Keywords: Convex optimization, optimal Control, polynomial systems, switched systems, stability analysis vi Simplicial partition and grid size . . . . .

a dilution rate u = 0.0 . . . . . . . . . . . . . . . . 11 Comparison of the system response computed with nonlinear and CPWL models to the dilution u = 0.031 . . . . . . . . . . . . . . . . 12 Transient response of cells and glucose concentration for u=0.036 for different start-up conditions . . . . . . . . .

With this method, and from a theoretical point of view, we provide an alternative way to search for a common Lyapunov function for switched nonlinear systems.

The main idea behind the proposed approach is to include in the system analysis the hidden constraints. We need to check negative semidefinitness of V with respect to the constrained set. In order to do that, we use the idea of penalization used in optimization theory with constraints. For that, we use a function λ(x, s), which can be interpreted as a penalization function or a Lagrange multiplier. This idea is based on some results for constrained control systems, where we can use the dissipation inequality concept using storage functions and supply rates. We then extend the results to a more general class of switched systems, those modeled by elementary and nested elementary functions. This class of functions is related to explicit symbolic derivatives, such as exponential, logarithm, power-law, trigonometric, and hyperbolic functions. For this aim, we transform, using a recasting process, the system obtained by the equivalent representation in a system with polynomial form, and then we use the results of the previous section for stability analysis.

Besides stability analysis, optimal control problems for switched nonlinear systems are also investigated. We propose an alternative approach for solving effectively the optimal control problem for an autonomous nonlinear switched system based on the v Generalized Maximum Principle (GMP). The essence of this method is the transformation of a nonlinear, non-convex optimal control problem, i.e., the switched system, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computing. Consequently, we propose to convexify the state and control variables by means of the method of moments obtaining SDP programs. A generalization to solve the optimal control problem of nonlinear switched systems based on the recasting process is investigated then.

Finally, we concentrate in the industrial application obtaining a piecewise-linear approximation of nonlinear cellular growth using orthonormal canonical piecewise linear functions, which is tested by a probing control strategy for the feed rate.

We deal with the mammalian cells BHK (Baby Hamster Kidney) in bioreactor in batch, fed-batch, and continuous mode operation. Simulation results show that this piecewise linear approximation is well suited for modeling such nonlinear dynamics.

Introductory Remarks and Motivation

Hybrid systems arise in a wide variety of practical systems. We start pointing out that the term hybrid system can be understood from several point of views. From the technological point of view, systems that contain analog and digital components, systems that comprise part of different physical natures such as biological, chemical, electrical, electronic, hydraulic and mechanical ones, and more generally, settings that involve the use of computers for control proposes are termed hybrid systems. From a mathematical modeling point of view, systems described in different forms, such as algebraic equations, difference equations, ordinary differential equations, logical equations, and partial differential equations, are hybrid systems. In the opening article of the European Journal of Control [START_REF] Blondel | Survey on the State of Systems and Control[END_REF] (1995), hybrid systems take a more prominent position. Hybrid systems are mentioned among the major open problems in systems and control theory by several respondents, including Lennart Ljung, Peter Caines, and Pravin Varaiya. The thrust of the thinking on the subject can be seen from Vidyasagar's remark:

Another interesting question is: 'How can one combine differential/difference equations with logical switches so as to enhance performance?' In some sense, this is the central question of intelligent control.

It seems therefore that by the mid-nineties, hybrid systems have been clearly identified as a major new research area for systems and control theory, and they still constitute a relatively new and very active area of current research (e.g., [START_REF] Antsaklis | A brief introduction to the theory and applications of hybrid systems[END_REF], [START_REF] Morse | Special issue on hybrid control systems[END_REF], [START_REF]Special issue on hybrid systems and applications[END_REF], [START_REF]Special issue on hybrid systems and applications (2)[END_REF], [START_REF]Special issue on hybrid systems and applications (3)[END_REF], [START_REF] Lakshmikantham | Special issue on hybrid systems and applications (4)[END_REF], [START_REF] Zaytoon | Special issue on hybrid systems and applications (5)[END_REF], [START_REF]Special issue on hybrid systems and applications (6)[END_REF], [START_REF]Special issue on hybrid systems and applications (7)[END_REF], [START_REF]Special issue on hybrid systems and applications (8)[END_REF]). In spite of this current interest in hybrid systems, they have 1 been with us at least since the days of the relay. The earliest reference we know of is the work of Witsenhausen from MIT, who formulated a class of hybrid-state continuous-time dynamic systems and investigated an optimal control problem [START_REF] Witsenhausen | A Class of Hybrid-State Continuous-time Dynamic Systems[END_REF].

