section, we give numerical simulations for the closed-loop system (5.1)-(5.4) for three dierent initial conditions z 0 = x -x 2 , 100z 0 , and 1000z 0 with the following reaction coecient λ = 20, the initial time t 0 = 0, and with the nonlinear control U (t) dened as in (5.20) rst for

K(V (z(t, •))) = c 2 V (z(t,
•)) α , with c = 0.5, α = 0.5, and σ = 2 (given in (5.5)), to ensure FTS and

then for K(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t, •)) β , with c 1 = 0.
(V (z(t, •))) = c 2 V (z(t,
•)) α is described at the bottom of Figure 5.1 for the initial condition z 0 = x -x 2 ) with K(V (z(t, •))) = cV (z(t, •)) α , for the initial condition z 0 = x -x 2 . Finally, Figure 5.1 shows on top right, in a logarithmic scale, the evolution of the norm z(t, •) 2 L 2 (0,1) of the closed-loop system (5.1)-(5.4) with the nonlinear control

U (t) in (5.20) with K(V (z(t, •))) = cV (z(t, •)) α in solid lines and with K(V (z(t, •))) = cV (z(t, •))
in dashed lines, and for three dierent initial conditions: z 0 = x -x 2 in blue lines, 100z 0 in red lines, and 1000z 0 in black lines. Hence, we can observe from the solid lines that the larger the initial condition, the larger the settling time (i.e. the times of convergence depend on the initial condition).

The case of xed-time stabilization

(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t, •)) β (
whose time evolution is presented at the bottom of Figure 5.1 for the initial condition z 0 = x -x 2 ), for the initial condition z 0 = x -x 2 . Figure 5.2 shows in a logarithmic scale the evolution of the norm z(t, •) 2 L 2 (0,1) of the closed-loop system (5.1)-(5.4) with the nonlinear control U (t) in (5.20) with

K(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t, •)) β in
solid lines and with K(V (z(t, •))) = cV (z(t, •)) in dashed lines, and for three dierent initial conditions: z 0 = x -x 2 in blue solid lines, 100z 0 in red solid lines, and 1000z 0 in black solid lines. Hence, we can observe that the settling time is upper bounded by a constant that does not depend on the initial conditions (i.e. the time of the convergence does not depend on the initial conditions).

Comments on possible extensions of our approach

In this section, we will provide insightful comments regarding potential extensions of our approach.

Finite/xed-time tracking of a trajectory

Using our approach it is also possible to design a control that ensures nite/xed-time tracking of a given trajectory z r (t, x) satisfying (5.1)-(5.4). In fact, by introducing an error variable zr (t, x) = z(t, x) -z r (t, x), the study of the problem of tracking becomes equivalent to studying the stability with the control U (t) in (5.20) with K(V (z(t, •))) = cV (z(t, •)) α (whose time evolution is given on the bottom) for the initial condition z(t 0 , x) = x -x 2 . On top right, the evolution of the norm z(t, •) 2 L 2 (0,1) of the closed-loop system (5.1)-(5.4) in a logarithmic scale in a blue line for the initial condition z(t 0 , x) = x -x 2 , in a red line for z(t 0 , x) = 100(x -x 2 ), and in a black line for z(t 0 , x) = 1000(x -x 2 ), where we used the nonlinear control U (t) given in (5.20) with K(V (z(t, •))) = cV (z(t, •)) α to get FTS as shown in solid lines, and with K(V (z(t, •))) = cV (z(t, •))

to get exponential stability shown in dashed lines.

of

           zr t (t, x) = zr xx (t, x) + λz r (t, x), t ≥ t 0 , x ∈ [0, 1], zr (t, 0) = 0, t ≥ t 0 , zr (t, 1) = U (t) -z r (t, 1) := Ũ (t), t ≥ t 0 , zr (0, x) = z 0 (x) -z r (0, x) := z0 (x), x ∈ [0, 1]. (5.27) (5.28) (5.29)
(5.30)

Attenuation/rejection of control-matched disturbances

Another interesting feature of our approach is its ability to cope with a control-matched disturbance d(t) ∈ R (i.e. z(t, 1) = U (t) + d(t)). Remarkably, it guarantees either attenuation of this type of disturbance or rejection when the disturbance is bounded. Such a fact is possible by considering the control U (t) := Ũ (t) -M sign(z x (t, 1)) for some M > 0. To design Ũ (t), we can

(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t,
•)) β (whose time evolution is given at the bottom) for the initial condition z(t 0 , x) = x -x 2 . On top right, the evolution of the norm z(t, •) 2 L 2 (0,1) of the closed-loop system (5.1)-(5.4) in a logarithmic scale in a blue line for the initial condition z(t 0 , x) = x -x 2 , in a red line for z(t 0 , x) = 100(x -x 2 ), and in a black line for z(t 0 , x) = 1000(x -x 2 ), where we used the nonlinear control U (t) given in (5.20) and we

took K(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t, •)) β to get FxTS as shown in solid lines. Then, we took K(V (z(t, •))) = c 1 V (z(t, •)) to get exponential stability shown in dashed lines.
follow the lines of Section 5.3, where we can start by computing the time derivative of V given in (5.5) along the solutions of (5.1)-(5.4) and then by integrating by parts twice, we recover

dV (z(t, •)) dt ≤ (σ 2 + 2λ)V (z(t, •)) + 2e σ z x (t, 1)z(t, 1) -σe σ |z(t, 1)| 2 , (5.31)
= (σ 2 + 2λ)V (z(t, •)) + 2e σ z x (t, 1) Ũ (t) + d(t) -M sign(z x (t, 1)) -σe σ Ũ (t) + d(t) -M sign(z x (t, 1)) 2 , (5.32)
By expanding this last inequality and using the fact that Ũ (t) is assumed to satisfy 2 , in addition, sign( Ũ (t)) = -sign(z x (t, 1)), we recover (5.15) which yields

dV (z(t, •)) dt ≤ (σ 2 + 2λ)V (z(t, •)) + 2e σ z x (t, 1) Ũ (t) -σe σ | Ũ (t)| 2 + 2e σ z x (t, 1)d(t) -2e σ M |z x (t, 1)| -2σe σ Ũ (t)d(t) + 2σe σ M sign(z x (t, 1)) Ũ (t) + 2σe σ M sign(z x (t, 1))d(t) -σe σ M 2 -σe σ |d(t)| 2 , (5.33)
≤ (σ 2 + 2λ)V (z(t, •)) + 2e σ z x (t, 1) Ũ (t) -σe σ | Ũ (t)| 2 + 2e σ |z x (t, 1)||d(t)| -2e σ M |z x (t, 1)| + 2σe σ | Ũ (t)||d(t)| + 2σe σ M sign(z x (t, 1)) Ũ (t) + 2σe σ M |d(t)| -σe σ M 2 -σe σ |d(t)| 2 , (5.34)
= (σ 2 + 2λ)V (z(t, •)) + 2e σ z x (t, 1) Ũ (t) -σe σ | Ũ (t)| 2 + 2e σ |z x (t, 1)||d(t)| -2e σ M |z x (t, 1)| + 2σe σ | Ũ (t)||d(t)| -2σe σ M | Ũ (t)| + 2σe σ M |d(t)| -σe σ M 2 -σe σ |d(t)| 2 , (5.35)
= (σ 2 + 2λ)V (z(t, •)) + 2e σ z x (t, 1) Ũ (t) -σe σ | Ũ (t)| 2 -σe σ |d(t)| -M 2 + 2e σ |z x (t, 1)| + σ| Ũ (t)| |d(t)| -M , (5.36)
≤ -2e σ σ 2 | Ũ (t)| 2 -z x (t, 1) Ũ (t) -B(V (z(t, •))) -K(V (z(t, •))) + 2e σ |z x (t, 1)| + σ| Ũ (t)| |d(t)| -M , (5.37)
where B(•) is given as in (5.8) and Ũ (t

) is chosen to satisfy σ 2 Ũ (t) 2 -z x (t, 1) Ũ (t)-B(V (z(t, •))) = 0, we get Ũ (t) = z x (t, 1) -sign(z x (t, 1)) z x (t, 1) 2 + 2σB(V (z(t, •))) σ , (5.38)
in order to obtain

dV (z(t, •)) dt ≤ -K(V (z(t, •))) + 2e σ |z x (t, 1)| + σ| Ũ (t)| |d(t)| -M , ∀σ > 0; ∀ M > 0. (5.39)
As we can observe, the last inequality ensures the attenuation of the control-matched disturbance d(t) or the rejection when sup t≥t0 |d(t)| ≤ M for some upper-bound M > 0.

Notice that

Ũ (t) = -sign(z x (t, 1)) z x (t, 1) 2 + 2σB(V (z(t, •))) -z x (t, 1) 2 σ , (5.40)
and that sign( Ũ (t)) = -sign(z x (t, 1)) as we assumed. Moreover, Ũ (t) stays bounded for all t ≥ t 0 .

Limitation of our approach

Despite the advantages of this approach, it is not straightforward to determine how to adapt it to the problem of nite/xed-time estimation when for instance z x (t, 1) is measured beforehand. In fact, to achieve this goal, the most logical step to do is to consider an observer of the form

     ẑt (t, x) = ẑxx (t, x) + λẑ(t, x) -P 1 (x, z x (t, 1) -ẑx (t, 1)), t ≥ t 0 , x ∈ [0, 1], ẑ(t, 0) = -P 2 (z x (t, 1) -ẑx (t, 1)), t ≥ t 0 , ẑ(t, 1) = U (t) -P 3 (z x (t, 1) -ẑx (t, 1)) t ≥ t 0 , (5.41) (5.42)
(5.43)

2 see (5.40)

Limitation of our approach 103

which yields an error system of the form

     zt (t, x) = zxx (t, x) + λz(t, x) + P 1 (x, zx (t, 1)), t ≥ t 0 , x ∈ [0, 1], z(t, 0) = P 2 (z x (t, 1)), t ≥ t 0 , z(t, 1) = P 3 (z x (t, 1)), t ≥ t 0 , (5.44) (5.45) (5.46)
for some observer-gain functions P 1 (•), P 2 (•), and P 3 (•). The next step is to use the Lyapunov function V given in (5.5) to recover some inequality of the form

dV (z(t, •)) dt ≤ [cV (z(t, •)) + K(V (z(t, •))) + Q 1 (z x (t, 1))] -K(V (z(t, •))), (5.47)
for some positive coecient c and some polynomial Q 1 (•) given in terms of P 1 (•), P 2 (•), and P 3 (•) and satisfying Q 1 (0) = 0. From this last inequality, we need to ensure that cV (z(t,

•)) + K(V (z(t, •))) + Q 1 (z x (t, 1)) = 0 to recover inequality (2.115) (i.e. d dt V (z(t, •)) ≤ -K(V (z(t, •))).
To achieve this, we need to express Q 1 (•) (in turn P 1 (•), P 2 (•), and P 3 (•)) in terms of z(t, •). However, this not possible as Q 1 (•) (in turn P 1 (•), P 2 (•), and P 3 (•)) should depend only on zx (t, 1) by denition.

Conclusion

In this chapter, we treated the problem of nonlinear boundary stabilization, with any predened type of convergence, for a class of 1D reaction-diusion systems. To achieve this, we used the spatially weighted L 2 -norm" as a Lyapunov functional candidate V . By taking the time derivative of this functional along the trajectories of the reaction-diusion PDE d dt V (z(t, •)), we noticed that we got an inequality that relates d dt V (z(t, •)) to a second-degree polynomial involving the control U (t) subtracted by the term K(V (z(t, •))) (for any continuous function K such that K(0) = 0). By computing the root of this polynomial, we managed to design a nonlinear control U (t) and to obtain the inequality d dt V (z(t, •)) ≤ -K(V (z(t, •))). Using this last inequality, for well-chosen expressions of K, we ensured nite/xed-time stabilization of the reaction-diusion PDE. The present chapter did not study the existence/uniqueness issues of the solutions of the closed-loop system. To this purpose, ideas contained in [START_REF] Showalter | Montone Operators in Banach Space and Nonlinear Partial Dierential Equations[END_REF], [START_REF] Miyadera | Nonlinear Semigroups (Translations of mathematical monographs[END_REF] can be used. However, the obtained stability estimates will certainly help the analysis.

Part II Time-varying feedbacks for prescribed-time stabilization in innite-dimensional settings This chapter deals with the problem of prescribed-time stabilization of controllable linear systems with distributed input delay. We model the input delay as a transport PDE and reformulate the original problem as a cascade ODE-PDE system while accounting for the innite dimensionality of the actuator. We build on reduction-based and backstepping-forwarding transformations to convert the system into a target system having the prescribed-time stability property. Then, we prove the bounded invertibility of the transformations, and hence we show that the prescribedtime stability property is preserved into the original problem. To better illustrate the ideas of this approach, we focus rst on the scalar case. Then, we give a sketch of the main lines for the general case. To this end, we choose the ODE dynamics of the target system to be a Linear Time-Varying system so that we can rely on recent developments which include a polynomial-based Vandermonde matrix and the generalized Laguerre polynomials that allow a compact formulation for the stability analysis. A simulation example is presented to illustrate the obtained results.

Introduction

In this chapter, we combine the ideas of [START_REF] Zhu | Delay-adaptive control for linear systems with distributed input delays[END_REF], [START_REF] Holloway | Prescribed-time output feedback for linear systems in controllable canonical form[END_REF], and [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF] to handle the problem of stabilization of LTI systems in the presence of distributed input delay. We model the input delay as a transport PDE and reformulate the original problem as a cascade ODE-PDE system while accounting for the innite dimensionality of the actuator. We build on the reduction-based and backsteppingforwarding transformations to transform the system into a target system having the prescribedtime stability property. We relate back to such property through a suitable study of the bounded invertibility of the aforementioned transformations. The resulting predictor-like feedback is made up of time-varying gains. To better illustrate the ideas of this approach, we focus rst on the scalar case. Then, we give the detailed steps for the general case.

Chapter 6. Prescribed-time stabilization of LTI systems with distributed input delay

This chapter is organized as follows. In Section 6.2, we introduce the studied system (an LTI system with distributed input delay). In Section 6.3, we use an ODE-PDE setting and suitable transformations to come up with a prescribed-time predictor-based controller. We focus on the scalar case to better communicate the main ideas of our approach. The generalization to the general case is provided in Section 6.3.2. In Section 6.4 we consider a numerical example to illustrate the results.

Problem statement

We consider the following controllable linear system with distributed input delay as stated in [START_REF] Zhu | Delay-adaptive control for linear systems with distributed input delays[END_REF]:

ż(t) = Az(t) + D 0 B(D -σ)U (t -σ)dσ, t ≥ t 0 , (6.1)
where z(t) ∈ R n is the system state, U (θ) ∈ R for θ ∈ [t -D, t] is the actuator state, U (t) ∈ R is the control input, D > 0 is a known constant delay and A, B are the system matrix and input vector of appropriate dimensions, respectively. The input vector B(•) is a continuous real-valued vector function dened on [0, D].

The objective of this chapter is to design predictor-like feedback achieving prescribed time stability (PTS) in light of Denitions 2.1 and 2.3, but whose specic notion is adapted to the current problem (which is innite-dimensional) as we will see in Section 6.3.1.2 and Section 6.3.2.3.

In order to better communicate the key ideas in our approach, we are going to deal rst with the analysis and design for a simple linear scalar equation with distributed input delay.

Prescribed-time predictor-based control: an ODE-PDE setting and Reduction-based and backstepping-forwarding transformations

We reformulate the system (6.1) into a cascade ODE-PDE setting (i.e. cascade linear hyperbolic PDE with an LTI system) introduced in [START_REF] Bekiaris-Liberis | Lyapunov stability of linear predictor feedback for distributed input delays[END_REF], [START_REF] Zhu | Delay-adaptive control for linear systems with distributed input delays[END_REF], and that employed a backstepping-forwarding transformation and a reduction-based change of variable. As in [START_REF] Espitia | Prescribed-time predictor control of LTI systems with input delay[END_REF], [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF], the main idea of our approach is to transform the original system into a target system that is PTS (in an appropriate sense in light of Denitions 2.1 and 2.3) and that we choose to satisfy the property of convergence in a prescribed time t 0 + D + T . Here, T is xed a priori, D is the known input delay, and for simplicity of notations, we take the initialization time t 0 = 0.

As previously mentioned, in an attempt to better illustrate the key ideas of our method, we rst deal with a scalar linear system with distributed input delay. The generalization to the ndimensional case (i.e. LTI systems of the form (6.1)) follows the same strategy as the one we used in the scalar case (which represents the heart of this chapter contribution) as well as some suitable changes of variables and some transformations in the framework of linear time-varying systems.

Prescribed-time predictor-based control: an ODE-PDE setting and

Reduction-based and backstepping-forwarding transformations 109

Scalar case

Let us consider the following scalar control system:

ż(t) = az(t) + b D 0 U (t -σ)dσ, t ≥ 0. (6.2)
which is a particular case of (6.1) in the problem statement, with the state z ∈ R, the delay D ≥ 0,

and the distributed control input D 0 U (t -σ)dσ which is delayed by σ ∈ [0, D] units of time. The input initial condition is U (t) = 0 for all t ∈ [-D, 0).
Following [START_REF] Zhu | Delay-adaptive control for linear systems with distributed input delays[END_REF], the system (6.2) can be rewritten as ODE-PDE system:

           ż(t) = az(t) + b D 0 u(t, σ)dσ, t ≥ 0, σ ∈ [0, D], u t (t, x) = u x (t, x), t ≥ 0, x ∈ [0, D], u(t, D) = U (t), t ≥ 0, (6.3) (6.4) (6.5)
where u(t, •) is the transport PDE state whose solution is given by u(t, x) = U (t+x-D) = u(t-σ), where σ = D -x. We aim at stabilizing (6.3)-(6.5) (in turn (6.2)) in a prescribed time D + T .

Reduction-based and backstepping-forwarding transformation

We consider the following reduction-based change of variables:

z(t) = z(t) + D 0 q(σ)u(t, σ)dσ. (6.6)
where q(•) and its domain are yet to be characterized while meeting the property q(0) = 0. In addition, we consider the following backstepping-forwarding transformation:

ω(t, x) = u(t, x) -Γ(t, x)z(t), (6.7)
where the function Γ is time-varying. The inverse transformation is given as follows:

u(t, x) = ω(t, x) + Γ(t, x)z(t). (6.8)
Under (6.6) and (6.7), we want to transform (6.3) into the following target system:

     ż(t) = -γ 2 (t)z(t), t ≥ 0, ω t (t, x) = ω x (t, x), t ≥ 0, x ∈ [0, D], ω(t, D) = 0, t ≥ 0, (6.9) (6.10) (6.11)
where ω : [0, ∞) × [0, D] → R is the transport PDE state and γ 2 is given as in (2.53) by:

γ 2 (t) = γ 2 2,0 T 2 (T -t) 2 .
(6.12) 110 Chapter 6. Prescribed-time stabilization of LTI systems with distributed input delay

In order to characterize the transformation (6.7), let us compute its time and spatial derivatives.

ω t (t, x) = u t (t, x) -Γ t (t, x)z(t) -Γ(t, x) ż(t), (6.13)
= u t (t, x) -Γ t (t, x)z(t) -Γ t (t, x) D 0 q(σ)u(t, σ)dσ -Γ(t, x) ż(t) -Γ(t, x) D 0 q(σ)u t (t, σ)dσ, (6.14)
= u t (t, x) -Γ t (t, x)z(t) -Γ t (t, x) D 0 q(σ)u(t, σ)dσ -aΓ(t, x)z(t) -bΓ(t, x) D 0 u(t, σ)dσ -Γ(t, x) D 0 q(σ)u t (t, σ)dσ. (6.15)
Next, by integrating the last term by parts, we get

ω t (t, x) = u t (t, x) -Γ t (t, x)z(t) -aΓ(t, x)z(t) -Γ t (t, x) D 0 q(σ)u(t, σ)dσ -bΓ(t, x) D 0 u(t, σ)dσ + Γ(t, x) D 0 q (σ)u(t, σ)dσ -Γ(t, x)q(D) u(t, D). (6.16)
Next, from (6.7) at x = D and after that using (6.6), we recover that

u(t, D) = Γ(t, D)z(t), (6.17)
= Γ(t, D)z(t) + Γ(t, D) D 0 q (σ)u(t, σ)dσ. (6.18)
Thus, the previous equality becomes:

ω t (t, x) = u t (t, x) + F (t, D, x)z(t) + D 0 G(t, D, x, σ)u(t, σ)dσ, (6.19)
where

F (t, D, x) = -Γ t (t, x) -a + Γ(t, D)q(D) Γ(t, x), (6.20)
and

G(t, D, x, σ) = -Γ t (t, x) + Γ(t, D)q(D)Γ(t, x) q(σ) + Γ(t, x) q (σ) -b . (6.21)
On the other hand, the space derivative of ω(t, x) is given by

ω x (t, x) = u x (t, x) -Γ x (t, x)z(t), (6.22)
= u x (t, x) -Γ x (t, x) z(t) + D 0 q(σ)u(t, σ)dσ . (6.23)
Subtracting (6.23) from (6.19), we get

ω t (t, x) -ω x (t, x) = u t (t, x) -u x (t, x) + D 0 (G(t, D, x, σ) + Γ x (t, x)q(σ))u(t, σ)dσ (6.24)
+ (F (t, D, x) + Γ x (t, x))z(t). Following the standard approach to nding the kernel equations, we can prove that the timevarying function Γ and the function q satisfy the following PDE system:

q (σ) + aq(σ) = b, Γ x (t, x) -Γ t (t, x) = (a + Γ(t, D)q(D))Γ(t, x), (6.26) (6.27)
where q and Γ are dened on the domains, respectively T q : {σ : 0 ≤ σ ≤ D} and T Γ : {(t, x) : 0 ≤ x ≤ D, 0 ≤ t < T + x -D}.