It is worthwhile mention in that there are various models for hybrid systems; due to its inherently interdisciplinary nature, the field has attracted the attention of people with diverse backgrounds, primarily computer scientists, applied mathematicians, and engineers [START_REF] Liberzon | Switching in Systems and Control[END_REF], [START_REF] Sun | Analysis and synthesis of switched linear control systems[END_REF], [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results[END_REF]. However, we consider continuous-time systems with discrete switching events, which consist of several subsystems and a switching law that determines the switching times and mode transitions. Such systems are called switched systems and can be viewed as higher-level abstraction of hybrid systems [START_REF] Liberzon | Switching in Systems and Control[END_REF]. Switched system modeling of any real-process dealing with physical variables are in agreement with the time-continuous and uniqueness principle, i.e., the value of every physical variable changes only continuously in time through every intermediate value (initial and final), and by possessing a unique value at a specific time and space.

Any synthesized control should be uniquely defined and continuous in time. Recent efforts in switched systems research have been typically focused on the analysis of dynamic behaviors, such as stability, controllability and observability, and optimal control, among others (e.g., [START_REF] Sun | Analysis and synthesis of switched linear control systems[END_REF], [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[END_REF], [START_REF] Liberzon | Switching in Systems and Control[END_REF]).

In this dissertation we deal with three different problems of switched systems:

• stability analysis under arbitrary switching,

• optimal control problem, and

• piecewise linear model and control of a bioreactor.

The first two problems are related by means of an equivalent polynomial representation. The third problem is an industrial application which uses a class of switched system when the switching law is decided by the partition of the state space. We present a brief introduction of each subject of the dissertation with its corresponding chapter.

Contributions, Literature Review, and Outline

For Stability Analysis

We deal with the stability analysis of switched non-linear systems under arbitrary switching. Most of the efforts in switched systems research have been typically focused on the analysis of dynamical behavior with respect to switching signals. Several methods have been proposed for stability analysis (see [START_REF] Liberzon | Switching in Systems and Control[END_REF], [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[END_REF], and references therein), but most of them have been focused on switched linear systems. Stability analysis under arbitrary switching is a fundamental problem for the analysis and design of switched systems. For this problem, it is necessary that all the subsystems are asymptotically stable. However, in general, the above stability condition is not sufficient to guarantee stability for the switched system under arbitrary switching. It is well known that if there exists a common Lyapunov function for all the subsystems, then the stability of the switched system is guaranteed under arbitrary switching. Previous attempts of general constructions of a common Lyapunov function for switched nonlinear systems have been presented in [START_REF] Dayawansa | A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching[END_REF] and [START_REF] Mancilla-Aguilar | A Converse Lyapunov Theorem for Nonlinear Switched Systems[END_REF], using converse Lyapunov theorems. Also in [START_REF] Vu | Common Lyapunov Functions for Families of Commuting Nonlinear Systems[END_REF], a construction of a common Lyapunov function is presented for the case when the individual systems are handled sequentially rather than simultaneously for a family of pairwise commuting systems. These methodologies are presented in a very general framework, and even though they are mathematically sound, they are too restrictive from a computational point of view, mainly because it is usually hard to check for the set of necessary conditions for a common function over all the subsystems (which might not exist). Also, these constructions are usually iterative, which involves running backward in time for all possible switching signals, being prohibitive when the number of modes increase.

The main contribution of Chapter 2 is twofold. First, we present a reformulation of the switched system as a differential continuous system on a constraint manifold.