Prescribed-time predictor-based control: an ODE-PDE setting and

Reduction-based and backstepping-forwarding transformations 111 Proposition 6.1

The system (6.26)-(6.27) has well-posed C ∞ solutions on T q and T Γ , given by

q(σ) = b a (1 -e -aσ ), (6.28)
Γ(t, x) = - a(a + γ 2 (t + x -D)) b(1 -e -aD ) e γ2,0T √ γ2(t)- √ γ2(t+x-D) , (6.29)
where γ 2 is dened by (6.12).

Proof. From (6.26), since q(0) = 0, we nd q(σ) = b a (1 -e -aσ ). Concerning (6.27), we know, thanks to the chosen z-dynamics of target system (6.9)-(6.11), that

-γ 2 (t)z(t) = ż(t) = ż(t) + D 0 q(σ)u t (t, σ)dσ, (6.30) = az(t) + b D 0 u(t, σ)dσ - D 0 q (σ)u(t, σ)dσ + q(D)u(t, D), (6.31) = az(t) + a D 0 q(σ)u(t, σ)dσ + q(D)Γ(t, D)z(t), (6.32)
= az(t) + q(D)Γ(t, D)z(t), (

= (a + q(D)Γ(t, D))z(t), (6.34) which leads to -γ 2 (t) = a + q(D)Γ(t, D). Γ(t, x) = e t 0 γ2(s)ds Γ(t, x), (6.36) in (6.27), we get Γx (t, x) -Γt (t, x) = 0, (6.37) whose solution is obtained by the method of characteristics as follows: (6.40)

Γ(t, x) = Γ(t + x -D, D) = e -t+x-D 0 γ2(s)ds Γ(t + x -D, D).
The only thing left is to compute Γ(t + x -D, D). To do this, we replace (6.28) in (6.35), From (6.7) and (6.6), at x = D, and using (6.28), (6.29) the boundary control is then,

Γ(t, D) = -a -γ 2 (t) q(D) = -a -γ 2 (t)
U (t) = -(a + γ 2 (t)) b a (1 -e -aD ) z(t) + b a D 0
(1 -e -aσ )u(t, σ)dσ , (6.43) where U (t + σ -D) = u(t, σ).

Equivalently, we have:

U (t) = -(a + γ 2 (t)) b a (1 -e -aD ) z(t) + b a D 0
(1 -e -aσ )U (t + σ -D)dσ .

(6.44)

Stability analysis

We rst study the stability of the target system and then we establish the bounded invertibility of the transformations by a suitable norm equivalence. Lemma 6.1

Let γ 2,0 > 0. Let T > 0 be xed and D > 0 be a known delay. Then, the dynamics of (6.9)

satises the following estimates: 1

|z(t)| 2 ≤ η z e -2γ2,0T √ γ2(t) |z 0 | 2 , (6.45)
for any z0 ∈ R and for all t ∈ [0, T ), where η z = e 2T γ 2 2,0 . In particular, |z(t)| 2 → 0 as t → T and |z(t)| ≡ 0 for t ≥ T . Moreover, the transport PDE ω of target system (6.9)-(6.11) is FxTS i.e. for any ω(0, x) ∈ L 2 (0, D), it holds that ω(t, •) L 2 (0,D) ≡ 0 for all t ≥ D.

Proof. From (6.9), we recover that z(t) = z0 e -t 0 γ2(s)ds , for any t ∈ [0, T ), (6.46) and z(t) = 0 for any t ≥ T.

(6.47)

Therefore, (6.46) leads to,

|z(t)| 2 ≤ η z e -2γ2,0T √ γ2(t) |z 0 | 2 , (6.48)
with η z = e T γ 2 2,0 . Then |z(t)| → 0 as t → T . On the other hand, by the method of the characteristics, the solution of (6.10)-(6.11) is FxTS. Indeed, ω(t, x) = ω 0 (t + x -D) for t ≤ D -x and ω(t, x) = 0 for t ≥ D -x. Hence, we can conclude that ω(t, •) L 2 (0,D) ≡ 0 for all t ≥ D.

Proposition 6.2

For the transformations (6.6) and (6.7), the following estimates hold:

u(t, •) 2 L 2 (0,D) ≤ 2 ω(t, •) 2 L 2 (0,D) + 2 Γ(t, •) 2 L 2 (0,D) |z(t)| 2 , (6.49)
1 e -2γ

|z(t)| 2 ≤ 2|z(t)| 2 + 2 q 2 L 2 (0,D) u(t, •) 2 L 2 (0,D) , (6.55)
which combined with (6.49) leads to (6.50).

Lemma 6.2

Let Γ be given by (6.29). Then, the following holds true:

lim t→D+T Γ(t, •) 2 L 2 (0,D) e -2γ2,0T √ γ2(t) = 0. (6.56) Proof. Let I(t) = Γ(t, •) 2 L 2 (0,D) e -2γ2,0T
√ γ2(t) . Using (6.27), we obtain

I(t) = D 0 e -2γ2,0T √ γ2(t) |Γ(t, x)| 2 dx, (6.57)
= D 0 e -2γ2,0T √ γ2(t) a 2 (a+γ2(t+x-D)) 2 b 2 (1-e -aD ) 2 e 2γ2,0T √ γ2(t)- √ γ2(t+x-D) dx, (6.58)
≤ D 0 2a 2 (a 2 + γ 2 (t + x -D) 2 ) b 2 (1 -e -aD ) 2 e -2γ2,0T √ γ2(t+x-D) dx, (6.59) = 2δ D 0 (a 2 + γ 2 (t + x -D) 2 )e -2γ2,0T √ γ2(t+x-D) dx, (6.60) ≤ F 1 (t -D) + F 2 (t -D), (6.61)
where δ =

F 1 (t -D) = 2δa 2 D 0 e -2γ2,0T √ γ2(t+x-D) dx, (6.62)
F 2 (t -D) = 2δ D 0 γ 2 (t + x -D) 2 e -2γ2
s = 2γ 2,0 T γ 2 (t + x -D) = 2(γ 2,0 T ) 2 T -t -x + D , (6.64)
we have

ds dx = 2(γ 2,0 T ) 2 (T -t -x + D) 2 = s 2 2(γ 2,0 T ) 2 , (6.65)
then, (6.63) becomes

F 2 (t -D) = 2δ 2γ2,0T √ γ2(t) 2γ2,0T √ γ2(t-D) s 4 (2γ 2,0 T ) 4 2(γ 2,0 T ) 2
s 2 e -s ds,

(6.66) = δ (2γ 2,0 T ) 2 2γ2,0T √ γ2(t) 2γ2,0T √ γ2(t-D)
s 2 e -s ds,

(6.67) = δ (2γ 2,0 T ) 2 -(s 2 + 2s + 2)e -s 2γ2,0T √ γ2(t) 2γ2,0T √ γ2(t-D) . (6.68)
Hence, we obtain,

F 2 (t -D) ≤ δ (2γ 2,0 T ) 2 4γ 2 2,0 T 2 γ 2 (t -D) 2 + 4γ 2,0 T γ 2 (t -D) + 2 e -2γ2,0T √ γ2(t-D) , (6.69)
and F 2 (t -D) → 0 as t → D + T .

On the other hand, using the previous change of variable (6.64) in (6.62), we obtain:

F 1 (t -D) = 2δ(aγ 2,0 T ) 2 2γ2,0T √ γ2(t) 2γ2,0T √ γ2(t-D)
e -s s 2 ds, The form of (6.71) allows to use the following generalized exponential integral:

E n (v) = v n-1 ∞ v e -s
s n ds, v > 0, n ∈ N.

0 < e -v 2 + v ≤ E 2 (v) ≤ e -v 1 + v . (6.73)
Then, we obtain that

F 1 (t -D) = 2δ(aγ 2,0 T ) 2 E2 2γ2,0T √ γ2(t-D) (2γ2,0T √ γ2(t-D)) , (6.74) ≤ 2δ(aγ 2,0 T ) 2 e -2γ 2,0 T √ γ 2 (t-D) (2γ2,0T √ γ2(t-D))(1+2γ2,0T √ γ2(t-D))
.

(6.75)

Hence, from (6.75), we get that F 1 (t -D) → 0 as t → D + T . Finally, we conclude from (6.61) that I(t) → 0 as t → D + T . Proposition 6.3 Let q be given by (6.28). Then, the following inequality holds: ≤ |z 0 | + q L 2 (0,D) u(0, •) L 2 (0,D) .

|z 0 | 2 ≤ 2(1 + q 2 L 2 (0,D) )(|z 0 | 2 + u(0, •) 2 L 2 (0,D) ).
(6.78)

Next, by Young's inequality, we obtain

|z 0 | 2 ≤ 2|z 0 | 2 + 2 q 2 L 2 (0,D) u(0, •) 2 L 2 (0,D) , (6.79)
from which (6.76) is deduced.

Theorem 6.1

Let γ 2 be given by (6.12). Let D > 0, γ 2,0 > 0, T > 0 xed. Let Γ be given by (6.29). Then, the solution of the closed-loop system (6.3)-(6.5) with prescribed-time predictor-based control (6.43) is PTS in the following sense: for any z 0 ∈ R and u(0, •) ∈ L 2 (0, D), the quantity |z(t)| 2 + u(t, •) 2 L 2 (0,D) remains bounded for t ∈ [0, max{D, T }], and for all t ∈ [max{D, T }, D + T ), the following estimate holds:

|z(t)| 2 + u(t, •) 2 L 2 (0,D) ≤ η z M e -2γ2,0T √ γ2(t) Γ(t, •) 2 L 2 (0,D) |z 0 | 2 + u(0, •) 2 L 2 (0,D) , (6.80) with M = 4(1 + 2 q 2 L 2 (0,D) )(1 + q 2 L 2 (0,D) ). In particular, |z(t)| 2 + u(t, •) 2 L 2 (0,D) → 0 as t → D + T, (6.81)
Moreover, |U (t)| → 0 as t → T .

Proof. By Proposition 6.2 we have for all t ∈ [0, D + T )

|z(t)| 2 + u(t, •) 2 L 2 (0,D) ≤ M 1 ω(t, •) 2 L 2 (0,D) + M 2 (t)|z(t)| 2 , (6.82)
with

M 1 = 2 + 4 q 2 L 2 (0,D) and M 2 (t) = 2 + M 1 Γ(t, •) 2 L 2 (0,D) .
Case 1: Assume that T ≥ D. By the fact that ω(t, •) L 2 (0,D) ≡ 0 as t ≥ D, it holds that for any t ∈ [D, D + T ),

|z(t)| 2 + u(t, •) 2 L 2 (0,D) ≤ M 2 (t)|z(t)| 2 , (6.83) = 2|z(t)| 2 + M 1 Γ(t, •) 2 L 2 (0,D) |z(t)| 2 .
(6.84) By Lemma 6.1, we have |z(t)| ≡ 0 for t ≥ T , and it holds for t ∈ [T, D + T ),

|z(t)| 2 + u(t, •) 2 L 2 (0,D) ≤ M 1 Γ(t, •) 2 L 2 (0,D) |z(t)| 2 , (6.85) ≤ η z M 1 e -2γ2,0T √ γ2(t) Γ(t, •) 2 L 2 (0,D) |z 0 | 2 . (6.86)
Notice that Γ(t, •) 2 L 2 (0,D) |z(t)| 2 does not vanish when t → T , even though z(t) vanishes. This is because, the rate of the growth-in-time of Γ(t, •) is the same the rate of the decreasing-time of z, on the interval [0, T ].

Case 2: Assume that T ≤ D. Then, using the fact that |z(t)| ≡ 0 for t ≥ T , then it holds for t ∈ [T, D + T ),

|z(t)| 2 + u(t, •) 2 L 2 (0,D) ≤ M 1 ω(t, •) 2 L 2 (0,D) + Γ(t, •)
|z(t)| 2 + u(t, •) 2 L 2 (0,D) ≤ η z M e -2γ2,0T √ γ2(t) Γ(t, •) 2 L 2 (0,D) (|z 0 | 2 + u(0, •) 2 L 2 (0,D) ), (6.88)
where M = M 1 (2 + 2 q 2 L 2 (0,D) ). We nally obtain by Lemma 6.2, that |z(t)| 2 + u(t, •) 2 L 2 (0,D) → 0, as t → D + T . It remains to show that |U (t)| → 0. Indeed, from the transformation (6.7) it holds

|U (t)| = |Γ(t, D)||z(t)| ≤ a|a + γ 2 (t)| b|1 -e -a | √ η z |z 0 |e -γ2,0T √ γ2(t) , (6.89)
from which we can conclude that |U (t)| → 0 as t → T .

General case

In this section, we present the design of the predictor-feedback prescribed-time stabilization of the general case (6.1). Let us consider the cascade ODE-PDE formulation of (6.1), i.e.

           ż(t) = Az(t) + D 0 B(σ)u(t, σ)dσ, t ≥ t 0 , σ ∈ [0, D], u t (t, x) = u x (t, x), t ≥ t 0 , x ∈ [0, D], u(t, D) = U (t), t ≥ t 0 . (6.90) (6.91) (6.92)
Let BD = D 0 e -A(D-y) B(y)dy. We assume that the pair (A, BD ) is controllable: Assumption 6.1

The controllability matrix C D = BD , A BD , . . . , A n-1 BD is of full rank n.

Reduction-based and backstepping-forwarding transformations

We consider the following reduction-based change of variables inspired from [START_REF] Zhu | Delay-adaptive control for linear systems with distributed input delays[END_REF]:

z(t) = P z(t) + D 0 σ 0 e -A(σ-y) B(y)dy u(t, σ)dσ , (6.93)
where P = g , (gA) , . . . , (gA n-1 ) , (6.94) and g the n-th row of the matrix C -1 D (existence of the inverse comes from Assumption 6.1).

In addition, consider the following backstepping-forwarding transformation ω(t, x) = u(t, x) -Γ(t, x) z(t), (6.95) where Γ(t, x) is a space and time-varying vector function such that The inverse transformation is given as follows:

Γ(t, x) z(t) = n i=1 Γ i (t, x)z i (t).
u(t, x) = ω(t, x) + Γ(t, x) z(t).

(6.97) Then, the system (6.90)-(6.92) is mapped into the following target system:

     ż(t) = C(t)z(t), t ≥ t 0 , ω t (t, x) = ω x (t, x), t ≥ t 0 , x ∈ [0, D], ω(t, D) = 0, t ≥ t 0 , (6.98)
(6.99) (6.100)

where C(t) is a companion canonical matrix, i.e.

C(t) =      0 1 • • • 0 . . . . . . . . . . . . 0 0 • • • 1 -p 0 (t) -p 1 (t) • • • -p n-1 (t)      , (6.101)
where the functions p i-1 , i ∈ {1, . . . , n} are dened as in [START_REF] Espitia | Prescribed-time predictor control of LTI systems with input delay[END_REF], [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF] by p 0 (t) = σn (r 1 , .., r n )γ 2 (t) n , (6.102) and for j = 1, . . . , n -1,

pj(t) = ( γ2(t)) n-j (γ2,0T ) n-j n k=j (-1) k-j σn-k (r1, ..., rn) k -1 j -1 k! j! γ2,0T γ2(t) n-k , (6.103)
where γ 2 (•) is given in (6.12), the coecients r 1 , • • • , r n are positive reals such that r i = r j for i = j in the range of n, and the polynomials σn-k (•) are the elementary symmetric polynomials Similar computations to the scalar case prove that the PDE equation of transformation (6.95) is as follows:

Γ x (t, x) -Γ t (t, x) = C(t) Γ(t, x), (6.108)
where Γ is dened on T Γ . Proposition 6.4

The system (6.108) has a well-posed C ∞ solution on T Γ , given by

Γ(t, x) = V (t) -D(t) -1 D(t + x -D)V (t + x -D) Γ(t + x -D, D), (6.109)
where D(s) = diag e -r1γ2,0T √ γ2(s) , . . . , e -rnγ2,0T √ γ2(s) , (6.110)

Γ i (t + x -D, D) = a i-1 -p i-1 (t + x -D), i ∈ {1, • • • , n}, (6.111
V (t) =      1 • • • 1 (δ 0 (-r 1 γ 2 ))(t) • • • (δ 0 (-r n γ 2 ))(t) . . . . . . . . . (δ n-2 (-r 1 γ 2 ))(t) • • • (δ n-2 (-r n γ 2 ))(t)      , (6.112)
with (6.113) where L

(δ k (-r i γ 2 ))(t) = -riγ2(t)( √ γ2(t)) k (γ2,0T ) k k!L (1) k r i γ 2,0 T γ 2 (t) , k ∈ {0, • • • , n -2},
(1) k (•) are the generalized Laguerre polynomials (see [START_REF] Espitia | Prescribed-time predictor control of LTI systems with input delay[END_REF], [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF], [START_REF] Kamen | Fundamentals of Linear Time-Varying Systems[END_REF]).

Proof. Using the following change of variable: Γ(t, x) = V (t) Γ(t, x), (6.114) we get Γx (t, x) -Γt (t, x) = D g (t) Γ(t, x), (6.115) where

D g (t) = diag [-r 1 γ 2 (t), -r 2 γ 2 (t), • • • , -r n γ 2 (t)] , (6.116)
is a diagonal matrix such that V satises (see [START_REF] Espitia | Prescribed-time predictor control of LTI systems with input delay[END_REF], [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF], [START_REF] Kamen | Fundamentals of Linear Time-Varying Systems[END_REF]): From (6.115)-(6.117), we have Γi,x (t, x) -Γi,t (t, x) = -r i c(t) Γi (t, x), (6.118) for all i ∈ {1, • • • , n}. The solution is given by, Γi (t, x) = e riγ2,0T √ γ2(t) e -riγ2,0T √ γ2(t+x-D)

V (t)D g (t) = C(t)V (t) -V (t).
Γi (t + x -D, D), (6.119) similar to (6.29) in the scalar case. As a result,

Γ(t, x) = D(t) -1 D(t + x -D) Γ(t + x -D, D), (6.120)
where D(t) is given in (6.110). Returning to the original variable Γ, we get

V (t) Γ(t, x) = D(t) -1 D(t + x -D)V (t + x -D) Γ(t + x -D, D), (6.121)
then Moreover, using the fact that u(t, D) = Γ(t, D) z(t) in (6.126), we obtain

Γ(t, x) = V (t) -D(t) -1 D(t + x -D)V (t + x -D) Γ(t + x -D, D).
C(t)z(t) = ż(t) (6.127) = A c z(t) + B c u(t, D) (6.128) = A c z(t) + B c Γ(t, D) z(t), (6.129)
where A c = P AP -1 is a Companion canonical matrix, i.e.

A c (t) =      0 1 • • • 0 . . . . . . . . . . . . 0 0 • • • 1 -a 0 -a 1 • • • -a n-1      , (6.130)
and B c = P BD = [0, ..., 0, 1] (see [START_REF] Antsaklis | Linear Systems[END_REF]Section 3]). Finally, we obtain,

B c Γ(t, D) = C(t) -A c , (6.131) and Γ i (t, D) = -(p i-1 (t) -a i ). (6.132)
This concludes the proof.

Prescribed-time predictor-based control

From (6.95), at x = D, and using (6.93), the boundary control is given as follows: U (t) = Γ(t, D) P z(t) + D 0 σ 0 e A(y-σ) B(y)dy u(t, σ)dσ , (6.133) where U (t + σ -D) = u(t, σ). Equivalently, we have:

U (t) = Γ(t,

Stability result

As in the scalar case, we start by studying the stability of the target system and then we establish the bounded invertibility of the transformations by a suitable norm equivalence. Proposition 6.5 There exists a polynomial function R 1 (•) in terms of γ 2 (t + x -D) such that: for all t ∈ [D, D + T ) the following estimate holds true:

|Γ(t, x) z(t)| 2 ≤ ηz R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) z(0) 2 2 .
(6.135)

Proof. Let us consider the following change of variables z(t) = V (t)z(t). Using this change of variables and using (6.98), we get

C(t)z(t) = ż(t), (6.137)
= V (t)z(t) + V (t) ż(t), (6.138) from which we recover that

C(t)V (t)z(t) = V (t)z(t) + V (t) ż(t), (6.139)
and thus, using (6.117), we get

ż(t) = D g (t)z(t), (6.140)
Solving this last equation, we obtain z(t) = D(t)D(0) -1 z(0).

(6.141)

From Proposition 6.4, we have

|Γ(t, x) z(t)| 2 = |Γ(t + x -D, D) V (t + x -D)D(t + x -D)D(t) -1 V (t) -1 z(t)| 2 .
(6.142)

Using (6.136), we get

|Γ(t, x) z(t)| 2 = |Γ(t + x -D, D) V (t + x -D)D(t + x -D)D(t) -1 z(t)| 2 . (6.143)
Using (6.141) followed by (6.136) for t = 0, we get

|Γ(t, x) z(t)| 2 = |Γ(t + x -D, D) V (t + x -D)D(t + x -D)D(t) -1 D(t)D(0) -1 z(0)| 2 , (6.144) = |Γ(t + x -D, D) V (t + x -D)D(t + x -D)D(0) -1 V (0) -1 z(0)| 2 .
(6.145)

Using the Cauchy-Schwartz inequality, we get

|Γ(t, x) z(t)| 2 ≤ Γ(t + x -D, D) 2 2 V (t + x -D) 2 2 D(t + x -D) 2 2 D(0) -1 2 2 × V (0) -1 2 2 z(0) 2 2 , (6.146)
= ηz Γ(t + x -D, D)

2 2 V (t + x -D) 2 2 D(t + x -D) 2 2 z(0) 2 2 , (6.147)
≤ ηz Γ(t + x -D, D)

2 2 V (t + x -D) 2 2 e -2rminγ2,0T √ γ2(t+x-D) z(0) 2 2 , (6.148)
where ηz = D(0

) -1 2 2 V (0) -1 2 2 .
Next, using the fact that for any s ∈ R n : s 2 ≤ s F (where • F is the Frobenius norm), we get

|Γ(t, x) z(t)| 2 ≤ ηz Γ(t + x -D, D) 2 2 V (t + x -D) 2 F e -2rminγ2,0T √ γ2(t+x-D) z(0) 2 2 .
(6.149)

Now, looking at the components of Γ and V , we can easily see that

Γ(t + x -D, D) 2 2 V (t + x -D) 2 F = R 1 ( γ 2 (t + x -D)), (6.150)
for some polynomial function R 1 (•) in terms of γ 2 (t + x -D). Then For the transformations (6.93) and (6.95), the following estimates hold:

|Γ(t, x) z(t)| 2 ≤ ηz R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) z(0) 2 2 .
u(t, •) 2 L 2 (0,D) ≤ 2 ω(t, •) 2 L 2 (0,D) + 2 Γ(t, •) z(t) 2 L 2 (0,D) , (6.152)
and

z(t) 2 2 ≤ N 1 z(t) 2 2 + M 1 u(t, •) 2 L 2 (0,D) , (6.153)
where

N 1 = 2 P -1 2 and M 1 = 2 e A• 2 L 2 (0,D) e -A• 2 L 2 (0,D) B(•) 2 L 2 (0,D) .
Proof.

z(t) 2 2 ≤ 2 P -1 2 2 z(t) 2 2 + 2 e A• 2 L 2 (0,D) e -A• 2 L 2 (0,D) B(•) 2 L 2 (0,D) u(t, •) 2 L 2 (0,D) , (6.160) = N 1 z(t) 2 2 + M 1 u(t, •) 2 L 2 (0,D) , (6.161)
which leads to (6.153). Lemma 6.3 Let Γ be given by (6.29). Then, the following property holds true:

lim t→D+T D 0 R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) dx = 0. (6.162) Proof. Let I(t) = D 0 R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) dx and let κ = deg(R). Notice that I(t) = κ i=0 p i J i (t) with {p i = R (i) (0)} {i=0,••• ,κ} are the polynomial coecients of R 1 (•) and J i (t) = D 0 γ 2 (t + x -D) i e -2rminγ2,0T
√ γ2(t+x-D) dx. By using this change of variables, we obtain

J j (t) = 2r min γ 2 2,0 T 2 (2r min γ 2,0 T ) j 2rminγ2,0T √ γ2(t) 2rminγ2,0T √ γ2(t-D)
s j-2 e -s ds,

(6.168) = γ 2,0 T (2r min γ 2,0 T ) 1-j 2rminγ2,0T √ γ2(t) 2rminγ2,0T √ γ2(t-D)
s j-2 e -s ds.

J i (t) = γ 2,0 T (2r min γ 2,0 T ) 1-i 2rminγ2,0T √ γ2(t) 2rminγ2,0T √ γ2(t-D) s i-2 e -s ds, (6.170) = γ 2,0 T (2r min γ 2,0 T ) 1-i -(i -2)! i-2 k=0 s k k! e -s 2rminγ2,0T √ γ2(t) 2rminγ2,0T √ γ2(t-D) (6.171)
≤ (i -2)!γ 2,0 T (2r min γ 2,0 T ) 1-i i-2 k=0 (2r min γ 2,0 T γ 2 (t -D)) k k! e -2rminγ2,0T
√ γ2(t-D) .

(6.172)

Note that proving (6.171) is straightforward by induction.

Case 2: Let i ∈ {0, 1} (i.e. i -2 < 0). Then, by using the generalized exponential, dened as

E n (r) = r n-1 ∞ r e -s s n ds, r > 0, n ∈ N * , (6.173)
and its property [202, Section 2]:

e -r n + r ≤ E n (r) ≤ e -r n -1 + r , (6.174)
we obtain,

J i (t) = γ 2,0 T (2r min γ 2,0 T ) 1-i 2rminγ2,0T √ γ2(t) 2rminγ2,0T √ γ2(t-D) e -s s 2-i ds, (6.175)
≤ γ 2,0 T (2r min γ 2,0 T )

:= γ 2,0 T (2r min γ 2,0 T ) 1-i (2r min γ 2,0 T γ 2 (t -D)) 1-i E 2-i (2r min γ 2,0 T γ 2 (t -D)), (6.177)
≤ γ 2,0 T (2r min γ 2,0 T ) 1-i e -2rminγ2,0T √ γ2(t-D)

1 -i + 2r min γ 2,0 T γ 2 (t -D) 2r min γ 2,0 T γ 2 (t -D)
I(t) ≤ R 2r min γ 2,0 T γ 2 (t -D) e -2rminγ2,0T √ γ2(t-D) , (6.179)
where

R(s) = γ 2,0 T 1 i=0 c i (2r min γ 2,0 T ) 1-i (1 -i + s)s 1-i + p i=2 c i (2r min γ 2,0 T ) 1-i (i -2)! i-2 k=0 s k k! .
Hence, we conclude that I(t) → 0 as t → D + T . Proposition 6.7

The following estimate holds:

z(t 0 ) 2 2 ≤ N 0 z(t 0 ) 2 2 + M 0 u(t 0 , •) 2 L 2 (0,D) , (6.180)
for any t 0 ≥ 0 where N 0 = 2 P 2 2 and M 0 = 2

P 2 2 e -A• 2 L 2 (0,D) e A• 2 L 2 (0,D) B(•) 2 L 2 (0,D) .
Proof. By replacing t = t 0 in (6.

≤ P 2 z(t 0 ) 2 + e -A• L 2 (0,D) u(t 0 , •) L 2 (0,D) e A• L 2 (0,D) B(•) L 2 (0,D) . (6.185)
Next, by Young's inequality, we obtain

z(t 0 ) 2 2 ≤ 2 P 2 2 z(t 0 ) 2 2 + 2 P 2 2 e -A• 2 L 2 (0,D) u(t 0 , •) 2 L 2 (0,D) e A• 2 L 2 (0,D) B(•) 2 L 2 (0,D) , (6.186)
≤ N 0 z(t 0 ) 2 2 + M 0 u(t 0 , •) 2 L 2 (0,D) , (6.187)
from which (6.180) is deduced. Theorem 6.2

Let γ 2 be given by (6.12) and let r min = min i∈{1,••• ,n} {r i } with r i > 0 involved in (6.102)-(6.103). Let D > 0, γ 2,0 > 0, T > 0 xed . Let γ be given by (6.109). Then, the solution of the closed-loop system (6.90)-(6.92) with prescribed-time predictor-based control (6.133) is PTS in the following sense: for any z 0 ∈ R n and u(0, •) ∈ L 2 (0, D), the quantity z(t) 2 + u(t, •) 2 L 2 (0,D) remains bounded for t ∈ [0, max{D, T }], and for all t ∈ [max{D, T }, D + T ), the following estimate holds: In particular, z(t) 2 2 + u(t, •) 2 L 2 (0,D) → 0 as t → D + T.

z(t) 2 2 + u(t, •) 2 L 2 (0,D) ≤ M D 0 R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) dx × z 0 2 2 + u(0, •) 2 L 2 (0,D) , (6.188
Moreover, |U (t)| → 0 as t → T .

Proof. From Proposition 6.6, we have

z(t) 2 2 + u(t, •) 2 L 2 (0,D) ≤ N 1 z(t) 2 2 + (M 1 + 1) u(t, •) 2 L 2 (0,D) , (6.190)
≤ N 1 z(t) 2 2 + 2(M 1 + 1) ω(t, •) 2 L 2 (0,D) + Γ(t, •) z(t) 2 L 2 (0,D) , (6.191)
where

N 1 = 2 P -1 2 and M 1 = 2 e A• 2 L 2 (0,D) e -A• 2 L 2 (0,D) B(•) 2 L 2 (0,D) .
Next, using Proposition 6.5, we obtain

z(t) 2 2 + u(t, •) 2 L 2 (0,D) ≤ N 1 z(t) 2 2 + 2(M 1 + 1) ω(t, •) 2 L 2 (0,D) + 2(M 1 + 1)η z × D 0 R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) dx z(0) 2 2 . (6.192)
Finally, by Proposition 6.7, we get

z(t) 2 2 + u(t, •) 2 L 2 (0,D) ≤ N 1 z(t) 2 2 + 2(M 1 + 1) ω(t, •) 2 L 2 (0,D) + 2(M 1 + 1)η z × D 0 R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) dx (6.193) × N 0 z(0) 2 2 + M 0 u(0, •) 2 L 2 (0,D) ,
where N 0 = 2 P 2 2 and M 0 = 2

P 2 2 e -A• 2 L 2 (0,D) e A• 2 L 2 (0,D) B(•) 2 L 2 (0,D) .
As in Theorem 6.1, we need to discuss the case T ≥ D and the case T ≤ D, after that we use the fact that ω(t, •) L 2 (0,D) ≡ 0 when t ≥ D and the fact that z(t) 2 ≡ 0 when t ≥ T to conclude that in both cases the following inequality holds:

z(t) 2 2 + u(t, •) 2 L 2 (0,D) ≤ M D 0 R 1 ( γ 2 (t + x -D))e -2rminγ2,0T √ γ2(t+x-D) dx × N 0 z(0) 2 2 + M 0 u(0, •) 2 L 2 (0,D) ,
with M = 2(M 1 + 1)η z max(N 0 , M 0 ). Finally, by using Lemma 6.3, we conclude that z(t) 6.199) for some positive polynomial function R 2 (•). In particular, we have U (t) → 0 as t → T .

|U (t)| ≤ R 2 ( γ 2 (t))e -2rminγ2,0T √ γ2(t) z(0) 2 (

Simulations

We consider a scalar linear equation with distributed input delay (6.2), with a = 2, b = 0.5, γ 2,0 = 2 and D = 1s. We x T = 4s. Numerical simulations were done by discretizing the cascade ODE-PDE system (6.3) and making use of transformation (6.7). Figure 6.1 shows on the top right, the evolution of z(t) the solution of the closed-loop system (6.3), in blue using the prescribed-time control (6.43), and in red dashed line using the following predictor feedback for exponential stabilization (see [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Zhu | Delay-adaptive control for linear systems with distributed input delays[END_REF]),

U (t) = k z(t) + b D 0 σ 0 e a(y-σ) dy u(t, σ)dσ , (6.200)
where, we chose k = -18.5 such that a + b × k < 0. On the top left it shows the evolution of u(t, x) the solution of the transport PDE. Finally, it shows on the bottom the evolution of the L 2 -norm of the closed-loop system (plotted in logarithmic scale) with the prescribed-time control U (t) given in (6.43) in solid lines and using the predictor feedback (6.200) for exponential stabilization in dashed lines, for 3 dierent initial conditions z 0 = 1, 100z 0 , and 10000z 0 . We can observe that under the prescribed-time control U (t), no matter what initial condition we take, the convergence to the origin is always ensured at time D + T = 5s.

Conclusion

In this chapter, we extended the existing results of prescribed-time delay-compensation and stabilization of LTI systems with input delay to distributed input delay. The main ideas were developed rst for a scalar LTI system with distributed input delay and after that generalized to the nth-dimensional LTI case. The prescribed-time predictor feedback design was achieved based on the backstepping approach using a time-varying backstepping-forwarding transformation and system, and the desired stability property is transferred back to the closed-loop system using the inverse transformation. The eectiveness of the proposed approach is demonstrated through numerical simulations.

Introduction

In this chapter, we address the problem of achieving prescribed-time boundary output-feedback stabilization for a class of linear 1D reaction-diusion partial dierential equations (PDEs) with input delay. We propose a novel approach, inspired by the employment of state predictions, as represented by [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]Chapter 11] to solve this problem. This approach is an advantageous alternative but requires radical advancements to be adjusted from nonlinear ODEs to linear PDEs. Our contribution builds upon the results of [START_REF] Espitia | On continuous boundary timevarying feedbacks for xed-time stabilization of coupled reaction-diusion systems[END_REF] and [START_REF] Steeves | Prescribed-time stabilization of reaction-diusion equation by output feedback[END_REF] for the case of delayed input and extends the results of [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diusion PDEs[END_REF] for the delay-dependent case to output-feedback stabilization.

The main idea of our approach is to transform the original PDE system into a PDE-PDE cascade system and then apply a time-varying backstepping transformation. Unlike [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diusion PDEs[END_REF] -where both the parabolic and hyperbolic parts of the cascade system are transformed using two different invertible backstepping transformations -we only transform the hyperbolic PDE state.

This transformation leads to a stable target system that ensures the desired prescribed-time convergence. Finally, by inverse transformation, we transfer the stability property and the desired Non-Asymptotic convergence back to the original closed-loop system. Moreover, unlike [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diusion PDEs[END_REF], our resulting predictor-based controller does not depend on a spatial derivative of the state and overcomes also the issue of incompatibility of boundary conditions of kernel equations (which arises when considering point-wise damping term in the design) as observed in [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diusion PDEs[END_REF]. In the course of developing a methodology alternative to [START_REF] Espitia | Prescribed-time predictor control of LTI systems with input delay[END_REF], [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diusion PDEs[END_REF], [START_REF] Steeves | Prescribed-time stabilization of reaction-diusion equation by output feedback[END_REF], we develop a number of technical innovations, located throughout the chapter and its appendices, and usable in future works on predictor-based prescribed-time stabilization.

This chapter is organized as follows. In Section 7.2, we introduce the unstable 1D reactiondiusion system with boundary input delay. In Section 7.4, we focus on the problem of prescribedtime stabilization of the original system using full-state feedback control, where we start by reformulating the system as a PDE-PDE cascade system; we use an invertible backstepping transformation to link the cascade systems to some well-chosen prescribed-time stable target systems.

We perform a stability analysis on the target systems. Then, by inverse transformation, we establish the boundedness of the state of the original systems and their convergence to the origin in a prescribed time using a suitable norm equivalence. In Section 7.4.4, we switch to the problem of prescribed-time output-feedback stabilization, where we adapt our approach to taking into account the dynamics of the proposed observer system. In Section 7.7, we consider a numerical example to illustrate the main results.

               z t (t, x) = z xx (t, x) + λz(t, x), t ∈ [t 0 , t 0 + D + T ), x ∈ [0, 1], z(t, 0) = 0, t ∈ [t 0 , t 0 + D + T ), z(t, 1) = U (t -D), t ∈ [t 0 , t 0 + D + T ), y(t) = z x (t, 1), t ∈ [t 0 , t 0 + D + T ), z(t 0 , x) = z 0 (x), x ∈ [0, 1], (7.1)
(7.2)

(7.3) (7.4) (7.5)
with the initial time t 0 ≥ 0, the reaction term λ ∈ R, the state z(t, x) ∈ R, the control U (t) ∈ R, with the following initial condition: U (t 0 + s) = 0 for all s ∈ [-D, 0], the collocated output y(t) ∈ R, and the initial condition z 0 ∈ L 2 (0, 1).

Our main goal is to design a predictor-based output-feedback controller achieving prescribedtime stabilization of the closed-loop system (7.1)-(7.5) in the following sense: there exist a class KL function β and a function µ : [t 0 , t 0 + D + T ) → R + , where µ tends to innity as t goes to t 0 + D + T , such that for any initial condition and for all t ∈ [t 0 , t 0 + D + T ), the following estimate

holds: z(t, •) L 2 (0,1) ≤ β z 0 L 2 (0,1) , µ(t -t 0 -D) .
In order to solve this problem we follow three steps:

Step 1: Prescribed-time full-state feedback stabilization: we start by solving the problem of prescribed-time stabilization by full-state feedback of the closed-loop system (7.1)-(7.5).

The idea is to rst represent the input delay as a linear transport PDE (inspired from [START_REF] Krstic | Control of an unstable reaction-diusion PDE with long input delay[END_REF]), so that the system (7.1)-(7.5) is rewritten as a parabolic-transport hyperbolic PDE-PDE cascade system. Then, we propose a novel innite-dimensional transformation (inspired from [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]Chapter 11,page 171]) to transform the resulting cascade system into a suitable target system that is prescribed-time stable. The target system is chosen to be a parabolictransport hyperbolic PDE-PDE cascade system with diusion dynamics being exactly as in (7.1) but whose Dirichlet right boundary condition is given not only in terms of a boundary term of transport PDE which vanishes after delay D but includes also a feedback term which renders the diusion PDE converging to zero after the delay. Such a feedback is borrowed from [START_REF] Espitia | On continuous boundary timevarying feedbacks for xed-time stabilization of coupled reaction-diusion systems[END_REF]. Finally, the stability property is transferred to (7.1)-(7.5) via the inverse transformation.

Step 2: Prescribed-time observer design: We employ the prescribed-time observer design from [START_REF] Steeves | Prescribed-time stabilization of reaction-diusion equation by output feedback[END_REF] to estimate the states of (7.1)-(7.5) in a prescribed time.

Step 3: Prescribed-time output-feedback stabilization: Finally, we combine the designed prescribed-time observer with the full-state control to ensure output-feedback stabilization in a prescribed time of the closed-loop system (7.1)-(7.5).

Remark 7.1 Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay

Extension of our result to the case of input-output delay is straightforward by using the following change of variables z(t, x) = z(t -D 2 , x), as it allows to obtain a system of the form of (7.1)-(7.5) (with D = D 1 + D 2 ) for which we can apply our approach to ensure prescribed-time stabilization.

           zt (t, x) = zxx (t, x) + λz(t, x), t ∈ [t 0 , t 0 + D 1 + D 2 + T ), x ∈ [0, 1], z(t, 0) = 0, t ∈ [t 0 , t 0 + D 1 + D 2 + T ), z(t, 1) = U (t -D 1 ), t ∈ [t 0 , t 0 + D 1 + D 2 + T ), y(t) = zx (t -D 2 , 1), t ∈ [t 0 , t 0 + D 1 + D 2 + T ), (7.6)
Remark 7.2

Note that in our approach, it is necessary to have a prescribed-time boundary controller for the delay-free case of system (7.1)-(7.5).

Prescribed-time boundary stabilization by full-state feedback in the delay-free case

Before presenting our approach, let us briey summarize the main results of [START_REF] Espitia | On continuous boundary timevarying feedbacks for xed-time stabilization of coupled reaction-diusion systems[END_REF] on prescribedtime boundary stabilization of system (7.1)-(7.5) when D = 0.

Consider the following blow-up function:

γ m (t -t0 ) := γ m m,0 T m ( t0 + T -t) m , (7.10)
for m ∈ N * , dened for all t ∈ [ t0 , t0 + T ) where T > 0 is a priori xed.

We recall the following time-varying boundary control:

U (t) = 1 0 K(1, s, t -t0 )z(t, s)ds, (7.11) where K is given explicitly in [163, Lemma 1] by K(x, s, t -t0 ) = - 1 2 γ 2 (t -t0 )s ∞ n=0 γ 2 (t -t0 ) n (n + 1)! x 2 -s 2 4T γ 2,0 n L (1) n -T γ 2,0 γ 2 (t -t0 ) , (7.12)
where L

n (•) are the generalized Laguerre polynomials. In addition, (7.12) can be simplied using the rst-order modied Bessel function I 1 (•) to get,

K(x, s, t -t0 ) = -γ 2 (t -t0 )s e √ γ 2 (t-t 0 )(x 2 -y 2 ) 4T γ 2,0 I 1 ( γ 2 (t -t0 )(x 2 -s 2 )) γ 2 (t -t0 )(x 2 -s 2 ) , (7.13) for (x, s, t) ∈ T := {(x, s, t) ∈ [0, 1] 2 × [ t0 , t0 + T ) : s ≤ x}.
Therefore, the closed-loop system (7.1)-( 7 More precisely, there exist positive constants c k > 0, c l > 0, and M > 0 such that for any initial condition z( t0 , •) at an initial time t0 , we have

z(t, •) L 2 (0,1) ≤ ξ 1 (t -t0 ) z( t0 , •) L 2 (0,1) , (7.15)
where

ξ 1 (t -t0 ) := M c l γ 2 (t -t0 )e -α0 √ γ2(t-t0) + e -γ2,0T √ γ2(t-t0) , (7.16)
with M := e (γ 2 2,0 +λ)T 1 + e 1 4T +γ2,0c k > 0 and α 0 :=

4γ 2 2,0 T 2 -1 4T γ2,0 > 0. Furthermore, |U (t)| ≤ ξ 2 (t -t0 ) z( t0 , •) L 2 (0,1) , (7.17)
with

ξ 2 (t -t0 ) := c l M γ 2 (t -t0 )e -α0
√ γ2(t-t0) .

(7.18)

In particular, z(t, •) L 2 (0,1) → 0, and |U (t)| → 0 when t → t0 + T .

Remark 7.3

To achieve exponential stabilization, it is sucient to replace the control gain K by

K exp (x, s) = -(λ + λ 0 )s I 1 ( (λ + λ 0 )(x 2 -s 2 )) (λ + λ 0 )(x 2 -s 2 ) , (7.19)
for λ 0 ≥ 0 and (x, s) ∈ T := {(x, s) ∈ [0, 1] 2 : s ≤ x}.