This representation opens several possibilities of analysis and design of switching systems in a consistent way, and also with numerical efficiency [START_REF] Mojica-Nava | Optimal Control of Switched Systems: A Polynomial Approach[END_REF], [START_REF] Mojica-Nava | Stability Analysis of Switched Polynomial Systems Using Dissipation Inequalities[END_REF], which is possible thanks to some tools developed in the last decade for polynomial differentialalgebraic equations analysis [START_REF] Ebenbauer | Stability Analysis of Constrained Control Systems: An Alternative Approach[END_REF], [START_REF] Papachristodoulou | Positive Polynomials in Control, ch. Analysis of Non-polynomial Systems Using the Sum of Squares Decomposition[END_REF], [START_REF] Kunkel | Stability Properties of Differential-Algebraic Equations and Spin-Stabilized Descretizations[END_REF]. The second contribution is to show an alternative method to search for a common Lyapunov function for switched systems with an efficient numerical method, using results from stability analysis of polynomial systems based on dissipativity theory [START_REF] Ebenbauer | Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares[END_REF], [START_REF] Mojica-Nava | Stability Analysis of Switched Polynomial Systems Using Dissipation Inequalities[END_REF]. We propose a methodology to construct common Lyapunov functions for switched non-linear systems, which provides a less conservative test for proving stability under arbitrary switching. It has been mentioned in [START_REF] Prajna | Analysis of Switched and Hybrid Systems -Beyond Piecewise Quadratic Methods[END_REF] that the sum of squares decomposition, presented only for switched polynomial systems, can sometimes be made for a system with a non-polynomial vector fields. These cases, where possible, are restricted to subsystems, which after the rendering in polynomial forms using auxiliary variables, preserve all the same dimensions. However, to our knowledge this has not been shown in the literature.

The methodology that we propose does not have the dimentionality limitation mentioned above. In a previous work [START_REF] Mojica-Nava | Stability Analysis of Switched Polynomial Systems Using Dissipation Inequalities[END_REF], we have presented the method only for the case when all the subsystems are in a polynomial form. Later, we extend some preliminary results to a more general non-linear case, and a representative example is presented to show the effectiveness of the methodology by reliable and efficient numerical methods. Basically, this theory is based in terms of an inequality involving a generalized system power input, or supply rate, and a generalized energy function, or storage function [START_REF] Willems | Dissipative Dynamical Systems Part I: General Theory[END_REF]. The interpretation of this storage function establishes the connection between Lyapunov stability and dissipativity. Stability problems can be solved once the dissipativity property is assured, and the storage function becomes a Lyapunov function, which can be used to construct Lyapunov functions for nonlinear dynamical systems. As for a common Lyapunov function, a single storage function for all subsystems is usually difficult to find or may not exist (computational problems arise when a common function needs to be found). However, thanks to the computational tools that have been developed lately, we are able to use dissipativity theory with efficient numerical methods to establish a common Lyapunov function for the equivalent polynomial system.

Alternatively, the authors in [START_REF] Liberzon | Switching in Systems and Control[END_REF] propose a Lie algebraic condition for switched LTI systems, which is based on the solvability of the Lie algebra generated by the set of state matrices. The Lie algebraic condition is also extended to switched nonlinear systems to obtain local stability results based on Lyapunov's first method. Most recently global stability properties for switched nonlinear systems are presented in [START_REF] Vu | Common Lyapunov Functions for Families of Commuting Nonlinear Systems[END_REF], [START_REF] Margaliot | Stability analysis of switched systems using variational principles: An introduction[END_REF], [START_REF] Margaliot | Lie-Algebraic Stability Conditions for Nonlinear Switched Systems and Differential Inclusions[END_REF], and a Lie algebraic global stability criterion is derived, based on Lie brackets of the nonlinear vector fields. This sort of analysis based on algebraic conditions and Lie algebra are not considered in this work.