Prescribed-time stabilization by full-state feedback for the input delay case

Let us now consider the PDE-PDE cascade representation of (7.1)-(7.5),

               z t (t, x) = z xx (t, x) + λz(t, x), t ∈ [t 0 , t 0 + D + T ), x ∈ [0, 1], z(t, 0) = 0, t ∈ [t 0 , t 0 + D + T ), z(t, 1) = v(t, 0), t ∈ [t 0 , t 0 + D + T ), v t (t, y) = v y (t, y), t ∈ [t 0 , t 0 + D + T ), y ∈ [0, D], v(t, D) = U (t), t ∈ [t 0 , t 0 + D + T ). ( 7
F(t + y -t 0 -D, ϕ(t, •, y)) := 1 0
K(1, s, t + y -t 0 -D)ϕ(t, s, y)ds (7.27) with K dened as in (7.13) (with t0 = t 0 + D), and the predictor ϕ is chosen to satisfy ϕ(t, x, y) = z(t + y, x) which means it is the solution of

           ϕ y (t, x, y) = ϕ xx (t, x, y) + λϕ(t, x, y), ϕ(t, 0, y) = 0, ϕ(t, 1, y) = v(t, y), ϕ(t, x, 0) = z(t, x). (7.28) (7.29) (7.30) (7.31) with (t, x, y) ∈ {(t, x, y) ∈ [t 0 , t 0 + D + T ) × [0, 1] × [0, D] : t + y ∈ [t 0 + D, t 0 + D + T )}.
Notice that ϕ can be computed explicitly (see [START_REF] Polyanin | Handbook of Linear Partial Dierential Equations for Engineers and Scientists[END_REF]Chapter 3, page 266]) as follows: n(-1) n+1 sin(nπx)e (λ-n 2 π 2 )(y-τ ) v(t, τ )dτ, (7.33) Now, by substituting (7.32) in (7.26), we obtain, ω(t, y) = v(t, y) - (7.37) where the new kernels γ and q are given by γ(y, s, t + y -t 0 -D) := 2 +∞ n=1 e (λ-n 2 π 2 )y sin(nπs) 1 0 K(1, x, t + y -t 0 -D) sin(nπx)dx, (7.38) 7.4. Prescribed-time stabilization by full-state feedback for the input delay case133 and q(y -τ, t + y -t 0 -D) := -2π +∞ n=1 e (λ-n 2 π 2 )(y-τ ) n(-1) n+1 1 0 K(1, x, t + y -t 0 -D) sin(nπx)dx, (7.39) Remark 7.4

ϕ(t, x, y) = 2 +∞ n=1 sin(nπx)e (λ-n 2 π 2 )y 1 0 sin(nπs)z(t, s)ds + nπ(-1) n+1 y 0 e -(λ-n 2 π 2 )τ v(t, τ )dτ , (7.32)
1 0 K(1, x, t + y -t 0 -D)ϕ(t, x, y)dx, (7.34) = v(t, y) - 1 0 K(1, x, t + y -t 0 -D) 1 0 2 +∞ n=1 sin(nπx) sin(nπs)e (λ-n 2 π 2 )y z(t, s)ds + y 0 2π +∞ n=1 n(-1) n+1 sin(nπx)e (λ-n 2 π 2 )(y-τ ) v(t, τ )dτ dx, (7.35) = v(t, y) - 1 0 2 +∞ n=1 sin(nπs)e (λ-n 2 π 2 )y 1 0 K(1, x, t + y -t 0 -D) sin(nπx)dx z(t, s)ds + y 0 2π +∞ n=1 n(-1) n+1 e (λ-n 2 π 2 )(y-τ ) 1 0 K(1, x, t + y -t 0 -D) sin(nπx)dx v(t, τ )dτ, (7.36) := v(t, y) - 1 0 γ2 (y, s, t + y -t 0 -D)z(t, s) ds + y 0 q(y -τ, t + y -t 0 -D)v(t, τ )dτ,
To obtain exponential stabilization of (7.1)-(7.5), it is sucient to replace the transformation (7.26) by ω(t, y) = v(t, y) -F exp (ϕ(t, •, y)), (7.40) where ϕ is generated as before, i.e., from (7.28)-(7.31), and F exp is dened as, F exp (ϕ(t, •, y)) := 1 0 K exp (1, s)ϕ(t, s, y)ds (7.41) with the gain K exp given as in (7.19). Moreover, using (7.32), the transformation (7.26) can be simplied as follows: n(-1) n+1 e (λ-n 2 π 2 )(y-τ ) 1 0 K exp (1, x) sin(nπx)dx, (7.49) hence, recovering the kernels of the backstepping transformation obtained in [START_REF] Krstic | Control of an unstable reaction-diusion PDE with long input delay[END_REF] for λ 0 = 0.

ω(t, y) = v(t, y) - 1 0 K exp (1, x)ϕ(t, x, y)dx, (7.42) = v(t, y) - 1 0 K exp (1, x)
               z t (t, x) = z xx (t, x) + λz(t, x), t ∈ [t 0 , t 0 + D + T ), x ∈ [0, 1], z(t, 0) = 0, t ∈ [t 0 , t 0 + D + T ), z(t, 1) = ω(t, 0) + F(t -t 0 -D, z(t, •)), t ∈ [t 0 , t 0 + D + T ), ω t (t, y) = ω y (t, y), t ∈ [t 0 , t 0 + D + T ), y ∈ [0, D], ω(t, D) = 0, t ∈ [t 0 , t 0 + D + T ), (7.50)
F(t -t 0 -D, z(t, •)) = 1 0 K(1, s, t -t 0 -D)z(t, s)ds. (7.55)
The transformation is realized by using the fact that ϕ(t, x, y) = z(t + y, x) for all (t, x, y) ∈ [t 0 , t 0 + D + T ) × [0, 1] × [0, D], and noticing that the time-varying transformation (7.26) satises (7.53) (i.e. ω t (t, y) = ω y (t, y)). Indeed, we have

ω t (t, y) = v t (t, y) - ∂F(t + y -t 0 -D, ϕ(t, •, y)) ∂t , (7.56)
= v y (t, y) -

1 0 K t (1, s, t + y -t 0 -D)ϕ(t,
= ω y (t, y)

where we have used the fact that:

K t (1, s, t + y -t 0 -D) = ∂(t + y) ∂t ∂K(1, s, t + y -t 0 -D) ∂(t + y) (7.63) = ∂K(1, s, t + y -t 0 -D) ∂(t + y) (7.64) = ∂(t + y) ∂y ∂K(1, s, t + y -t 0 -D) ∂(t + y) (7.65) = K y (1, s, t + y -t 0 -D), ( 7
U (t) = v(t, D) = F(t -t 0 , ϕ(t, •, D)) = 1 0 K(1, s, t -t 0 )ϕ(t, s, D)ds, (7.73)
where ϕ is the solution of (7.28)-(7.31) and K is given in (7.13). Equivalently, using (7.67) at y = D,

U (t) = v(t, D) = F(t -t 0 , ψ(t, •, D)), (7.74)
where ψ is the solution of (7.69)-(7.72).

From (7.38), the expression of the control U (t) in (7.73) can be simplied as follows:

U (t) = 1 0 γ(s, D, t -t 0 )z(t, s) ds + D 0 q(D -τ, t -t 0 )v(t, τ )dτ, (7.75)
where γ and q are given in (7.38)-(7.39).

Remark 7.6

In light of Remark 7.4, the expression of the control U exp (t) that achieves exponential stabilization of (7.1)-(7.5) is given as follows:

U exp (t) = 1 0 γexp (s, D)z(t, s)ds + D 0 qexp (D -τ )v(t, τ )dτ, (7.76)
where γexp and qexp are given in (7.48) and (7.49), respectively. Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay

Stability analysis

In this subsection, we start by performing the stability analysis on the target system (7.50)-(7.54).

Then, by the inverse transformation (7.69)-(7.72) we establish the boundedness of the state of the original system (7.20)-(7.24) and its convergence to zero in a prescribed time using a suitable norm equivalence.

Proposition 7.1

Let γ 2,0 satisfy (7.14). There exists a polynomial function Q

(•) in terms of γ 2 (t + y -t 0 -D) such that, for all t ∈ [t 0 + D, t 0 + D + T ): D 0 ξ 2 (t + y -t 0 -D) 2 dy ≤ Q 2α 0 γ 2 (t -t 0 -D) e -2α0 √ γ2(t-t0-D) . (7.77)
where ξ 2 is dened in (7.18).

Proof. Let A be dened as follows:

A = D 0 ξ 2 (t + y -t 0 -D) 2 dy, (7.78)
:= c l M (2α 0 ) 4 D 0 (2α 0 ) 4 γ 2 (t + y -t 0 -D) 2 e -2α0 √ γ2(t+y-t0-D) dy, (7.79)
= c l M (2α 0 ) 4 D 0 (2α 0 γ 2 (t + y -t 0 -D)) 4 e -2α0
√ γ2(t+y-t0-D) dy.

where M , c k , and α 0 are all dened in (7.16).

Next, let us consider the following change of variables:

s = 2α 0 γ 2 (t + y -t 0 -D), (7.81)
= 2α 0 γ 2,0 T (t 0 + D + T -t -y) , (7.82)
from which we recover,

ds dy = 2α 0 γ 2,0 T (t 0 + D + T -t -y) 2 , (7.83) = 2α 0 γ 2,0 T γ 2 (t + y -t 0 -D), (7.84)
= s 2 2α 0 γ 2,0 T .

A = γ 2,0 c l M T 8α 3 0 2α0 √ γ2(t-t0) 2α0 √ γ2(t-t0-D)
s 2 e -s ds (7.86) Finally, by integrating by parts twice, we recover

A = γ 2,0 c l M T 8α 3 0 -(s 2 + 2s + 2)e -s 2α0 √ γ2(t-t0) 2α0 √ γ2(t-t0-D) , (7.87) = -Q 2α 0 γ 2 (t -t 0 ) e -2α0 √ γ2(t-t0) + Q 2α 0 γ 2 (t -t 0 -D) e -2α0 √ γ2(t-t0-D) , (7.88) ≤ Q 2α 0 γ 2 (t -t 0 -D) e -2α0 √ γ2(t-t0-D) , ( 7
+ D + T ): v(t, •) L 2 (0,D) ≤ ξ 3 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) . (7.90)
where .91) where Q(•) is given as in Proposition 7.1. In particular, it holds v(t, •) L 2 (0,D) → 0 for all t → t 0 + D + T .

ξ 3 (t -t 0 -D) := Q 2α 0 γ 2 (t -t 0 -D) 1/2 e -α0 √ γ2(t-t0-D) . ( 7
Proof. From (7.67) and using the fact that ω(t, y) = 0, ∀(t, y)

∈ [t 0 + D, t 0 + D + T ) × [0, D], we recover that v(t, y) = F(t + y -t 0 -D, ψ(t, •, y)). (7.92)
Next, by squaring the previous equality, using the fact that ψ(t, •, y) = z(t + y, •), and using (7.17) for t0 := t 0 + D, we get,

|v(t, y)| 2 ≤ ξ 2 (t + y -t 0 -D) 2 z(t 0 + D, •) 2 L 2 (0,1) . (7.93)
Now, by integrating from 0 to D with respect to y and using (7.77) in Proposition 7.1, we obtain,

v(t, •) 2 L 2 (0,D) ≤ D 0 ξ 2 (t + y -t 0 -D) 2 dy z(t 0 + D, •) 2 L 2 (0,1) , (7.94) ≤ ξ 3 (t -t 0 -D) 2 z(t 0 + D, •) 2 L 2 (0,1) , (7.95)
where

ξ 3 (t -t 0 -D) = Q 2α 0 γ 2 (t -t 0 -D) 1/2 e -α0 √ γ2(t-t0-D) , (7.96)
Now, by passing to the square roots, we recover (7.90). In particular, we can clearly see that ξ 3 (t -t 0 -D) → 0 as t → t 0 + D + T . As a result, we obtain that v(t, •) L 2 (0,D) → 0 when t → t 0 + D + T .

Let us now introduce our rst main result.

Theorem 7.1

Let γ 2,0 be chosen such that (7.14) is ensured. Let T > 0, D > 0 and t 0 ≥ 0. Then, the solution of the closed-loop system (7.20)-(7.24) with the prescribed-time time-varying controller (7.73) (or (7.74)) is prescribed-time stable in the following sense: For any initial condition z 0 , the quantities z(t, •) L 2 (0,1) and v(t, •) L 2 (0,D) remain bounded for all t ∈ [t 0 , t 0 + D]; and for all t ∈ [t 0 + D, t 0 + D + T ), the following norm

I(t) = z(t, •) L 2 (0,1) + v(t, •) L 2 (0,D) satises, I(t) ≤ B D ξ 4 (t -t 0 -D) z 0 L 2 (0,1) , (7.97)
with ξ 4 = ξ 1 + ξ 3 and B D = 2e λ(t0+D) e -π 2 t0 + ≤ 2 z 0 L 2 (0,1) e λ(t0+D) e -π 2 t0 + +∞ 1 e -x 2 π 2 t0 dx ,

≤ 2 z 0 L 2 (0,1) e λ(t0+D) e -π 2 t0 + +∞ 0 e -x 2 π 2 t0 dx ,

= 2 z 0 L 2 (0,1) e λ(t0+D) e -π 2 t0 + 1 2 √ πt 0 .

(7.105)

Finally, by squaring and integrating with respect to the variable x from 0 to 1, we get, for all t ∈ [t 0 , t 0

+ D] z(t, •) L 2 (0,1) ≤ B D z 0 L 2 (0,1) , (7.106)
with B D := 2 z 0 L 2 (0,1) e λ(t0+D) e -π 2 t0 + 1 2 √ πt0 .

• PTS of the closed-loop system of (7.20)-(7.24): Using (7.15) and (7.90) from Proposition 7.2, we have

I(t) = z(t, •) L 2 (0,1) + v(t, •) L 2 (0,D) , (7.107) ≤ ξ 1 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) + ξ 3 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) , (7.108) = ξ 4 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) , (7.109)
for all t ∈ [t 0 + D, t 0 + D + T ), where ξ 4 (•) := ξ 1 (•) + ξ 3 (•).

Next, using inequality (7.106) for t = t 0 + D, we recover (7.97). In particular, we have that I(t) → 0 when t → t 0 + D + T . Furthermore, we deduce that z(t, •) L 2 (0,1) is bounded for all 7.5. Prescribed-time output boundary feedback stabilization in the delay-free case 139 t ∈ [t 0 , t 0 + D + T ).

• Boundedness of v(t, •) L 2 (0,D) in [t 0 , t 0 + D]: Notice that v is given in [t 0 , t 0 + D] × [0, D] by v(t, y) = 0, t ∈ [t 0 , t 0 + D -y], U (t + y -D), t ∈ [t 0 + D -y, t 0 + D]. Seeing that z(t, •) L 2 (0,1) is bounded for all t ∈ [t 0 , t 0 +D+T ) and that K(1, •, t+y-t 0 -D) L 2 (0,1) is bounded in [t 0 , t 0 + D], we deduce that v(t, •) L 2 (0,D) is bounded for all t ∈ [t 0 , t 0 + D].

• Convergence of the control to the origin in a prescribed time: From the equations (7.74) and (7.93), we have for all t ≥ t 0 + D |U (t)| = |F(t -t 0 , ψ(t, •, D))| (7.117) ≤ ξ 2 (t -t 0 ) z(t 0 + D, •) L 2 (0,1) . (7.118) In particular, it is clear that U (t) → 0 when t → t 0 + T . This concludes the proof. K(1, s, t + y -t 0 -D) φ(t, s, y)ds (7.138) and φ is chosen to satisfy φ(t, x, y) = ẑ(t + y, x), and therefore, is the solution of the following parabolic PDE: where ω : [t 0 , t 0 + D + T ) × [0, D] → R is the transport PDE state.

              
Note that using the fact that ϕ(t, x, y) = z(t + y, x) and φ(t, x, y) = ẑ(t + y, x) for all (t, x, y) ∈ [t 0 , t 0 + D + T ) × [0, 1] × [0, D], it is clear that (7.137) veries (7.146)-(7.147).

Time-varying innite-dimensional inverse transformation

The inverse transformation is given by, v(t, y) = ω(t, y) + F(t + y -t 0 -D, ψ(t, •, y)), (7.151) where F(t + y -t 0 -D, ψ(t, •, y)) := 1 0 K(1, s, t + y -t 0 -D) ψ(t, s, y)ds (7.152) 7.6. Prescribed-time stabilization by output feedback for the input delay case 143 where ψ is the solution of

                    
ψy (t, x, y) = ψxx (t, x, y) + λ ψ(t, x, y) + P (x, t + y -t 0 , D + T ) × ψ x (t, 1, y) -ψx (t, 1, y) , ψ(t, 0, y) = 0, ψ(t, 1, y) = ω(t, y) + F(t + y -t 0 -D, ψ(t, •, y)), ψ(t, x, 0) = ẑ(t, x). K(1, s, t -t 0 ) φ(t, s, D)ds, (7.161) from (7.137) at y = D, where φ is generated from (7.28)-(7.31) and K is given in (7.13). Likewise from (7.151) at y = D, we can get U (t) = v(t, D) = F(t -t 0 , ψ(t, •, D)), (7.162) where ψ is generated from (7.153)- (7.156). Remark 7.8 To achieve exponential output-feedback stabilization, it is sucient to replace the control gain K by (7.19) as in Remark 7.4 and the observer gain P by

P exp (x) = -(λ + λ 0 )x I 1 ( (λ + λ 0 )(1 -x 2 )) (λ + λ 0 )(1 -x 2 ) , (7.163)
for λ 0 ≥ 0 and x ∈ [0, 1]. In this subsection, we start by performing a stability analysis on the target system (7.143)- (7.147).

Then, by the inverse transformation (7.151), we establish the boundedness of the state of the original system (7.20)- (7.24) and its convergence to zero in a prescribed time using a suitable norm equivalence. Proposition 7.3 Let γ 2,0 and γ 3,0 satisfy (7.14) and (7.135) respectively. Then, there exists a rational function Q 4 (•) in terms of γ 1 (t -t 0 -D) such that, (α 3 γ 1 (t -t 0 -D)) j j! e -α3γ1(t-t0-D) . := 1 (α 3 γ 1 (t -t 0 -D)) 1-i E 2-i (α 3 γ 1 (t -t 0 -D)), (7.178)

≤

e -α3γ1(t-t0-D) (1 -i + α 3 γ 1 (t -t 0 -D))(α 3 γ 1 (t -t 0 -D)) 1-i . (7.179) Finally, by using (7.174) from Case 1 and (7.179) from Case 2, we recover, B ≤ Q 4 (α 3 γ 1 (t -t 0 -D))e -α3γ1(t-t0-D) , (7.180) where

Q 4 (s) = α 3 γ 1,0 T   1 i=0 c i (1 -i + s)s 1-i + p i=2 c i (i -2)! i-2 j=0
s j j!   . (7.181) This concludes the proof. Proposition 7.4 For the transport PDE v(t, x) satisfying (7.23), there exists a positive polynomial function Q 4 (•) in terms of γ 1 (t -t 0 -D) such that the following estimate holds for t ∈ [t 0 + D, t 0 + D + T ): v(t, •) L 2 (0,D) ≤ ζ 4 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0,1) , (7.182) where ζ 4 (t -t 0 -D) 2 := Q 4 (α 3 γ 1 (t + y -t 0 -D))e -α3γ1(t+y-t0-D) , (7.183) with γ 1 (•) given in (7.10). In particular, it holds v(t, •) L 2 (0,D) → 0 for all t → t 0 + D + T . [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diusion PDEs[END_REF] Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay Proof. Let t ∈ [t 0 +D, t 0 +D +T ). Then, from (7.151) and using the fact that ω(t, y) = 0, ∀(t, y) ∈ [t 0 + D, t 0 + D + T ) × [0, D], we recover v(t, y) = F(t + y -t 0 -D, ψ(t, •, y)).

In particular, we get v(t, •) L 2 (0,D) → 0 as t → t 0 + D + T .

Let us now give our second main result, Theorem 7.2

Let γ 2,0 > 0, and γ 3,0 > 0 be chosen such that (7.14) and (7.135) are ensured. Let T > 0, D > 0 and t 0 ≥ 0. Then, the solution of the closed-loop system (7.20)- (7.24) with the observer (7.132)-(7.134) and the prescribed-time time-varying output control (7.161) (or (7.162)) is prescribed-time stable in the following sense: For any initial conditions z 0 and ẑ0 , the quantities z(t, •) L 2 (0,1) , ẑ(t, •) L 2 (0,1) , and v(t, •) L 2 (0,D) remain bounded for all t ∈ [t 0 , t 0 + D]; and for all t ∈ [t 0 + D, t 0 + D + T ), the following norm I(t) = z(t, •) L 2 (0,1) + ẑ(t, •) L 2 (0,1) + v(t, •) L 2 (0,D) satises, I(t) ≤ L D ζ 5 (t -t 0 -D) z 0 L 2 (0,1) + ẑ0 L 2 (0,1) , As in the proof of Theorem 7.1, it is clear that: z(t, •) L 2 (0,1) ≤ B D z 0 L 2 (0,1) , (7.191) ≤ B D ( z 0 L 2 (0,1) + ẑ0 L 2 (0,1) ), (7.192) 7.6. Prescribed-time stabilization by output feedback for the input delay case 147 in [t 0 , t 0 + D], with B D = 2e λ(t0+D) e -π 2 t0 + 1 2 √ πt0 . Moreover, using (7.136) and (7.192), we obtain ẑ(t, •) L 2 (0,1) ≤ z(t, •) L 2 (0,1) + z(t, •) -ẑ(t, •) L 2 (0,1) , (7.193) ≤ B D ( z 0 L 2 (0,1) + ẑ0 L 2 (0,1) ) + ζ 1 (t -t 0 -D) z 0 -ẑ0 L 2 (0,1) , (7.194) ≤ BD ( z 0 L 2 (0,1) + ẑ0 L 2 (0,1) ), • PTS of the closed-loop system of (7.20)-(7. Next, using (7.192) and (7.195) for t = t 0 +D, we recover (7.190). In particular, we have that I(t) → 0 when t → t 0 +D +T . Furthermore, we deduce that z(t, •) L 2 (0,1) and ẑ(t, •) L 2 (0,1) are bounded for all t ∈ [t 0 , t 0 + D + T ).

• Boundedness of v(t, •) L 2 (0,D) in [t 0 , t 0 + D]:

Notice that v is given in [t 0 , t 0 + D] × [0, D] by v(t, y) = 0, t ∈ [t 0 , t 0 + D -y], U (t + y -D), t ∈ [t 0 + D -y, t 0 + D]. Seeing that ẑ(t, •) L 2 (0,1) is bounded for all t ∈ [t 0 , t 0 + D + T ) and that K(1, ≤ ζ 3 (t -t 0 ) z(t 0 + D, •) L 2 (0,1) .

(7.207)

In particular, it is clear that U (t) → 0 when t → t 0 + T . This concludes the proof.

Numerical results

In this section, we give numerical simulations for the closed-loop system (7.20)-(7.24) with prescribedtime predictor-based output-feedback controller U (t) given in (7.161) used to attain prescribedtime stabilization. We take the delay D = 0.5s, the reaction coecient λ = 11, the initial time t 0 = 0, the prescribed time T = 1s and the initial conditions: z 0 (x) = 10(x -x 2 ), ẑ0 (x) = 0, v 0 (y) = 0.

We take the initial conditions of the blow-up functions γ 2 and γ 3 , given in (7.10), respectively as γ 2 (0) = 3.3 and γ 3 (0) = 2.2.

For the numerical simulations, we implement an implicit Euler scheme for the parabolic subsystems combined with the two-step LaxWendro method for the hyperbolic subsystems. The discretization with respect to space and time is done with steps ∆x = 10 -2 , ∆y = 2×10 -3 , ∆t = 10 -3 for the interval [t 0 , 0.7(t 0 +D+T )) and ∆t = 5×10 -5 for the interval [0.7(t 0 + D + T ), t 0 + D + T ). Finally, in the upper left and right of Figure 7.3, we give a comparison between the norm of the closed-loop system (7.20)-(7.24), for a delay D = 0.5s, using the prescribed-time predictor-based output-feedback controller U (t) given in (7.161) (in red solid line) and the same norm using the exponential predictor-based controller U exp (t) given in (7.164) with the constant-gain λ + λ 0 = 11 involved in the control and observer kernels K exp and P exp given in Remark 7.8 (in black dashed line). In particular, we compare the case with a higher constant-gain λ + λ 0 = 28 (in blue dashed line). On the bottom of Figure 7.3, we give a comparison between the controllers U (t) (in red solid line) and U exp (t) with the gain λ + λ 0 = 11 (in black dashed line) and with a higher gain λ + λ 0 = 28 (in blue dashed line).

As it can be observed, at the delay D = 0.5s, the norm of the closed-loop system (7.20)-(7.24) using the exponential controller U exp (t) (with the higher gain λ + λ 0 = 28) exhibits the "peaking phenomenon" [START_REF] Sussmann | The peaking phenomenon and the global stabilization of nonlinear systems[END_REF] (see the blue dashed curve in the upper left plot of Figure 7.3). After the delay D = 0.5s, the norm outpaces the same norm using the prescribed-time controller U (t). However, as time progresses, the curves of the two norms cross, and from then on the norm of the closedloop system (7.20)-(7.24) using the prescribed-time controller U (t) outperforms the same norm using the exponential controller U exp (t). This is due to the fact that the exponential controller starts with an aggressive control eort, because of the high gain λ + λ 0 , but with time its eort diminishes (see bottom left of Figure 7.3). In contrast, the prescribed-time controller U (t) starts with a moderate eort to avoid peeking and then gradually increases its control eort towards the end of the simulation to ensure that the convergence is completed in the prescribed time (see the upper left and bottom plots of Figure 7.3). ) for λ 0 = 0 in dashed lines (logarithmic scale), for the initial conditions z(t 0 , x) = 10(x -x 2 ), ẑ(t 0 , x) = 0, and v(t 0 , y) = 0, and for 3 dierent delays: D = 0.5s in blue; D = 0.6s in red; D = 0.7s in black.

predictor-based concepts, we transformed the PDE-PDE unstable system into a well-chosen PTS target system. Using the inverse transformation we ensured the desired prescribed-time stability property for the original system. More precisely, the main contributions of this thesis are as follows:

In Chapter 3, we dealt with the problem of nite-time estimation of linear time-invariant (LTI) systems, in the observable form, with delayed output. The main ideas relied on rewriting the system into an ODE-PDE cascade setting, where the PDE part modeled the eect of the delay on the output. The nonlinear gains were designed such that the error observer system is either FTS or FxTS. To achieve this, we used the backstepping approach where we chose a suitable nonlinear target system satisfying a chosen nite/xed-time convergence property. Finally, we used the invertibility of the backstepping transformation to pass this property to the error system.

In Chapter 4, we solved the problem of nite/xed-time stabilization of a chain of integrators with input delay. The chain of integrators was rewritten into an ODE-PDE setting, where the PDE part modeled the eect of the delay on the input. To solve this problem, we extended the well-developed nite-dimensional homogeneity-based tools to t into the innite-dimensional settings and combined them with the PDE backstepping approach. The predictor-based controller was designed using a nonlinear innite-dimensional backstepping transformation that linked the ODE-PDE setting to the well-chosen nite/xed-time stable target system.

An interesting direction, would be to extend the results of Chapters 3, 4, 6 and 7 to more complex interconnected dynamics namely nonlinear ODEs or linear parabolic PDEs that are interconnected with hyperbolic quasilinear/semilinear PDEs with non-local terms for which we can also use the results in [START_REF] Irscheid | Prescribed-time control for a class of semilinear hyperbolic PDE-ode systems[END_REF].

It could be also interesting to continue working on ODEs with input/output delay but consider other types of delays including stochastic, time-variant, state-dependent or inputdependent delays, where the lastly mentioned type of delays is dened implicitly through an integral of the past input values. This type of delay arises for instance in microuidic processes involving the Zweifach-Fung eect (see [START_REF] Petit | Control of a microuidic separation process governed by the zweifach-fung eect[END_REF], [START_REF] Bekiaris-Liberis | Predictor-feedback control of a model of microuidic process with hydraulic input-dependent input delay[END_REF]).

Another interesting application would be to address the design and analysis of multi-variable nite/xed-time extremum seeking for static maps subject to arbitrarily long time delays using the techniques developed in Chapter 4 and extending the results [START_REF] Poveda | Fixed-time extremum seeking[END_REF] to delay-dependent case inspired by [START_REF] Oliveira | Extremum seeking for static maps with delays[END_REF].

Finally, it could be interesting to investigate in depth the Lyapunov-based method introduced in Chapter 5, solve its issue and limitations, and extend it to more general complex high-order

PDEs starting with more general classes of nonlinear reaction-diusion-advection systems.

One can also try to apply this approach for global stabilization by boundary feedback of nonlinear parabolic PDEs that may blow up in the absence of the control. One could also attempt to mix the results of Chapter 7 with the nite/xed-time controller introduced in Chapter 5 to deal with the problem of nite/xed-time delay compensation for these classes of PDEs.

  preliminary denitions used throughout the thesis are as follows:

; 1 0 2 ; 2 L 2 (

 1222 For all a ∈ R and all b ≥ a, we denote by C([a, b]) the set of all continuous functions dened on the closed interval [a, b]; For all a ∈ R and all b ≥ a, we denote by C n ([a, b]), n ∈ N * , the set of functions that have continuous derivatives up to the n-th order on the closed interval [a, b]; For all a ∈ R and all b ≥ a, we denote by M n,m (R), n, m ∈ N * , the set of all m-by-n real matrices with the norm M 2 = sup { M z 2 : z ∈ R 2 with z 2 = 1}; • F denotes the Frobenius norm and is dened for any M ∈ M n,m (R) as M F = n i=1 m j=1 |M i,j | 2 ; xiii xiv Notation For all m ∈ N * and n ∈ N * , we denote by 0 m×n is the (m, n)-zero matrix which is the matrix with all entries equal to zero; For all n ∈ N * , we denote by Īn be the identity matrix of dimension n; For all m ∈ Z, I m (•), J m (•), denote the modied Bessel and (nonmodied) Bessel functions of the rst kind; For all n ∈ N * , we denote by L (α) n (•) the generalized Laguerre polynomials; For all n ∈ N * , we denote by σn (•) the elementary symmetric polynomials; For all a and b in R + , we denote by Γ(a) = ∞ 0 t a-1 e -t dt the Gamma Function; For all a and b in R + , we denote by I(a, b, r) = r 0 t a-1 (1-t) b-1 dt t a-1 (1-t) b-1 dt the regularized Incomplete Beta Function which is dened for all r ∈ [0, 1]; For all a ∈ R and all b ≥ a, we denote by L 1 (a, b) the set L 1 (a, b) := {f : [a, b] → R : b a |f (x)|dx < ∞}; For all a ∈ R and all b ≥ a, we denote by L 1 loc (a, b) the set L 1 loc (a, b) := {f : [a, b] → R : d c |f (x)|dx < ∞, ∀(c, d) ∈ R 2 : a ≤ c ≤ d ≤ b}; For all a ∈ R and all b ≥ a, we denote by L 2 (a, b) the set L 2 (a, b) := {f : [a, b] → R : b a |f (x)| 2 dx < ∞} with the scalar product f, g L 2 (a,b) := b a f (x)g(x)dx, and the norm f L 2 (a,b) := ( b a f (x) 2 dx) 1 For all a ∈ R and all b ≥ a, we denote by H 1 (a, b) the set H 1 (a, b) := f ∈ L 2 (a, b); f ∈ L 2 (a, b) , with the scalar product f, g H 1 (a,b) := f, g L 2 (a,b) + f , g L 2 (a,b) and with the norm f H 1 (a,b) := ( f 2 L 2 (a,b) + f

1 .L

 1 stabilisation non-asymptotique (en temps ni, xe et prescrit) pour certaines classes de systèmes dynamiques de dimension innie. Principalement, nous étudions les équations diérentielles ordinaires (EDO) linéaires, invariantes dans le temps, aectées par des retards d'entrée ou de sortie (ponctuels ou distribués), ainsi que les équations aux dérivées partielles (EDP) à une dimension de type réaction-diusion avec et sans retards d'entrée. Pour résoudre les problèmes d'estimation et de stabilisation, nous étendrons les outils et résultats approfondis développés dans le contexte des systèmes de dimension nie aux cas des systèmes de dimension innie. Plus précisément, nous nous focaliserons sur deux types de résultats : les résultats d'homogénéité qui produisent des commandes par retour d'état non linéaires (non régulières), et sur les résultats des commandes par retour d'état qui utilisent dans leurs formulations des fonctions gains qui divergent en temps ni et qui aident à la stabilisation en temps prescrit. Dans ce contexte, nous abordons tout d'abord le problème de la compensation, en temps ni et xe, des retards d'entrée ou de sortie pour les systèmes linéaires invariants dans le temps. En reformulant cette classe de systèmes en système cascade EDO-EDP, où la partie EDP correspond à une équation de transport modélisant l'eet du retard sur la sortie, nous utiliserons des techniques des systèmes de dimension innie. En eet, nous exploiterons l'approche du backstepping maintenant dans un contexte non linéaire ou temps variant. Ensuite, nous aborderons le problème de la stabilisation frontière en temps ni ou xe d'une classe d'EDP de type réaction-diusion. À notre connaissance, les tentatives existantes de stabilisation frontière pour cette classe de systèmes utilisent des commandes qui dépendent de l'état du système, mais aussi d'une fonction du temps qui diverge en temps ni. Cependant, l'utilisation de commandes par retours d'état dépendants de l'état seul (éventuellement non linéaires ou non régulières) n'a pas été considérée jusqu'à présent (comme nous l'expliquerons au chapitre 2). Nous aborderons ce problème complexe à l'aide de méthodes classiques liées aux Fonctions de Lyapunov de Contrôle (CLF). À la n, nous soulignerons les avantages et les limitations de cette approche et donnerons quelques indications sur l'extension de cette méthode au problème de stabilisation entrée-état (ISS) et au problème du suivi en temps ni ou xe pour les EDP de réaction-diusion. Ce manuscrit est divisé en deux grandes parties. La première partie concerne les outils non linéaires (non réguliers) qui permettent d'obtenir une stabilisation et une estimation en temps ni ou xe pour des systèmes de dimension innie. La deuxième partie est consacrée à l'exploitation d'outils basés sur l'utilisation de gains variant dans le temps et divergeant en temps prescrit dans des contextes de dimension innie. Nous commencerons, au chapitre 2, par rappeler certains résultats notables sur la stabilisation non-asymptotique pour certaines classes de systèmes de dimension innie. Nous introduirons ensuite les notions nécessaires et les principaux outils (y compris les diérentes notions de stabilité et leurs caractérisations avec les fonctions de Lyapunov) qui sont essentiels pour élaborer nos contributions. Enn, nous positionnerons nos contributions par rapport aux travaux les plus pertinents dans la littérature. Le reste de la thèse sera divisé comme suivant: Partie I C1) Dans le chapitre 3, nous présenterons notre première contribution qui vise à résoudre le problème d'estimation en temps ni ou xe d'états d'une classe des systèmes linéaire invariants 1ż(t) = Az(t) + BU (t -D), t ≥ t0 ,(3)où t0 ≥ 0 est le temps initial, z(t) = [z 1 (t), ..., z n (t)] ∈ R n , n ∈ N * , est l'état du systèmes U (t) ∈ R est la commande, et D > 0 est un retard constant connu à l'avance. Ici, la matrice du système et le vecteur d'entrée sont données respectivement par A := {A i,j } ∈ M n (R), où pour i, j ∈ {1, • • • , n}, A i,j = 1 si j = i + 1 et A i,j = 0 sinon, B = e n := 0 • • • 0 'idée de cette contribution est similaire à notre première contribution. Pour être précis, nous commencerons par reformuler la chaîne d'intégrateurs avec un retard en un système cascade ODE-PDE (une cascade d'une EDP de transport linéaire modélisant, l'eet du retard sur l'entrée, avec la chaîne d'intégrateurs). Cependant, dans cette situation, plusieurs dés supplémentaires s'ajoutent, car nous utiliserons une transformation backstepping non linéaire pour convertir le système cascade en un système cible stable en temps ni ou xe. Nous eectuerons l'analyse de stabilité sur le système cible en utilisant des outils classiques non asymptotiques tels que l'homogénéité et les fonctions GKL. Enn, nous utiliserons la transformation inverse pour transférer la propriété de stabilité au système en boucle fermée. C3) Dans le chapitre 5, nous présenterons notre troisième contribution qui vise à résoudre le problème de stabilisation frontière en temps ni ou xe d'une classe des EDP linéaire de réaction-diusion :

L

  'idée centrale de cette contribution est basée sur la technique du backstepping avec des gains variant dans le temps. Pour mettre en ÷uvre cette approche, nous commencerons par modéliser l'eet de retard distribué par une EDP de transport, puis nous reformulerons le problème original sous la forme d'un système en cascade ODE-EDP. En parallèle avec la transformation backstepping, nous introduirons des transformations de réduction des systèmes de dimension nie. Cela nous permettra de convertir le système en cascade en un système cible composé d'une équation de transport qui s'annule en un temps xe (égal au retard D) et d'une EDO linéaire avec une matrice variant dans le temps. Cette matrice est basée sur les développements récents concernant les matrices de Vandermonde et les polynômes généralisés de Laguerre. Ensuite, nous démontrerons que la propriété de stabilité en temps prescrit est transférée au système cascade en utilisant les transformations inverses. C5) Finalement, dans le chapitre 7, nous présenterons notre cinquième contribution qui vise à résoudre le problème de stabilisation frontière d'une classe des système de réaction-diusion à une dimension (1D) soumis à un retard d'entrée :

6 Chapter 1 .

 61 IntroductionDierential equations are examples of mathematical models that can be used to express the relation between the dierent physical components of systems. Dierential Equations can be classied according to their properties. For instance, they can be Ordinary Dierential Equations (in short ODEs), Partial Dierential Equations (in short PDEs), or Time-Delay systems (in short TDS).

Figure 1 . 1 :

 11 Figure 1.1: Torricelli's law (taken from Wikipedia)

  a known constant delay, A, B are respectively the system matrix and input vector of appropriate dimensions. The input vector B(•) is a continuous real-valued vector function dened on [0, D].The idea of this contribution is based on the backstepping technique, we start by modeling the input delay as a transport PDE and reformulating the original problem as a cascade ODE-PDE system while accounting for the innite dimensionality of the actuator. We introduce reduction-based and backstepping-forwarding transformations to convert (1.8) into a target system composed of a homogeneous transport equation that vanishes in a xed time equal to 1.4. Publications 13 the delay D and a Linear Time-Varying ODE that relies on the recent developments which include a polynomial-based Vandermonde matrix and the generalized Laguerre polynomials.

Figure 2 . 1 :

 21 Figure2.1: The evolution of (2.8) (in logarithmic scale on the right) for t0 = 0, α = 0.8, c = 15, and 4 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, and in black using z 0 = 1000. We can observe that the larger the initial condition, the larger the

Figure 2 . 2 :

 22 Figure2.2: The evolution of (2.11) (in logarithmic scale on the right) for t0 = 0, α 1 = 0.25, α 2 = 1.75, c 1 = c 2 = 5, and 4 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, and in black using z 0 = 1000. We can observe that the settling time is

Figure 2 . 3 :

 23 Figure 2.3:The evolution of (2.14) (in logarithmic scale on the right) for t0 = 0, α 1 = 0.5, T c = 0.3, and 5 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, in black using z 0 = 1000, and in orange using z 0 = 10000. We can observe that the

  .

Figure 2 . 4 :

 24 Figure2.4: The evolution of (2.17) (in logarithmic scale on the right) for t0 = 0, T = 1, c = 1, and 4 dierent initial conditions in blue using z 0 = 1, in red using z 0 = 10, in green using z 0 = 100, and in black using z 0 = 1000.

Theorem 2 . 1 (

 21 Lyapunov Stability Theorem [68, Theorem 4.1])

5 6 |z 1 | 6 5 + 1 2 |z 2 | 2 >

 5622 0 for all z = [0, 0]. Moreover, by computing the time derivative of V along the trajectories of (2.26)-(2.27), we get

Theorem 2 . 5 ([ 86 ,

 2586 Lemma 1])

  (3.12)(3.13) 50 Chapter 3. Finite/xed-time estimation of LTI systems with pointwise input delay from which the following error system is derived:

  initial condition z(θ) = z0 (θ) = [z 1,0 (θ), . . . , zn,0 (θ)] for all θ ∈ [t 0 -D, t 0 ].

  ) from [69, Chapter 3] which transforms (3.14)-(3.16) into the following target system:

1

 1 Characterisation of the observer-gain functions for second-order LTI systems

Remark 3. 2

 2 Note that, the idea behind choosing L 1 and L 2 as in(3.35)-(3.36) is to make the ODE part of the target error system (i.e. (3.21)-(3.23)) FTS when ω(t, 0) becomes zero. Similarly, if we want to ensure xed-time stability (FxTS) of (3.21)-(3.23) it is sucient to replace L 1 and L 2 respectively by

) ≤ k 2

 2 t0+D t0 |ω(s, 0)|ds + Dk 2 + |a 0 | t0+D t0 |ω(s, 0)|ds

(3. 86 )

 86 Using the fact that t 0 ≤ t ≤ t 0 + D, we get|z 1 (t)| ≤ t0+D t0 |z 2 (s)|ds + k 1 t0+D t0 |ω(s, 0) + z1 (s)| α1 ds + |a 1 | that ∀[a, b] ∈ R 2 + , ∀β ∈ (0, 1) : (a + b) β ≤ a β + b β and since α 1 ∈ (0, 1), we obtain |z 1 (t)| ≤t0+D t0 |z 2 (s)|ds + k 1 t0+D t0 |ω(s, 0)| α1 ds + k 1 t0+D t0 |z 1 (s)| α1 ds + |a 1 | t0+D t0 |ω(s, 0)|ds + |z 1,0 |.(3.88) 56 Chapter 3. Finite/xed-time estimation of LTI systems with pointwise input delay

(3. 124 )

 124 Employing (3.12)-(3.13), we nd z(t 0 + D)2 2 ≤ G 2 ( z 0 -ẑ0 2 ).

(3. 125 )where G 2 . 5 For

 12525 is the function dened by(3.110).Remark 3.4 Note that if we replace the nite-time observer-gain functions(3.35)-(3.36) by the xed-time observer-gain functions (3.35)-(3.36), the previous Lemmas can be easily adapted. Remark 3the general case and in both cases of the nite-time observer-gain function (3.44) and the xedtime observer-gain function (3.55), Lemmas 3.1-3.3 can be deduced using the ISS property of the (3.21)-(3.23) (see[START_REF] Zimenko | Homogeneity based nite/xedtime observers for linear mimo systems[END_REF] Corollaries 3 and 4]) by seeing the term ω(t, 0) as a bounded disturbance that vanishes after a certain amount of time.Lemma 3.4 For any t 0 , D ≥ 0, the target system (3.37)-(3.40) is FTS. Moreover, there exist positive scalar parameters δ, P min and P max such that for t ≥ t 0 + D z(t)

2

 2

(3. 137 ) 2 -

 1372 The function V is 1, α+1 homogeneous of degree ν = α + 1 and positive-denite because the 62 Chapter 3. Finite/xed-time estimation of LTI systems with pointwise input delay matrix P is also positive denite. In fact, the principal minors of P are positive

3. 3 .

 3 Stability analysis 63 value of a, b, and c from (3.137), we obtain ā = -ak 1 (α + 1) -2ck 2 , (3.153)

and k 2

 2 are given in (3.37)-(3.40), and with some constant C > 2+2α 3+α 3+α 2+2α

  ) and the Cauchy-Schwartz inequality, we obtain|ũ(t, x)| ≤ |ω(t, x)| + Ce Ax z(t) .

(3. 180 ) 66 Chapter 3 .

 180663 Finite/xed-time estimation of LTI systems with pointwise input delay

Figure 3 . 1 ,

 31 Figure3.1, we plot the states z 1 and z 2 of the system (3.4)-(3.7) in solid blue lines and the estimated states ẑ1 and ẑ2 of (3.8)-(3.10) in dashed red lines, with feedback U (t) = -7z 1 (t)-5z 2 (t), delay D = 1s, and initial condition z(0) =[START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF][START_REF] Liu | Networked control systems: A time-delay approach[END_REF] . Figure3.2, we have on the left the evolution of the error states of the ODE part of the error system(3.14) multiplied by e AD with feedback U (t) = -7z 1 (t) -5z 2 (t) and delay D = 1s. On the right hand, the evolution of the error state

Figure 3 . 1 :

 31 Figure 3.1:The evolution of the states z 1 and z 2 of the system (3.4)-(3.7) in solid blue lines and the estimated states ẑ1 and ẑ2 of (3.8)-(3.10) in dashed red lines, with feedback U (t) = -7z 1 (t)-5z 2 (t), delay D = 1s, and initial condition z(0) =[START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF][START_REF] Liu | Networked control systems: A time-delay approach[END_REF] .

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: On the left hand, we have the evolution of the error states of the ODE part of the error system (3.14) multiplied by e AD with feedback U (t) = -7z 1 (t) -5z 2 (t) and delay D = 1s. On the right hand, the evolution of the error state of the PDE part of the error system (3.15)-(3.16).