For the Optimal Control Problem

A Brief Literature Review for Optimal Control of Hybrid Systems

The earliest reference we know of optimal control for hybrid systems is the seminal paper of Witsenhausen [106](1966), where an optimal terminal constraint problem was considered on his hybrid systems model. Later in [START_REF] Seidman | Optimal Control for Switching Systems 1[END_REF], an optimal control for switching systems was presented, followed by the influential work [START_REF] Branicky | Algorithms for optimal hybrid control[END_REF], and [START_REF] Branicky | A unified framework for hybrid control: Model and optimal control theory[END_REF] where the authors compared several algorithms for optimal control and discuss general conditions for the existence of optimal control laws. Eventually, necessary optimality conditions for hybrid systems were derived using general versions of the maximum principle [START_REF] Sussmann | A maximum principle for hybrid optimal control problems[END_REF], [START_REF] Riedinger | Linear quadratic optimization for hybrid systems[END_REF], [START_REF] Riedinger | Suboptimal switched controls in context of singular arcs[END_REF] and more recently in [START_REF] Garavello | Hybrid Necessary Principle[END_REF], and in particular for switching systems in [START_REF] Bengea | Optimal control of switching systems[END_REF] and [START_REF] Wei | Applications of numerical optimal control to nonlinear hybrid systems[END_REF], where the switched system was embedded into a larger family of systems and the optimization problem was formulated. In some recent papers, [START_REF] Spinelli | A note on optimal control of autonomous switched systems on a finite time interval[END_REF] and [START_REF] Das | Optimally switched linear systems[END_REF], we can find some work related with embedding approach for the linear case.

However, they do not provide further insights on how to find the optimal switching strategy. For general hybrid systems, with nonlinear dynamics in each location and with autonomous and controlled switchings, necessary optimality conditions were recently presented in [START_REF] Shaikh | On the Hybrid Optimal Control Problem: Theory and Algorithms[END_REF], and using these conditions, algorithms based on the hybrid maximum principle were derived. An approach based on the parameterization of the switching instants and the differentiation of the cost function was presented in [START_REF] Xu | Optimal control of switched systems via nonlinear optimization based on direct differentiations of value functions[END_REF], [START_REF] Xu | Quadratic optimal control problems for hybrid linear autonomous systems with state jumps[END_REF], [START_REF] Xu | Optimal control of switched systems based on parameterization of the switching instants[END_REF]. The algorithm proposed is based on a two-stage optimization problem.

However, the method encounters major computational difficulties when the number of available switches increases.

Lincoln and Rantzer [START_REF] Lincoln | Relaxing dynamic programming[END_REF] presented the method dubbed relaxing dynamic programming to approximate hybrid optimal control laws and to compute lower and upper bounds of the optimal cost, while the case of piecewise-affine systems was discussed by Rantzer and Johansson [START_REF] Rantzer | Piecewise linear quadratic optimal control[END_REF]. For determining the optimal feedback control law, these techniques require the discretization of the state space in order to solve the corresponding Hamilton-Jacobi-Bellman equations that make it intractable numerically.

For discrete-time linear hybrid systems, Bemporad and Morari introduced a hybrid modeling framework that, in particular, handles both internal switches (i.e., caused by the state reaching a particular boundary) and controllable switches [START_REF] Bemporad | Control of systems integrating logic, dynamics, and constraints[END_REF].

The authors also showed how mixed-integer quadratic programming (MIQP) can be used to determine optimal control sequences. On the other hand, it is generally perceived that the best numerical methods available for hybrid optimal control problems involve mixed integer programming (MIP). While great progress has been made in recent years in improving these methods, the MIP is an NP-hard problem, so scalability is problematic [START_REF] Wei | Applications of numerical optimal control to nonlinear hybrid systems[END_REF]. Bemporad and Morari have worked in model predictive control with different problems (e.g., constrained finite time optimal control (CFTOC), constrained infinite time optimal control (CITOC)) [START_REF] Bemporad | The explicit solution of model predictive control via multiparametric quadratic programming[END_REF]. In [START_REF] Morari | Recent developments in the control of constrained hybrid systems[END_REF] Morari and Baric presented the recent developments in control constrained hybrid systems, in which the control paradigm is focused on MPC, with the emphasis on explicit solution. Nonlinear parametric optimization using cylindrical algebraic decomposition is presented in [START_REF] Fotiou | Nonlinear parametric optimization using cylindrical algebraic decomposition[END_REF], [START_REF] Fotiou | Parametric optimization and optimal control using algebraic geometry methods[END_REF].