Figure 3 . 4 :

 34 Figure 3.4: Two-links manipulator

  z = [z 1 , z 2 ] , z = [z 3 , z 4 ], and for i ∈ {1, 2}, δ i is now given by

  B, and C are given in (3.2)-(3.3). The observers are then given by,

To simulate the systems ( 3 .

 3 201)-(3.204), (3.205)-(3.208), (3.209)-(3.211), and (3.212)-(3.214),

  2034 and δ = 1.075 which is calculated using the Matlab function fmincon. We can clearly see that these two values are far from the settling time observed in the simulations (see Figures 3.6, 3.7 and 3.8) and our estimates using (3.128) (T 1,max = 3.3396s and T 2,max = 3.5380s).

72 Chapter 3 .Figure 3 . 5 :

 335 Figure 3.5: The evolution of the states z 1 , z 2 , z 3 , and z 4 of the system (3.194)-(3.197) in solid blue lines and the estimated states ẑ1 , ẑ2 , ẑ3 , and ẑ4 in dashed red lines, with feedback τ in(3.190) and without noise measurement, where the delay D = 0.04s, the initial positions [z 1 (0), z 3 (0)] =[START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF] 2] , the initial velocities [z 2 (0), z 4 (0)] = [0, 0] m/s, and the input w 1 (t) = 2 sin(10t) and w 2 (t) = 1.4 sin(20t).

Figure 3 . 6 :Figure 3 . 7 :Figure 3 . 8 :

 363738 Figure 3.6: On the left hand, we have the evolution of the error states of the ODE part of the error system linked to the subsystem (3.201)-(3.204) multiplied by e AD with feedback τ in (3.190) and without noise measurement, where D = 0.04s and T 1,max = 3.3396s. On the right hand, the evolution of the error state of the PDE part of the error system with feedback τ in (3.190) and without noise measurement.

Figure 3 . 9 :Figure 3 . 10 :Figure 3 . 11 :Figure 3 . 12 :

 39310311312 Figure3.9: The evolution of the states z 1 , z 2 , z 3 , and z 4 of the system (3.194)-(3.197) in blue lines and the estimated states ẑ1 , ẑ2 , ẑ3 , and ẑ4 in red lines, with feedback τ in(3.190) and noise of power 0.0001, where the delay D = 0.04s, the initial positions [z 1 (0), z 3 (0)] =[START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF] 2] , the initial velocities [z 2 (0), z 4 (0)] = [0, 0] m/s, and the input w 1 (t) = 2 sin(10t) and w 2 (t) = 1.4 sin(20t).

(4. 21 )

 21 Next, using (4.10) we transform the system (4.5)-(4.8) into the following nonlinear target system:

1 F

 1 22)-(4.25) is FTS (resp. FxTS), let us assume that F satises the following assumption (for the ODE part of (4.22)-(4.25)): Assumption 4.is a continuous nonlinear function, dierentiable everywhere except on S = n i=1 {s = [s 1 , • • • , s n ] ∈ R n , s i = 0} , such that F(0, • • • , 0) = 0 and the origin of (4.1)-(4.2) with D = 0:

(4. 41 )

 41 Or form(4.26), at x = D, U (t) = u(t, D) = F(ψ 1 (t, D), . . . , ψ n (t, D)),(4.42)

  •) and T max are given in Assumption 4.1.

(4. 46 )

 46 Next, we use inequality (4.33) from Assumption 4.1 to obtain,

(4. 57 )

 57 Using inequality (4.50) from Proposition 4.2, we obtain,u(t, •) L 2 (0,D) ≤ √ 2Dβ 2 ( z(t 0 + D) 2 , t -t 0 -D), ∀t ≥ t 0 + D.

  initial condition U t0 : s ∈ [-D, 0] → U (t 0 + s) be dened and bounded in L 2 (-D, 0). Let D > 0 and t 0 ≥ 0. Then, the solution of the closed-loop system (4.5)-(4.8) with nitetime (resp. xed-time) predictor-based controller (4.40) (or (4.42)) is FTS (resp. FxTS) in the following sense: For any initial condition z 0 ∈ R n , the quantity

( 4 . 60 )

 460 = e A(t-t0) z 0 + t t0 e A(t0-y) BU (y -D)dy .

( 4 . 61 )

 461 Chapter 4. Finite/xed-time stabilization of LTI systems with pointwise input delayUsing the change of variables s = y -D -t 0 , we obtainz(t) = e A(t-t0) z 0 + t-t0-D -De A(-s-D) BU (t 0 + s)ds .

(4. 62 )≤ e D z 0 2 +

 622 Using e A(t-t0)2 ≤ e D and e A(-s-D) 2 ≤ e D , we getz(t) 2 ≤ e D z 0 2 + e 2D De 2D U t0 ∞ ,(4.65)= B D ( z 0 2 , U t0 ∞ ) .

(4. 66 )

 66 Then, z(•)

( 4 . 73 ) 4 . 4 .

 47344 Some characterizations of F for the design of nite/xed-time predictor-based controllers 89

For the target system ( 4 .

 4 22)-(4.25) with n = 2, we propose the following characterization of F

  22)-(4.25) with n = 2 and ω ≡ 0: for example by selecting r > -2κ and r 1 = r, r 2 = r + κ, α 1 = r + 2κ r , α 2 = r + 2κ r + κ . Hence, we can realize the resulting nonlinear predictor-based controller U (t) (4.40), with F having the structure in (4.75), stabilizing the system (4.5)-(4.8) in nite time. Let us give now a characterization of F to get a FxTS counterpart: Proposition 4.4 The z-subsystem of the target system (4.22)-(4.25) with n = 2 is FxTS when F is selected as follows:

d

  (s) = e G d s = diag [e r1s , . . . , e rns ].

  (4.22)-(4.25), let F(z) = z r+nκ d kd (-ln z d ) z,(4.79)

( 4 .

 4 [START_REF] D'apice | Packet ow on telecommunication networks[END_REF])-(4.25) with w ≡ 0 is globally FTS for κ < 0 and the settling time is given by

  0s , . . . , e rn,0s ] and d ∞ (s) = e G d∞ s = diag [e r1,∞s , . . . , e rn,∞s ]. Let a 0 , b 0 , a ∞ , b ∞ be chosen positive reals. For the z-subsystem of the target system (4.22)-(4.25), let

  X 0 and X ∞ in R n×n , y 0 and y ∞ in R 1×n (where k 0 = y 0 P 0

(4. 88 )

 88 where a 11 , a 21 and a 22 are real constants, D 1 and D 2 are positive known delays.

Chapter 4 .

 4 Finite/xed-time stabilization of LTI systems with pointwise input delay delays D 1 = 0.75s, D 2 = 1s, and the parameters as follows: κ = -0.2, r = 3, k 1 = 10 and k 2 = 11.

Figure 4 .

 4 Figure 4.1 shows on the left the evolution of the states z 1 and z 2 of the ODE part of the closedloop system (4.93)-(4.96) with predictor-based controller U (t) in (4.97) (whose time evolution is described in Figure 4.3 alongside of the time evolution of the norm of (4.93)-(4.96) for dierent values of the delays, using the expression of F in (4.75) to get FTS. On the right hand we can see the numerical solution u(t, x) of the PDE part of with respect the initial conditions z(t 0 ) = [5, 3] and u(t 0 , x) = 0, x ∈ [0, D 1 + D 2 ]. Finally, Figure 4.2 shows in a logarithmic scale the evolution of the norm z(t) 2 2 of the closed-loop system (4.87)-(4.88) with predictor-based controller U (t) in (4.97) on the left using the expression of F in (4.75) and on the right using the expression of F in (4.76). As it can be observed on the left, the times of convergence depend on the initial conditions (the larger the initial condition, the larger the settling time). On the right-hand side, we can observe that the times of the convergence do not depend on the initial conditions (the settling time is upper bounded by a constant independent of the initial conditions).

Figure 4 . 1 :Figure 4 . 2 : 2 2Figure 4 . 3 : 2 L 2

 414224322 Figure 4.1: On the left: the evolution of the states z 1 (t), z 2 (t) of the ODE part of the closed-loop system (4.93)-(4.96) with predictor-based controller U (t) in (4.97) and using the expression of F in (4.75) to get FTS, in blue solid lines for the initial condition z(t 0 ) = [5, 3] , in red dashed lines for z(t 0 ) = [0, 30] , and in black dotted lines for z(t 0 ) = [500, 300] , with the delays D 1 = 0.75s and D 2 = 1s. On the right: the evolution of u(t, x) the state of the PDE part of (4.93)-(4.96) for only the initial condition z(t 0 ) = [5, 3] and u(t 0 , x) = 0, x ∈ [0, D 1 + D 2 ].

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Problem statement: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Application to nite/xed-time stabilization . . . . . . . . . . . . . . . . 98 5.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.5.1 The case of nite-time stabilization . . . . . . . . . . . . . . . . . . . . . . . 99 5.5.2 The case of xed-time stabilization . . . . . . . . . . . . . . . . . . . . . . . 99 5.6 Comments on possible extensions of our approach . . . . . . . . . . . . 99 5.6.1 Finite/xed-time tracking of a trajectory . . . . . . . . . . . . . . . . . . . 99 5.6.2 Attenuation/rejection of control-matched disturbances . . . . . . . . . . . . 100 5.7 Limitation of our approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 , c 2 = 1 ,

 521 Figure 5.1 shows on top left the evolution of the state z(t, x) of the closed-loop system (5.1)-(5.4) with the control U (t) in (5.20) (whose time evolution for K(V (z(t, •))) = c 2 V (z(t, •))α is described at the bottom of Figure5.1 for the initial condition z 0 = x -x 2 ) with K(V (z(t, •))) = cV (z(t, •)) α , for the initial condition z 0 = x -x 2 . Finally, Figure5.1 shows on top right, in a logarithmic scale, the evolution of the norm z(t, •)2

Figure 5 .

 5 Figure 5.2 shows on top left the evolution of the state z(t, x) of the closed-loop system (5.1)-(5.4) with the control U (t) in (5.20) with K(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t, •)) β (whose time evolution is presented at the bottom of Figure5.1 for the initial condition z 0 = x -x 2 ), for the initial condition z 0 = x -x 2 . Figure5.2 shows in a logarithmic scale the evolution of the norm z(t, •) 2 L 2 (0,1) of the closed-loop system (5.1)-(5.4) with the nonlinear control U (t) in (5.20) withK(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t, •)) β insolid lines and with K(V (z(t, •))) = cV (z(t, •)) in dashed lines, and for three dierent initial conditions: z 0 = x -x 2 in blue solid lines, 100z 0 in red solid lines, and 1000z 0 in black solid lines. Hence, we can observe that the settling time is

Figure 5 . 1 :

 51 Figure 5.1: On top left: the evolution of the state z(t, x) of the closed-loop system (5.1)-(5.4)with the control U (t) in (5.20) with K(V (z(t, •))) = cV (z(t, •)) α (whose time evolution is given on the bottom) for the initial condition z(t 0 , x) = x -x 2 . On top right, the evolution of the norm z(t, •)2

5. 6 .Figure 5 . 2 :

 652 Figure 5.2: On top left, the evolution of the state z(t, x) of the closed-loop system (5.1)-(5.4) with the control U (t) in (5.20) with K(V (z(t, •))) = c 1 V (z(t, •)) α + c 2 V (z(t, •)) β (whose time evolution is given at the bottom) for the initial condition z(t 0 , x) = x -x 2 . On top right, the evolution of the norm z(t, •)2

( 6 . 35 )

 635 Moreover, by using (6.35) alongside the following change of variables:

( 6 . 38 )

 638 As a result,Γ(t, x) = Γ(t + x -D, D)e -t+x-D t γ2(s)ds ,

  Γ(t, x) = -a -γ 2 (t + x -D) b a (1 -e -aD ) e γ2,0T √ γ2(t)-√ γ2(t+x-D) .

(6. 42 ) 112 Chapter 6 .

 421126 Prescribed-time stabilization of LTI systems with distributed input delayPrescribed-time predictor-based control

a 2 b 2

 2 (1-e -aD ) 2 , and

(6. 72 )

 72 Moreover, the following inequality holds [202, Section 2]:

(6. 76 ) 6 . 3 .

 7663 Prescribed-time predictor-based control: an ODE-PDE setting and Reduction-based and backstepping-forwarding transformations 115 Proof. By replacing t = 0 in (6.6) and by the Cauchy-Schwarz inequality, we get |z 0 | ≤ |z 0 | + D 0 |q(σ)||u(0, σ) dσ, (6.77)

(6. 96 ) 6 . 3 .

 9663 Prescribed-time predictor-based control: an ODE-PDE setting and Reduction-based and backstepping-forwarding transformations 117

dened by σ0 (r 1

 1 , ..., r n ) = 1,(6.104) σk (r 1 , ..., r n ) = 1≤i1≤i2≤...i k ≤n r i1 r i2 . . . r i k , k ∈ {1, • • • , n -1}, (6.105) σn (r 1 , ..., r n ) = n i=1 r i , (6.106) σk (r 1 , ..., r n ) = 0, k ∈ {n + 1, n + 2, • • • }.

(6. 122 )σ 0 e=

 1220 Now, let us calculate Γ(t, D). By computing the time derivative of (6.93) and by integration by parts, we get ż(t) = P ż(t) + D 0 -A(σ-y) B(y)dy u t (t, σ)dσ , P AP -1 z(t) + D 0 e -A(D-y) B(y)dy u(t, D) ,(6.125) = P AP -1 z(t) + P BD u(t, D).

(6. 126 ) 6 . 3 .

 12663 Prescribed-time predictor-based control: an ODE-PDE setting and Reduction-based and backstepping-forwarding transformations 119

  D) P z(t) + D 0 σ 0 e -A(σ-y) B(y)dy U (t + σ -D)dσ .

(6. 136 ) 120 Chapter 6 .

 1361206 Prescribed-time stabilization of LTI systems with distributed input delay

(6. 151 ) 6 . 3 .

 15163 Prescribed-time predictor-based control: an ODE-PDE setting and Reduction-based and backstepping-forwarding transformations 121 Proposition 6.6

(6. 163 ) 122 Chapter 6 .

 1631226 Let us consider the following change of variables:s = 2r min γ 2,0 T γ 2 (t + x -D), (6.164) = 2r min γ 2 2,0 T 2 D + T -t -x , (6.165) Prescribed-time stabilization of LTI systems with distributed input delay from which we recover, ds dx = 2r min γ 2 2,0 T 2 (D + T -t -x) 2 ,

( 6 . 169 )

 6169 Case 1: Let i ∈ {2, 3, • • • , p} (i.e. i -2 ≥ 0). Then, by using multiple integrations by parts, we obtain,

σ 0 e 2 , 2 ,

 022 [START_REF] Sánchez-Torres | A discontinuous recurrent neural network with predened time convergence for solution of linear programming[END_REF] and by the Cauchy-Schwarz inequality and the Minkowski inequality, we get z(t 0 ) 2 = P z(t 0 ) + D 0 -A(σ-y) B(y)dy u(t 0 , σ)dσ (σ-y) B(y)dy u(t 0 , σ)dσ Ay 2 B(y) 2 dy |u(t 0 , σ)|dσ ,

) 124 Chapter 6 .

 1246 Prescribed-time stabilization of LTI systems with distributed input delay where M = 2(M 1 + 1)η z max(N 0 , M 0 ).(6.189)

a reduction-based change of variables. 126 Chapter 6 .Figure 6 . 1 :

 126661 Figure 6.1: On the top right, the evolution of the solution of the closed-loop system (6.3) (logarithmic scale) in blue line using the prescribed-time predictor-based control (6.43) with delay D = 1s, settling time T = 4s and initial condition z 0 = 1, and in red dashed line using the predictor feedback (6.200) for exponential stabilization. On the top left, the numerical solution of the transport PDE u(t, x). On the bottom, The evolution of the L 2 -norm of the closed-loop system (6.3) (logarithmic scale) in blue sold lines using the prescribed-time predictor-based control(6.43) and in dashed lines using the classical predictor feedback(6.200) for exponential stabilization, with delay D = 1s, prescribed time T = 4s and for 3 initial conditions z 0 = 1, 100z 0 , and 10000z 0 .

7. 2 . Problem statement 129 7. 2

 21292 Problem statementLet us consider the following reaction-diusion equation with a constant reaction term and a known constant boundary input delay D > 0:

  .5) with D = 0 and under the time-varying boundary feedback (7.11)-(7.13) is prescribed-time stable (PTS) [163, Theorem 3] provided that 2γ 2,0 T > 1.

(7. 14 ) 7 . 4 .

 1474 Prescribed-time stabilization by full-state feedback for the input delay case131

  ) sin(nπs)e (λ-n 2 π 2 )y z(t, s)ds +

1 ) 1 0K 1 ) 1 0K 1 0K

 11111 ) sin(nπs)e (λ-n 2 π 2 )y z(t, s)ds n+1 sin(nπx)e (λ-n 2 π 2 )(y-τ ) v(t, τ )dτ dx, )e (λ-n 2 π 2 )y exp (1, x) sin(nπx)dx z(t, s)ds (7.45) n+1 e (λ-n 2 π 2 )(y-τ ) exp (1, x) sin(nπx)dx v(t, τ )dτ, , s)z(t, s) ds + y 0 qexp (y -τ )v(t, τ )dτ,(7.47) where the kernels γexp and qexp are given by, γexp (y, s) := 2 +∞ n=1 e (λ-n 2 π 2 )y sin(nπs) exp (1, x) sin(nπx)dx, (7.48) and qexp (y -τ ) := -2π +∞ n=1

(7. 85 )

 85 Now, by using (7.82)-(7.85) in (7.78) (along with dy = 2α 0 γ 2,0 T ds s 2 ), we obtain

( 7 . 110 )

 7110 Thus, for all (t, y)∈ [t 0 , t 0 + D] × [0, D], we have |v(t, y)| ≤ |U (t + y -D)|.

( 7 . 111 ) 0 K( 1 , 0 K( 1 ,

 71110101 Next, by squaring the previous inequality, integrating with respect to y from 0 to D, and using the Cauchy-Schwartz inequality, we getv(t, •) 2 L 2 (0,D) ≤ D 0 |U (t + y -D)| 2 dy, + y -t 0 -D, ψ(t, •, y))| 2 dy, s, t + y -t 0 -D)ψ(t, s, y) ds •, t + y -t 0 -D) 2 L 2 (0,1) ψ(t, •, y) 2 L 2 (0,1) dy, •, t + y -t 0 -D) 2 L 2 (0,1) z(t + y, •) 2L 2 (0,1) dy.

7. 5

 5 Prescribed-time output boundary feedback stabilization in the delay-free caseAs before, let us rst summarize the main results of[START_REF] Steeves | Prescribed-time stabilization of reaction-diusion equation by output feedback[END_REF] on prescribed-time output boundary feedback stabilization of the delay-free case system (7.1)-(7.5) when D = 0. Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay where F has the same structure as in (7.27), i.e., F(t + y -t 0 -D, φ(t, •, y)) := 1 0

  y (t, x, y) = ψ xx (t, x, y) + λψ(t, x, y), ψ(t, 0, y) = 0,ψ(t, 1, y) = ω(t, y) + F(t + y -t 0 -D, ψ(t, •, y)), ψ(t, x, 0) = z(t, x).

  the inverse transformation (7.151), we recover (7.23)-(7.24).

7. 6 . 4

 64 Prescribed-time predictor-based output controllerAs in Subsection 7.4.4, we recover the expression of the boundary control U (t) as follows:U (t) = v(t, D) = F(t -t 0 , φ(t, •, D)) := 1 0

D 0 ζ 3 0 ζ 3 0 Q 3 α

 030303 (t + y -t 0 -D) 2 dy ≤ Q 4 (α 3 γ 1 (t -t 0 -D))e -α3γ1(t-t0-D) .(7.166) where γ 1 (•) and ζ 3 (•) given in(7.10) and(7.131) respectively.Proof. Let B be dened as follows,B = D (t + y -t 0 -D) 2 dy, 3 γ 1 (t + y -t 0 -D) e -α3γ1(t+y-t0-D) dy, (7.168)where ζ 3 is given in(7.129).Next, let us consider the following change of variables:s = α 3 γ 1 (t + y -t 0 -D), = α 3 γ 1,0 T (t 0 + D + T -t -y) ,(7.169)from which we recover,ds dy = α 3 γ 1,0 T (t 0 + D + T -t -y) 2 , = α 3 γ 1,0 T γ 2 (t + y -t 0 -D),using (7.169)-(7.170) in (7.168) (along with dy = α 3 γ 1,0 T ds s 2 ), we obtain B = α 3 γ 1,0 T α3γ1(t-t0) α3γ1(t-t0-D) Q 3 (s) s 2 e -s ds.

(7. 171 )B = α 3 γ 1

 1711 By noticing that the polynomial function Q 3 (•) (of degree p ∈ N * ) can be expressed as Q 3 (s) = p i=0 c i s i with some positive coecients c i > 0, we get, 2 e -s ds.

(7. 172 ) 145 Case 1 :

 1721451 Now, let us calculate each sub-integral of the previous expression. To do that, let us consider the following two cases: 7.6. Prescribed-time stabilization by output feedback for the input delay case Let i ∈ {2, 3, • • • , p} (i.e. i -2 ≥ 0). Then, by using multiple integrations by parts, we obtain,

(7. 174 )Case 2 :

 1742 Note that proving(7.173) is straightforward by induction. Let i ∈ {0, 1} (i.e. i -2 < 0). Then, by using the generalized exponential, dened asE n (r) = r n-1 ∞ r e -s s n ds, r > 0, n ∈ N * ,(7.175)and its property [202, Section 2]:e -r n + r ≤ E n (r) ≤ e -r n -1 + r ,

(7. 184 ) 0 ζ 3 2 ,( 7 . 186 )

 1840327186 Next, by squaring the previous equality and using the fact that ψ(t, •, y) = ẑ(t + y, •) and using(7.130) for t0 := t 0 + D, we obtain,|v(t, y)| 2 ≤ ζ 3 (t + y -t 0 -D) 2 z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0integrating from 0 to D with respect to y and using (7.166) in Proposition 7.3, we obtain,v(t, •) 2 L 2 (0,D) ≤ D (t + y -t 0 -D) 2 dy z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0,1) ≤ ζ 4 (t -t 0 -D) 2 z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0,1)2 .(7.187) where ζ 4 (t -t 0 -D) 2 := Q 4 (α 3 γ 1 (t -t 0 -D))e -α3γ1(t-t0-D) .

(7. 188 )

 188 Finally, by passing to the square roots,v(t, •) L 2 (0,D) ≤ ζ 4 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0,1) .

(7. 190 )

 190 where ζ 5 (•) := ζ 2 (•) + ζ 4 (•), L D := (B D + BD ) > 0, B D := 2e λ(t0+D) e -π 2 t0 + 1 2 √ πt0andBD := B D + sup s∈[t0,t0+D]

ζ 1 (

 1 s -t 0 -D). In particular, I(t) → 0 as t → t 0 + D + T and |U (t)| → 0 as t → t 0 + T . Proof. • Boundedness of the two norms z(t, •) L 2 (1,0) and ẑ(t, •) L 2 (1,0) in [t 0 , t 0 + D]:

(7. 195 )

 195 with BD := B D + sup s∈[t0,t0+D] ζ 1 (s -t 0 -D).

  24):Using(7.128) and (7.182) from Proposition 7.4, we haveI(t) = z(t, •) L 2 (0,1) + ẑ(t, •) L 2 (0,1) + v(t, •) L 2 (0,D) , (7.196) ≤ ζ 2 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0,1) + ζ 4 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0,1) , (7.197) = ζ 5 (t -t 0 -D) z(t 0 + D, •) L 2 (0,1) + ẑ(t 0 + D, •) L 2 (0,1) , (7.198) for all t ∈ [t 0 + D, t 0 + D + T ), where ζ 5 (•) := ζ 2 (•) + ζ 4 (•).

( 7 . 199 )

 7199 Thus, for all (t, y)∈ [t 0 , t 0 + D] × [0, D], we have |v(t, y)| ≤ |U (t + y -D)|.

( 7 . 200 ) 0 K( 1 , 0 K( 1 ,

 72000101 Next, by squaring the previous inequality, integrating with respect to y from 0 to D, and passing to the square roots, we getv(t, •) 2 L 2 (0,D) ≤ D 0 |U (t + y -D)| 2 dy, + y -t 0 -D, ψ(t, •, y))| 2 dy, s, t + y -t 0 -D) ψ(t, s, y) ds •, t + y -t 0 -D) 2 L 2 (0,1) ψ(t, •, y) 2 L 2 (0,1) dy, •, t + y -t 0 -D) 2 L 2 (0,1) ẑ(t + y, •) 2L 2 (0,1) dy.

Figure 7 .

 7 Figure 7.1 shows on the top left the evolution of z(t, x), the state of reaction-diusion PDE (7.20)-(7.22), on the top right the evolution of ẑ(t, x), the observer state of reaction-diusion PDE (7.132)-(7.134), on the bottom left the evolution of v(t, y), the state of the transport PDE (7.23)-(7.24) and on the bottom right the evolution of ω(t, y) the state of the transport PDE (7.53)-(7.54), with the prescribed-time predictor-based output controller U (t) given in(7.161). Figure7.2 shows in a logarithmic scale the evolution of the norm z(t, •) 2 L 2 (0,1) + ẑ(t, •) 2 L 2 (0,1) + v(t, •) 2

2 L 2

 22 Figure 7.1 shows on the top left the evolution of z(t, x), the state of reaction-diusion PDE (7.20)-(7.22), on the top right the evolution of ẑ(t, x), the observer state of reaction-diusion PDE (7.132)-(7.134), on the bottom left the evolution of v(t, y), the state of the transport PDE (7.23)-(7.24) and on the bottom right the evolution of ω(t, y) the state of the transport PDE (7.53)-(7.54), with the prescribed-time predictor-based output controller U (t) given in (7.161). Figure7.2 shows in a logarithmic scale the evolution of the norm z(t,•) 2 L 2 (0,1) + ẑ(t, •) 2 L 2 (0,1) + v(t, •) 2 L 2 (0,D)of the closed-loop system (7.20)-(7.24) with the prescribed-time predictor-based output-feedback controller U (t) given in (7.161) for the initial condition 10z 0 (x) and for 3 dierent delays: D = 0.5s, D = 0.6s, and D = 0.7s. As it can be observed, the norm of the closed-loop system (7.20)-(7.24) converges to the origin in a prescribed time equal to t 0 + D + T no matter what delay we take.

Figure 7 . 1 :

 71 Figure 7.1: On the top left: the evolution of the state z(t, x) of the parabolic PDE (7.20)-(7.22) for the initial condition ẑ(t 0 , x) = 10(x-x 2 ), x ∈ [0, 1]. On the top right: the evolution of the observer state ẑ(t, x) of the parabolic PDE (7.132)-(7.134) for the initial condition ẑ(t 0 , x) = 0, x ∈ [0, 1]. On the bottom left: the evolution of v(t, y) the state of the hyperbolic PDE (7.23)-(7.24) with prescribed-time predictor-based controller U (t) in (7.161) for the initial condition v(t 0 , y) = 0, y ∈ [0, D], with a delay D = 0.5s. On the bottom right: the evolution of ω(t, y) the state of the transport PDE (7.53)-(7.54).

Figure 7 . 3 : 2 L 2

 7322 Figure 7.3: On the top: The evolution of the norm z(t, •) 2 L 2 (0,1) + ẑ(t, •) 2 L 2 (0,1) + v(t, •) 2 L 2 (0,D)of the closed-loop system (7.20)-(7.24) with prescribed-time predictor-based controller U (t) (7.161) in red solid line and with exponential predictor-based controller U exp (t) (7.164) for λ 0 = 0 in black dashed line and λ 0 = 17 in blue dashed line (normal scale on the left and logarithmic scale on the right), for the initial conditions z(t 0 , x) = 10(x -x 2 ), ẑ(t 0 , x) = 0, and v(t 0 , y) = 0, and for a delay D = 0.5s. On the bottom, the evolution of the applied prescribed-time predictor-based controller U (t)(7.161) in red solid line along with the evolution of the exponential predictor-based controller U exp (t) (7.164) for λ 0 = 0 in black dashed line and λ 0 = 17 in blue dashed line.

  2 avec poids en Résumé long 3 tant que fonction de Lyapunov pour construire une commande non linéaire garantissant que le système en boucle fermée est stable non-asymptotiquement. À la n, nous soulignerons les avantages et les limitations de cette approche et donnerons quelques indications sur l'extension de cette méthode au problème de stabilisation entrée-état (ISS) et au problème du suivi en temps ni ou xe pour les EDP de réaction-diusion.

	Partie II
	C4) Dans le chapitre 6, nous présenterons notre quatrième contribution visant à résoudre le
	problème de stabilisation en temps prescrit d'une classe des systèmes linéaires invariants
	dans le temps et soumis à un retard distribué d'entrée :

  1.3. Problems addressed in the thesis, structure and main contributions 11 systems within a PDE backstepping perspective, which reformulates the problem as stabilization of a cascade ODE-PDE system. Indeed, we exploit the backstepping approach (now in a nonlinear, non-smooth, and time-varying context), which brings additional challenges and issues. On the other hand, for reaction-diusion PDEs, the question of boundary nite/xed-time stabilization has remained open in the literature. As we will see (more particularly in Chapter 2), the seminal attempts for boundary stabilization in nite-time use time-varying tools, but using statedependent feedbacks (possibly non-smooth) have not been considered so far in the literature to the best of our knowledge. We attempt not only to tackle this challenging problem using old ideas in relation to Control Lyapunov Functions but also to point out the advantages and limitations of our approach. Finally, we consider the problem of input delay compensation of reaction-diusion systems in prescribed time by output feedback. This problem is challenging, as one deals with observer and control design with time-varying gains. This requires introducing novel innite time-

varying backstepping transformations in conjunction with advanced predictor-based concepts, now for parabolic PDEs. We believe that addressing this problem has paved the way to consider Non-Asymptotic stabilization for more complex dynamics like coupled parabolic-hyperbolic PDEs and other classes of PDE systems.

The thesis is divided into two parts. The rst part concerns non-smooth tools, which allow for obtaining nite/xed-time stabilization and estimation in innite-dimensional settings. The second part of the thesis is devoted to exploiting time-varying tools in innite-dimensional settings.

  ⊂ R n as an open connected set containing the origin, with n ∈ N * , and ∂Ω its boundary. Consider the following ODE system:

	Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings
	2.2 Mathematical background and main tools
	Now that we have a better understanding of asymptotic and nite/xed/predened/prescribed-
	time stability concepts, let us proceed rst to dene mathematically each stability notion for
	ODEs.
	Let us consider the domain Ω
	for prescribed-
	time stabilization of LTI systems with diusive actuator dynamics.

  , [112, Denitions 2.3, 2.4, and 2.5], and[START_REF] Moulay | Finite time stability conditions for non-autonomous continuous systems[END_REF].

	To illustrate these notions, let us give some preliminary examples accompanied by some nu-
	merical simulations,
	Example 2.1 (ES)

  We recall that Ω ⊂ R n is an open connected set containing the origin, with n ∈ N * .

	2.2.1.1 Lyapunov-based characterization of nite/xed/prescribed-time concepts
	Previous Lyapunov results have only addressed asymptotic stability, while it was not until 1995, to
	the best of our knowledge, that the rst Lyapunov-based result contributing to nite-time stability
	(FTS) was introduced by S.P. Bhat and D.S. Bernstein in their paper [73], which was later on
	rened in [76] where in addition an estimate of the settling time was provided. In this subsection,
	we revisit this pivotal result and explore related contributions within the same framework, focusing
	particularly on Lyapunov-based characterization of xed-time, predened-time, and prescribed-
	time stability concepts.
	we recall this important result alongside some other related results that contributed in this
	framework (mainly for xed-time and prescribed-time and predened-time)
	Theorem 2.3 ([76, Theorem 4.2.])

  1 , and α 2 > 1. Similarly to Example 2.8, we can prove that the conditions of Theorem 2.5 are satised for the same choice of the function V

	given in (2.37). As a result, we conclude that, (2.47)-(2.48) is FxTS.
	Next, we present a Lyapunov-based characterization of the predened-time stability (PdTS)
	concept.
	Proposition 2.1 ([97, Theorem 1])

  Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings nearly xed-time stable (nearly FxTS) if and only if it is stable and for any > 0 there exists 0 < T < +∞ such that z (t, •) L 2 (a,b) ≤ for all t ≥ t0 + T and all z 0 ∈ L 2 (a, b); xed-time stable (FxTS) if and only if it is FTS and sup

the settling time of the system (2.97); 2 one can use KL-function (see [178, Denition 2.8.]) Chapter 2.

z0∈L 2 (a,b)

  123) Remark 2.15 Note that, in Example 2.13, (2.107)-(2.108) can be proven to be FxTS either by computing the Chapter 2. Preliminaries of Non-Asymptotic concepts and control design in nite and innite-dimensional settings explicit solution via the characteristic method or using a Lyapunov function (see [160] for more details). However, nding a Lyapunov function satisfying the conditions of Proposition 2.5 is still an open problem.

	Example 2.15 (PTS)

  Positioning of our contributions with respect to results for systems which are of the same class as those addressed in this thesis 43

	K PTS

using a time-varying backstepping approach. This approach proposes to replace the constant reaction term in the target system

(

2.136)-(2.138) by a strictly increasing time-varying reaction term t → γ m (t -t0 ) that goes to innity when t approaches the terminal time t0 + T (see Example 2.15) and to replace the kernels K exp (x, y) and L exp (x, y) by 2.4. 2 (1, y, t -t0 ) and L PTS 2 (1, y, t -t0 ) in the backstepping transformations (2.139) and (2.140). By doing this, the closed-loop system (2.132)-(2.135) with the resulting boundary time-varying feedback

Table 2 .

 2

	Part I
	Non-smooth tools for
	nite/xed-time stabilization and
	estimation in innite-dimensional
	settings

1: Positioning of our contributions with respect to existing Non-Asymptotic stabilization/estimation results and open problems for some classes of dierential equations.

Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Characterisation of the observer-gain functions for second-order LTI systems 51 3.2.2 Characterisation of the observer-gain functions for nth-order LTI systems . 51 3.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.1 An academic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.2 Application to a simplied robot manipulator . . . . . . . . . . . . . . . . . 67

  )[START_REF] Bresch-Pietri | Delay-adaptive predictor feedback for systems with unknown long actuator delay[END_REF] Chapter 3. Finite/xed-time estimation of LTI systems with pointwise input delay

where d is a dilation with the generator

  .103) 58 Chapter 3. Finite/xed-time estimation of LTI systems with pointwise input delay Looking at (3.82) and (3.101), we can clearly see that the solution of the z-dynamics is also bounded for all t ∈ [t 0 , t 0 + D].

  .119) 60 Chapter 3. Finite/xed-time estimation of LTI systems with pointwise input delay Using Jensen's inequality and Young's inequality, we nd

  .121) Using the transformation (3.17) and Lemma 3.1, we have ω being bounded for all t ∈ [t 0 , t 0 + D], then, there exists a positive constant M such that

	sup
	t∈[t0,t0+D]

|ũ(t, 0)| ≤ M . (3.122)

Using this property in the previous inequality, we get

  .[START_REF] Lopez-Ramirez | Finite-time and xedtime observer design: Implicit lyapunov function approach[END_REF] Proof. By the method of the characteristics, the solution of the ω-dynamics in (3.37)-(3.40) for t ≥ t 0 +D is equal to zero, then ω(t, 0) = 0 for all t ≥ t 0 +D. Hence, the target system (3.37)-(3.40) becomes

  .168) 64 Chapter 3. Finite/xed-time estimation of LTI systems with pointwise input delay Integrating this inequality between t 0 + D and t, leads us to

  1 (t) -5z 2 (t) and the nonlinear observer (3.8)-(3.10) with n = 2 and the FxTS observer gain functions (3.41)-(3.41). We take the parameters of the observer gain function L given in (3.41)-(3.42) as follows:

  .161) which is calculated in Matlab using the function fmincon, the state z -ẑ at time t 0 + D: z(t 0 + D) -ẑ(t 0 + D) = [0.9933, -0.3206, 2.0048, 0.2360] . Therefore, the settling-time of each subsystem is numerically estimated using (3.128): T 1,max ( z(t 0 + D) -ẑ(t 0 + D) 2 ) = 3.2996s and T 2,max ( z(t 0

The rest of the parameters are estimated numerically as follows: the eigenvalues of the matrix P in (3.136), P min = 0.2150, P max = 2.2162, the coecient δ = 5.9525 in

[START_REF] Milton | Time delays and the control of biological systems: An overview[END_REF]

  F is a suitable nonlinear function to be characterized later on, and ϕ 1 , . . . , ϕ n are the solutions of

	Chapter 4. Finite/xed-time stabilization of LTI systems with pointwise input delay
	(4.10)

and [69,

Chapter 10]

, we consider the following nonlinear innite-dimensional backstepping transformation: ω(t, x) = u(t, x) -F(ϕ 1 (t, x), . . . , ϕ n (t, x)), where

  R is the transport PDE state. The nonlinear function F, to be specied latter on (see Subsection 4.3.3), is suitably chosen to get FTS/FxTS of the target system when ω(t, 0) becomes zero (this key feature of the transport PDE ω is discussed in Subsection 4.3.4).

	4.3. Finite/xed-time predictor-based controller via PDE-based backstepping approach	83
		(4.23)
		(4.24)
		(4.25)

with ω : [0, +∞) × [0, D] →

  1), β > 1,(4.39) could be an alternative yielding the nite/xed time stability property to the target system and thereby the original one. However, unfortunately, for a cascade nonlinear ODE -transport PDE system (such as (4.22)-(4.25)), it is still unclear whether one can construct such a Lyapunov function (without even the PDE part). See also the open question discussed in Remark 2.15 in Chapter 2. This is one reason why, our approach relies on GKL-class functions β and estimates on the

	solutions.
	4.3.3.2 Finite/xed-time predictor-based controller

Under Assumption 4.1, and from (4.10) at x = D, and using (4.15), the boundary control is then,

  From the transformation (4.26), the following estimate holds for t ≥ t 0 + D: |F(ψ 1 (t, x), . . . , ψ n (t, x))|, ∀t ≥ t 0 .

	4.3. Finite/xed-time predictor-based controller via PDE-based backstepping approach	87
	Proposition 4.3
	√
	u(t, •) L 2 (0,D) ≤	2Dβ 2 ( z(t 0 + D) 2 , t -t 0 -D),	(4.53)
	where β 2 is a class GKL function given in Proposition 4.2.
	Proof. Using (4.26), we have
	|u(t, x)| ≤ |ω(t, x)| + (4.54)
	Next, squaring the previous inequality and using Young inequality, we get,

|u(t, x)| 2 ≤ 2|ω(t, x)| 2 + 2|F(ψ 1 (t, x), . . . , ψ n (t, x))| 2 . (4.55) Now, using ω(t, x) = 0, ∀t ≥ t 0 + D, ∀x ∈ [0, D], we obtain |u(t, x)| 2 ≤ 2|F(ψ 1 (t, x), . . . , ψ n (t, x))| 2 ∀t ≥ t 0 + D, ∀x ∈ [0, D].

  2 is bounded in [t 0 , t 0 + D].

Next, let us prove inequality (4.59). Let t ≥ t 0 + D. Using (4.53) from Proposition 4.3, we have I(t) ≤ z(t) 2 2 + 2Dβ 2 ( z(t 0 + D) 2 , t -t 0 -D) 2 . (4.67) By inequality (4.33) from Assumption 4.1, we get

  From the equation (4.40), we have, |U (t)| is bounded for all t ∈ [t 0 , t 0 + D] (because ψ(t, D) = z(t + D) bounded for all t ≥ t 0 ).

	|U (t)| = |F(ψ 1 (t, D), . . . , ψ n (t, D))|,	(4.72)
	and that By Proposition 4.2, we obtain from inequality (4.49),

  a 11 a 22 ϕ 1 (t, x) + (a 11 + a 22 )ϕ 2 (t, x) + a 21 ϕ 1 (t, x -D 1 ) + u(t, x).

	(4.98)
	(4.99)

with x ∈ [0, D 1 + D 2 ] and u(t, x) the solution of the PDE part of (4.93)-(4.96).

Let us now give numerical simulations for the closed-loop system (4.93)-(4.96) with predictorbased controller U (t) in (4.97). First, using F given in (4.75) to attain FTS where we choose the

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3 Prescribed-time predictor-based control: an ODE-PDE setting and Reduction-based and backstepping-forwarding transformations . . . . . 108

	Chapter 6
	Prescribed-time stabilization of LTI
	systems with distributed input delay
	Contents
	6.1

6.3.1 Scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3.2 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

  + q L 2 (0,D) u(t, •) L 2 (0,D) .

	6.3. Prescribed-time predictor-based control: an ODE-PDE setting and Reduction-based and backstepping-forwarding transformations	113
	and
	|z(t)| 2 ≤ 2 + 4 Γ(t, •) 2 L 2 (0,D) q 2 L 2 (0,D) |z(t)| 2 + 4 q 2 L 2 (0,D) ω(t, •) 2 L 2 (0,D) .	(6.50)
	Proof. On one hand, from (6.7), we have
		|u(t, x)| ≤ |ω(t, x)| + |Γ(t, x)||z(t)|,	(6.51)
	then
		|u(t, x)| 2 ≤ 2|ω(t, x)| 2 + 2|Γ(t, x)| 2 |z(t)| 2 ,	(6.52)
	from which (6.49) is obtained.
	On the other hand, by the Cauchy-Schwarz inequality, we get
		D
		|z(t)| ≤ |z(t)| +	|q(σ)||u(t, σ)|dσ,	(6.53)
		0
		≤ |z(t)| (6.54)
	Using Young's inequality, we obtain
	e -2γ 2,0 T	2,0 T √ γ 2 (t) ≡ 0, ∀t ≥ T (see e.g. [201]). √ γ 2 (t) is a monotonically decreasing smooth bump-like	function having the property

  Prescribed-time stabilization of LTI systems with distributed input delayIt can be shown that F 1 (t -D) and F 2 (t -D) converge to zero in time D + T . Then, I(t) → 0 as t → D + T . Let us study rst F 2 (t -D). We introduce the following change of variables in(6.63)

	114	Chapter 6.
		√
		,0T	γ2(t+x-D) dx.	(6.63)

  2 L 2 (0,D) |z(t)| 2 . Prescribed-time stabilization of LTI systems with distributed input delay Using the fact that ω(t, •) L 2 (0,D) ≡ 0 as t ≥ D, the inequality (6.86) holds for t ∈ [D, D + T ).