For cases where online optimization is not viable, Seatzu et al. proposed a multiparametric programming for solving in state-feedback form the infinite-time hybrid optimal control, by showing that the resulting optimal control law is piecewise affine [START_REF] Seatzu | Optimal control of continuous-time switched affine systems[END_REF]. They considered the optimal control of continuous-time switched affine systems with a piecewise-quadratic cost function by two methods: i) the so-called master-slave procedure (MSP), and ii) the switching table procedure (STP). The drawback of all these approaches is that they take a lot of computing time.

Focusing on real-time application, Egerstedt et al. [START_REF] Egerstedt | Optimal control of switching times in switched dynamical systems[END_REF] considered an optimal control problem for switched dynamical systems, where the objective is to minimize a cost functional defined on the state, and where the control variable consists of the switching times. A gradient-descent algorithm is proposed based on an especially simple form of the gradient of the cost functional. In [START_REF] Axelsson | Optimal mode-switching for hybrid systems with unknown initial state[END_REF] and [START_REF] Boccadoro | Beyond the construction of optimal switching surfaces for autonomous hybrid systems[END_REF] the authors deal with the problems of mode-switching with an unknown initial state and the construction of a surface for optimality. Such systems change modes whenever the state intersects certain surfaces that are defined in the state space.

In [START_REF] Li | Control parameterization enhancing transform for optimal control of switched systems[END_REF] a control parameterization enhancing transform is presented with prespecified order of the sequence of subsystems, where the switching instants are included in the cost functional. Both the switching instants and the control function are to be chosen in a way that the cost functional is minimized. In [START_REF] Attia | Sub optimal control of switched nonlinear systems under location and switching constraints[END_REF], [START_REF] Alamir | An efficient algorithm to solve optimal control problems for nonlinear switched hybrid systems[END_REF] an algorithm based on strong variations to handle constraints on both locations and switching instants is proposed for switched nonlinear systems. With the advent of differential inclusion theory, some results using this technique are presented by Vinter in [START_REF] Galbraith | Optimal control of hybrid systems with an infinite set of discrete states[END_REF], in which the continuous subsystems are modeled as differential inclusions. A distinctive feature of the analysis is that it permits an infinite set of discrete states.

On the other hand, the H ∞ control problem for nonlinear switched systems is addressed in [START_REF] Zhao | Hybrid h ∞ control based on multiple lyapunov functions[END_REF] where, based on multiple Lyapunov functions, a sufficient condition for the problem to be solved is derived in terms of partial differential inequalities. The continuous controllers for each subsystem and the switching law are simultaneously designed.

Contributions on Optimal Control of Switched Systems

The main contribution of Chapter 3 is an alternative approach to solve effectively the optimal control problem for an autonomous nonlinear switched system based on the Generalized Maximum Principle (GMP) introduced in [START_REF] Young | Lectures on the Calculus of Variations and Optimal Control Theory[END_REF], and later used in [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF] and [START_REF] Muñoz | A Refinement on Existence Results in Nonconvex Optimal Control[END_REF] to establish existence conditions for an infinite-dimensional linear program over a space of measure. At a first stage, we focus our analysis on vector fields and running costs that are of polynomial form. However, it is well known that functions called nested elementary functions can be recasted exactly in polynomial systems with a larger state dimension. We will therefore use the fact that all system data are polynomial after the recasting process. We will then apply the Theory of Moments, a method previously introduced for global optimization in [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], [START_REF] Lasserre | An explicit equivalent positive semidefinite program for nonlinear 0-1 programs[END_REF], [START_REF] Lasserre | A Semidefinite Programming Approach to the Generalized Problem of Moments[END_REF], [START_REF] Meziat | The method of moments in global optimization[END_REF], and for variational calculus in [START_REF] Meziat | Analysis of Nonconvex Polynomial Programs by the Method of Moments[END_REF] and with some previous results recently introduced for optimal control problems in [START_REF] Lasserre | Nonlinear Optimal Control: Numerical Approximations via Moments and LMI-Relaxations[END_REF][START_REF] Lasserre | A Semidefinite Programming Approach to the Generalized Problem of Moments[END_REF][START_REF] Pedregal | Existence Results for Optimal Control Problems with Some Special Nonlinear Dependence on State and Control[END_REF][START_REF] Meziat | An alternative approach for non-linear optimal control problems based on the method of moments[END_REF][START_REF] Henrion | Nonlinear Optimal Control Synthesis via Occupation Measures[END_REF][START_REF] Lasserre | Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations[END_REF][START_REF] Mojica-Nava | Optimal Control of Switched Systems: A Polynomial Approach[END_REF]. The moment approach for global polynomial optimization based on semidefinite programming (SDP) is consistent, as it simplifies and/or has better convergence properties when solving convex problems [START_REF] Lasserre | Convexity in Semi-algebraic Geometry and Polynomial Optimization[END_REF]. This approach works properly when the control variable (i.e., the switching signal), and the state variables can be expressed as polynomials. The essential of this method is the transformation of a nonlinear, non-convex optimal control problem (i.e., the switched system), into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computing. In other words, we transform a given controllable switched system into a controllable continuous polynomial system with a linear and convex structure. Should we use a nonlinear, non-convex form of the control variable, we would not be able to use the Hamilton equations of the maximum principle, and nonlinear mathematical programming techniques. That would entail severe difficulties, either in the integration of the Hamilton equations or in the search method of any numerical optimization algorithm.