	116 Chapter 6. Now, from Proposition 6.3, we obtain
	(6.87)

) 118

 118 Chapter 6. Prescribed-time stabilization of LTI systems with distributed input delay p i-1 dened by (6.102)-(6.103), and V is polynomial-based Vandermonde matrix given as follows:

  , t 0 + D + T ) × [0, D] → R is the transport PDE state, and

	(7.51)
	(7.52)
	(7.53)
	(7.54)
	where ω : [t 0

  For the transport PDE v(t, x) satisfying (7.23), the following estimates holds for t ∈ [t 0 + D, t 0

	7.4. Prescribed-time stabilization by full-state feedback for the input delay case137
	where Q(s) =	γ2,0c l M T 0 8α 3	(s 2 + 2s + 2).
	This concludes the proof.
	Proposition 7.2
			.89)

  . Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay Proof. • Boundedness of the norm z(t, •) L 2 (1,0) in [t 0 , t 0 + D]:

	Using the fact that v(t, 0) = U (t -D) = 0 for all t ∈ [t 0 , t 0 + D], the solution of (7.20)-(7.24)
	is given explicitly from [206, Chapter 3] by
	z(t, x) = 2	+∞	e (λ-n 2 π 2 )t sin(nπx)	1	sin(nπy)z 0 (y)dy.	(7.98)
			n=1			0
	Then, we have
	|z(t, x)| ≤ 2	+∞	e (λ-n 2 π 2 )t | sin(nπx)|	1	| sin(nπy)||z 0 (y)| dy,	(7.99)
		n=1				0
	≤ 2		1	|z 0 (y)| dy e λt	+∞	e -n 2 π 2 t .	(7.100)
		0			n=1
	Next, using the Cauchy-Schwartz inequality, we obtain
						1 √ 2 πt0

. In particular, I(t) → 0 as t → t

0 + D + T and |U (t)| → 0 as t → t 0 + T |z(t, x)| ≤ 2 z 0 L 2 (0,1) e λt +∞ n=1 e -n 2 π 2 t . (7.101)

Now, using the fact that t ∈ (t 0 , t 0 + D] and the fact that +∞ n=1 e -n 2 π 2 t is a convergent series, we obtain |z(t, x)| ≤ 2 z 0 L 2 (0,1) e λ(t0+D) +∞ n=1 e -n 2 π 2 t0 ,

(7.102)

  φy (t, x, y) = φxx (t, x, y) + λ φ(t, x, y) + P (x, t + y -t 0 , D + T )× [ϕ x (t, 1, y) -φx (t, 1, y)] ,

			(7.139)
		φ(t, 0, y) = 0,	(7.140)
		φ(t, 1, y) = v(t, y),	(7.141)
		φ(t, x, 0) = ẑ(t, x).	(7.142)
	with ϕ being generated from (7.28)-(7.31).
	7.6.2 Target System
	       	z (7.143) (7.144)
			(7.145)
	      		(7.146) (7.147)
	and
	
	      
			(7.148)
	      	(7.149) (7.150)

Using

(7.137)

, we transform respectively (7.20)-

(7.24

) and (7.132)-(7.134) into the two following target systems:

t (t, x) = z xx (t, x) + λz(t, x), t ∈ [t 0 , t 0 + D + T ), x ∈ [0, 1], z(t, 0) = 0, t ∈ [t 0 , t 0 + D + T ), z(t, 1) = ω(t, 0) + F(t -t 0 -D, ẑ(t, •)), t ∈ [t 0 , t 0 + D + T ), ω t (t, y) = ω y (t, y), t ∈ [t 0 , t 0 + D + T ), y ∈ [0, D], ω(t, D) = 0, t ∈ [t 0 , t 0 + D + T ),

ẑt (t, x) = ẑxx (t, x) + λẑ(t, x)

+ P (x, t -t 0 , D + T ) × [z x (t, 1) -ẑx (t, 1)] , t ∈ [t 0 , t 0 + D + T ), x ∈ [0, 1], ẑ(t, 0) = 0, t ∈ [t 0 , t 0 + D + T ),

ẑ(t, 1) = ω(t, 0) + F(t -t 0 -D, ẑ(t, •)), t ∈ [t 0 , t 0 + D + T ).

  The expression of the exponential predictor-based output-feedback controller U exp (t) is then given by φ and ψ are respectively generated from (7.139)-(7.142) and (7.153)-(7.156) using the observer gain P exp (x)(7.163). Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay 7.6.5 Stability analysis

	1
	U exp (t) =	K exp (1, s) φ(t, s, D)ds,	(7.164)
	0

or U exp (t) = 1 0 K exp (1, s) ψ(t, s, D)ds,

(7.165)

with

  •, t -t 0 -D) L 2 (0,1) is bounded in [t 0 , t 0 + D], we deduce that v(t, •) L 2 (0,D) is bounded for all t ∈ [t 0 , t 0 + D]. Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay • Convergence of the control to the origin in a prescribed time: From the expressions (7.162) and (7.185), we have for all t ≥ t 0 + D |U (t)| = |F(t -t 0 , ψ(t, •, D))|,

	(7.206)

  150 Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay Figure 7.2: The evolution of the norm z(t, •) 2 L 2 (0,1) + ẑ(t, •) 2 L 2 (0,1) + v(t, •) 2 L 2 (0,D) of the closed-loop system (7.20)-(7.24) with prescribed-time predictor-based controller U (t) (7.161) in solid lines and with exponential predictor-based controller U exp (t) (7.76

	10	0
	10	-2
	10	-4
	10	-6
	10	-8
			0.2	0.4	0.6	0.8	1	1.2	1.4	1.6

Acknowledgements

First and foremost, I would like to express my deepest gratitude to my thesis supervisors Wilrd Perruquetti and Nicolas Espitia. I am truly honored to have been chosen by them to undertake this research journey. I am immensely grateful to them for the countless hours they invested in my growth as a researcher. I would also like to thank them for their valuable insights, feedbacks, for their unwavering interest and enthusiasm in the ideas presented by me, for believing in my potential, for never letting go, and for their constant guidance over these three years. Thank you both for your patience and encouragement; I have learned a lot through your guidance.

I would like to thank Reviewer Pr. Nikolaos Bekiaris-Liberis and Dr. Frederic Mazenc. Thank

Problem statement:

We consider the following reaction-diusion equation with constant reaction term and Dirichlet actuation:

z(t, 0) = 0, t ≥ t 0 , z(t, 1) = U (t), t ≥ t 0 , z(t 0 , x) = z 0 (x),

x ∈ [0, 1],

(

(5.2)

(5.3) (5.4) where t 0 ≥ 0 is the initial time, λ ∈ R is the reaction coecient, z(t, x) ∈ R is the state, U (t) ∈ R is the control, z 0 ∈ L 2 (0, 1) is the initial condition.

The goal of this chapter is to design a nonlinear control U (t) and a Lyapunov functional V (z(t, •)) such that the time derivative of V along the solutions of (5.1)-(5.4) satises (2.115) in Proposition 2.5 for any continuous function K : R + → R + such that K(0) = 0. As an application, we choose the function K (satisfying d dt V (z(t, •)) ≤ -K(V (z(t, •))) in (2.115)) such that the closedloop system (5.1)-(5.4) with the control U (t) is FTS or FxTS in light of the notions presented in Chapter 2.

Stability analysis

In this section, we rst introduce the Lyapunov functional candidate and we give some of its properties. Then, by computing its time derivative along the solutions of (5.1)-(5.4), we design a nonlinear control U (t) that will ensure inequality (2.115) for all t ≥ t 0 and all continuous function K : R + → R + such that K(0) = 0.

Let us consider the following spatially weighted L 2 -norm 1 as a Lyapunov function candidate:

e σx |z(x)| 2 dx, σ > 0.

(5.5)

We can clearly see that V satises for any σ > 0 the following property: z(t, •) 2 L 2 (0,1) ≤ V (z(t, •)) ≤ e σ z(t, •) 2 L 2 (0,1) .

(5.6) Thus V is coercive (see Denition 2.4 in Chapter 2). Moreover, by computing the time derivative of V along the solutions (5.1)-(5.4), we can establish the following proposition Proposition 5.1

Let K : R + → R + be a continuous function such that K(0) = 0. Then, the functional V given in (5.5) satises the following inequality for every t ≥ t 0 ≥ 0 and every σ > 0:

where B(•) is given by

(5.8)

1 similar functionals have been used in the framework of exponential stabilization (e.g. for linear conservation laws in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] or for a transport PDE with a zero input at the boundary in [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]Chapter 11,page 178]).

Stability analysis 97

Proof. Let us start by computing the time derivative of V in (5.5) along the solutions of (5.1)-(5.4),

= 2λ

e σx z(t, x)z xx (t, x)dx.

(5.10)

Next, by integration by parts on the last term, we get,

(5.11)

Then, we get

e σx ∂|z(t, x)| 2 ∂x dx.

(5.13) Now, by a second integration by parts on the last term, we obtain,

(5.17)

Then, using the fact that a ≤ max(0, a) for any a ∈ R, we get,

with B(•) being given in (5.8).

Let us now give the rst main result of our chapter, Theorem 5.1

Let t 0 ≥ 0, σ > 0. Let K : R + → R + be a continuous function such that K(0) = 0. Let B(•) be given as in (5.8). Then, under the following control:

or In this chapter, we consider a 1D reaction-diusion system with boundary input delay and propose a general method for studying the problem of prescribed-time boundary stabilization.

To achieve this objective, we rst reformulate the system as a PDE-PDE cascade system (i.e., a cascade of a linear transport partial dierential equation (PDE) with a linear reaction-diusion PDE), where the transport equation represents the eect of the input delay. We then apply a time-varying innite-dimensional backstepping transformation to convert the cascade system into a prescribed-time stable (PTS) target system. The stability analysis is conducted on the target Chapter 7. Prescribed-time boundary output-feedback stabilization of a class of reaction-diusion with input delay

Observer design

Assume that D = 0. The following observer system was proposed in [START_REF] Steeves | Prescribed-time stabilization of reaction-diusion equation by output feedback[END_REF]: with observer gain P given by

and γ 3 dened in (7.10). The observer state ẑ(t, •) converges to z(t, •) within the prescribed terminal time t0 + T provided that γ 3,0 T >

More precisely, there exist a positive constant α 1 and a positive polynomial function Q 1 (•) in terms of γ 2 (t -t0 ) such that, for any initial conditions z( t0 , •) and ẑ( t0 , •), the following inequality holds for all t ∈ [ t0 , t0 + T ):

where

In particular, z(t, •) -ẑ(t, •) L 2 (0,1) → 0 as t → t0 + T .

Control design

We recall the following time-varying boundary output control:

where the control gain K is as in (7.13) (subject to (7.14)) and ẑ(t, •) is generated from (7.119)- (7.122).

Using the control (7.127), the closed-loop PDE system (7.1)-(7.5) is prescribed-time stable in the following sense: there exist two positive constants α 2 and α 3 and two positive polynomial functions Q 2 (•) and Q 3 (•) dened in terms of γ 1 (t -t0 ) such that for any for any initial conditions z( t0 , •) and ẑ( t0 , •) at initial time t0 , the following inequality holds:

for all t ∈ [ t0 , t0 + T ), where

(7.129) 7.6. Prescribed-time stabilization by output feedback for the input delay case 141

Furthermore, we have

for all t ∈ [ t0 , t0 + T ), with

In particular, ẑ(t, •) L 2 (0,1) → 0, z(t, •) L 2 (0,1) → 0, and |U (t)| → 0 when t → t0 + T .

Prescribed-time stabilization by output feedback for the input delay case

Following the same lines of Section 7.4, let us adapt our approach to design an observed-based control of (7.73) for (7.1)-(7.5) and its PDE-PDE cascade representation (7.20)- (7.24). We propose the following observer for (7.1)-(7.5):

with the observer gain P is given as in (7.123) where we replace T by D + T in the expression of γ 3 (t -t 0 ) to ensure that the convergence of ẑ(t, •) to z(t, •) is achieved in t 0 + D + T instead of t 0 + T . Note that (7.134) can be always expressed using v(t, •) (i.e. ẑ(t, 1) = v(t, 0)) as it was done in (7.22).

Remark 7.7

As our goal is to design an output-feedback control U (t) for (7.20)-(7.24), we do not need to estimate the dynamics of (7.23)-( 7.24) because it is expressed in terms of the control U (t) which, in turn, is in terms of the observed state ẑ(t, •). Consequently, it is clear that, if γ 3,0 satises the following condition:

then, from (7.125), the following inequality holds:

for t ∈ [t 0 , t 0 + D + T ) where z 0 = z(t 0 , •) and ẑ0 = ẑ(t 0 , •). In particular, z(t, •) L 2 (0,1) → ẑ(t, •) L 2 (0,1) as t → t 0 + D + T .

Time-varying innite-dimensional backstepping transformation

As in Subsection 7.4.1, we consider the following time-varying innite-dimensional backstepping transformation:

Chapter 8. Conclusion and perspectives

The nite/xed-time stability property was transferred back to the original system by the inverse transformation and using GKL functions.

In Chapter 5, we treated the problem of nonlinear boundary stabilization, with any predened type of convergence, for a class of 1D reaction-diusion systems. To achieve this, we used the spatially weighted L 2 -norm" as a Lyapunov functional candidate V . By taking the time derivative of this functional along the trajectories of the reaction-diusion PDE d dt V (z(t, •)), we noticed that we got an inequality that relates d dt V (z(t, •)) to a second-degree polynomial involving the control U (t) subtracted by the term K(V (z(t, •))) (for any continuous function K such that K(0) = 0). By computing the root of this polynomial, we managed to design a nonlinear control U (t) and to obtain the inequality d dt V (z(t, •)) ≤ -K(V (z(t, •))). Using this last inequality, for well-chosen expressions of K, we ensured nite/xed-time stabilization of the reaction-diusion PDE. The present chapter did not study the existence/uniqueness issues of the solutions of the closed-loop system. To this purpose, ideas contained in [START_REF] Showalter | Montone Operators in Banach Space and Nonlinear Partial Dierential Equations[END_REF], [START_REF] Miyadera | Nonlinear Semigroups (Translations of mathematical monographs[END_REF] can be used. However, the obtained stability estimates will certainly help the analysis.

In Chapter 6, we extended the existing results of prescribed-time delay-compensation and stabilization of LTI systems with input delay to distributed input delay. The main ideas were developed rst for a scalar LTI system with distributed input delay and after that generalized to the nth-dimensional LTI case. The prescribed-time predictor feedback design was achieved based on the backstepping approach using a time-varying backstepping-forwarding transformation and a reduction-based change of variables.

Finally, in Chapter 7, we extended the results of the previously presented delay compensation techniques from nonlinear ODEs with input delay to linear PDEs with boundary input delay.

Using the developed approach, we tackled the problem of output-feedback stabilization in prescribed time of a linear 1D reaction-diusion PDE with boundary input delay. To apply this approach, the PDE was rewritten into a cascade of the parabolic reaction-diusion PDE with a hyperbolic PDE. Using an invertible time-varying innite-dimensional backstepping transformation coupled with advanced predictor-based concepts, we transformed the PDE-PDE unstable system into a well-chosen PTS target system. Using the inverse transformation we ensured the desired prescribed-time stability property for the original system.

Perspectives

In conclusion, this thesis has made signicant progress in studying the Non-Asymptotic stabilization and estimation problems of some classes of innite-dimensional systems. However, it is important to note that there are still many open problems worth exploring. We list some of the possible directions that can be further studied in the future.

In light of the results obtained in Chapter 4, it is natural to consider extending the results of Chapter 6 from prescribed-time stabilization to nite/xed-time stabilization of LTI systems with distributed delay. Afterward, one can try to combine the state-dependent tools, developed for instance in Chapter 4, with time-varying tools to the challenging problem of the "robustication" of the predictor-based prescribed-time controllers, with respect to external disturbances, for the distributed delay case as well as for the pointwise delay case studied in [START_REF] Espitia | Predictor-feedback prescribed-time stabilization of LTI systems with input delay[END_REF], following the same lines of [START_REF] Orlov | Time space deformation approach to prescribed-time stabilization: Synergy of time-varying and non-lipschitz feedback designs[END_REF].

Appendix

Due to the diculty of constructing Lyapunov functions for nonlinear systems, especially in the framework of Non-Asymptotic stability, alternative techniques such as homogeneity theory can be valuable tools for stability analysis. Homogeneity theory is a technique that allows local properties of a system to be extended globally. It is used to prove stability without the need for a Lyapunov function. It is based on the idea that the behavior of a nonlinear system can be characterized by its degree of homogeneity.

for all z ∈ R n , all real number λ = 0 and some real number k ∈ R, then f is said to be a homogeneous function of degree k.

Example A.1

All linear functions are homogeneous of degree 1.

Example A.2

The scalar function f : R → R dened by

is homogeneous of degree 0.

Example A.3

The function f : R 2 → R dened by

Theorem A.1 ( see [START_REF] Hahn | Stability of Motion[END_REF], [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF])

A.2 Weighted Homogeneity

In the late 1950s, weighted Homogeneity appeared for the rst time in V. I. Zubov's paper [START_REF] Zubov | On systems of ordinary dierential equations with generalized homogenous right-hand sides[END_REF].

In 1986, this concept reappeared in H. Hermes' paper [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], and later on in many other papers including [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to nite-time stability[END_REF]. Weighted homogeneity enlarges the concept of standard homogeneity by allowing the multiplicative factor λ to have dierent powers for each coordinate. Therefore, the dilation is dierent from the one used in the standard case.

Now, let us introduce the following denitions:

Denition A.2

Given some vector of weights r = [r 1 , . . . , rn ] ∈ R n + , the following continuous mapping:

where ρ ≥ rmax := max 1≤j≤n rj (r min := min 1≤j≤n rj ) is called the r-homogeneous norm. When the value of ρ is omitted (i.e. when the following notation z r is used to represent the r-homogeneous

Example A.4

The function f : R 2 → R dened by

Moreover, f is also [1, 2 3 ]-homogeneous of degree 2. This shows that the homogeneity degree and weights are not unique.

A.2. Weighted Homogeneity 159

Denition A.4

A vector eld K : R n → R n is said to be r-homogeneous of degree ν ∈ R if

where the matrix Λ r (λ) = diag [λ r1 , • • • , λ rn ] is called the dilation matrix associated to the vector of weights r.

Remark A.1

Notice that in the previous Denition, saying that the vector eld K : R n → R n is r-homogeneous of degree ν is equivalent to say that the functions K i are r-homogeneous of degree ν + ri , for each i ∈ {1, • • • , n}.

Example A.5

The vector eld K : R 2 → R 2 dened by

is [START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF][START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF]-homogeneous of degree -1 since for i ∈ {1, 2}

Notice that there is an incompatibility between the standard homogeneity and the weighted homogeneity concepts when the weights are all taken to be equal to 1. For instance, by looking at the previous example, we can spot this incompatibility since the vector eld K is homogeneous of degree 0 with respect to the standard homogeneity given in Denition A.1 but as we proved is [START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF][START_REF] Rigatos | Boundary control of the blackscholes pde for option dynamics stabilization[END_REF]-homogeneous of degree -1 with respect to the weighted homogeneity.

Theorem A.3 ([74], [START_REF] Nakamura | Lyapunov functions for homogeneous dierential inclusions[END_REF], [START_REF] Nakamura | Smooth lyapunov functions for homogeneous dierential inclusions[END_REF]) If the system (2.1) is r-homogeneous of degree ν and AS at the origin, then it is 1. globally ES at the origin if ν = 0, 2. globally FTS at the origin if ν < 0, 3. globally nearly FxTS if ν > 0.

Another fundamental result in the study of nite-time stability has been proven by Malkin in 1959 [START_REF] Malkin | Theory of stability of motion[END_REF] and Krasovskii in 1963 [START_REF] Krasovskii | Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay[END_REF] for standard homogeneous systems and it was generalized to weighted homogeneous systems in [START_REF] Bhat | Finite-time stability of homogeneous systems[END_REF] by Bhat and Bernstein.

Theorem A.4 ([74])

Let K 1 , • • • , K p be continuous homogeneous vector elds of degrees k 1 < k 2 < • • • < k p and denote

Assume moreover that K(0) = 0. If the origin is globally AS under K 1 then the origin is locally AS under K. Moreover, if the origin is FTS under K 1 then the origin is FTS under K.

A.3 Linear homogeneity

Homogeneity is a symmetry: scaling time and state leave the dynamical system unchanged. This property can be checked via some algebraic relations to be tested. Moreover, conventional (Euler)

⊂ weighted ⊂ linear ⊂ geometric (coordinate-free) homogeneity (⊂ means "is a subclass of"). Let us recall denitions of linear homogeneity: Let us dene a more general notion of dilation than weighted dilation (see [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] for more details).

Denition A.5 (see [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]) A mapping d

• Limit property: lim s→-∞ d(s)z 2 = 0 and lim s→+∞ d(s)z 2 = +∞ uniformly on the unit sphere. Remark A.3

Note that we can recover:

1. standard dilation when d(s) = e s I; 2. weighted dilation when d(s) = diag[e r1s , . . . , e rns ], r i > 0, ∀i ∈ {1, . . . , n}.

Canonical homogeneous norm plays a central role but it requires the dilation to be monotone. This issue is recalled as follows: Denition A.6 (see [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]) The dilation d is monotone in R n if d(s) 2 < 1, ∀s < 0. Then, the continuous mapping z → z d = e sz where s z ∈ R : d(-s z )z 2 = 1, is called the canonical homogeneous norm. The homogeneous unit sphere is then dened as S d = {z ∈ R n : z d = 1}.

Remark A.4

The canonical homogeneous norm is positive denite, d-homogeneous of degree 1 (see below for the denition of d-homogeneity). Denition A.7 (see [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF])

The dilation d is a linear dilation if and only if d(s

1 all eigenvalues have positive real parts; this is the origin of ż = -G d z is GAS.

A.3. Linear homogeneity 161

For any linear monotone dilation in R n , the canonical homogeneous norm is continuous on R n and locally Lipschitz continuous on R n \{0}. Moreover, it is dierentiable on R n \{0} provided that • is induced by P > 0, P G d + G d P > 0, (see [168]): These concepts simplify the nite/xed-time stability analysis: if the origin of a d-homogeneous system of degree κ ∈ R is globally AS, then if κ < 0 it is also globally FTS, κ = 0 it is globally ES, κ > 0 it is also nearly FxTS (a prerequisite for FxTS). The next denition is adapted from [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF].

Denition A.9

The vector eld K : R n → R n has a (d a , κ a , K a )homogeneous approximation at a ∈ {0, ∞} if and only if there exists a real constant κ a such that Notice that the approximation at 0 is useful for nite-time stability, the approximation at ∞ is useful for nearly xed-time stability, and both approximations are useful for xed-time stability.

From [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF]Theorem 2.20], one can deduce: Lemma A.1 Assume that (2.1) is globally AS.

If K has a (d 0 , κ 0 , K 0 )homogeneous approximation at 0 with K 0 globally AS and κ 0 < 0, then the origin of (2.1) is globally FTS.

In addition, if K has a (d ∞ , κ ∞ , K ∞ )homogeneous approximation at ∞ with K ∞ globally AS and κ ∞ > 0 then the origin of (2.1) is globally FxTS. And then there exists a bi-limit homogeneous Lyapunov function for (2.1).