Consequently, we propose to convexify the state and control variables by using the method of moments in the polynomial expression in order to deal with this kind of problems. Finally, we use our previous work, where we have limited our analysis to vector fields and running costs of polynomial form, to extend the result to a more general, nonlinear switched systems by means of the ideas introduced in [START_REF] Savageau | Recasting Nonlinear Differential Equations as S-systems: A Canonical Nonlinear Form[END_REF] that help us to cope with these non-polynomial terms, which are based on the recasting process of a specific kind of non-polynomial functions.

For the Piecewise Linear Model and Control of a Bioreactor

Mammalian cells of Baby Hamster Kidney (BHK) are used in the production of the vaccine against the foot-and-mouth disease. These cells display multiple steady states with widely varying concentrations of cell mass, desired products, and also waste metabolites [START_REF] Namjoshi | Unveiling steady-state multiplicity in hybridoma cultures: The cybernetic approach[END_REF], [START_REF] Mojica | Desarrollo y simulación de un modelo para un cultivo celular en bioreactor[END_REF], [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF]. It means that for identical input conditions to a fed-batch reactor, the outlet conditions change depending on how the culture is made fedbatch. These multiple states are manifestations of the complex interaction between cells and their environment. What make this process difficult is the additional level of complexity present in biological systems because of the genetic information in living cells. Several nonlinear models have been developed for mammalian cells (see [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF] and references therein), but most of them arise in computational problems. Usually, for nonlinear models from the point of view of control design, details about intracellular metabolism are omitted. The models are based on macroscopic mass balance, and include only the more relevant biological reactions. In spite of these attempts to find simple but useful nonlinear models, the modeling of the reaction kinetics are generally represented by rational functions of the state and numerous studies have shown that modeling of the kinetics is a very difficult task [START_REF] Bastin | Nonlinear and Adaptive Control in Biotechnology: A Tutorial[END_REF].

The peculiar features of mammalian cells growth in a fed-batch operating condition are addressed. The task of the controller is to determine, at every instant, the best feed substrate, using the compilation of information online from the sensor. The determination of an optimal strategy of feed substrate using the nonlinear modeling even if the kinetics are known, is not a straightforward matter and is often further complicated by the presence of constraints imposed on the state variables [START_REF] Bastin | Nonlinear and Adaptive Control in Biotechnology: A Tutorial[END_REF]. All of these difficulties in the modeling and control design of a biological process using nonlinear models lead to the search for new and more efficient tools for both modeling and control design. In this context, hybrid systems, i.e., systems including both continuous and discrete dynamics, open several possibilities for both modeling and control design. Chapter 4 is related with a modeling class of hybrid systems, viz., piecewise-linear (PWL) systems. The PWL approximation, i.e., systems which are linear or affine on each of the components of a polyhedral partition of the state space [START_REF] Sontag | Nonlinear regulation: The piecewise linear approach[END_REF], have shown advantages on implementation, performance analysis, and calculations [START_REF] Hassibi | Quadratic Stabilization and Control of Piecewise-Linear Systems[END_REF], [START_REF] Rantzer | Piecewise linear quadratic optimal control[END_REF], [START_REF] Rodrigues | Observer-Based Control of Piecewise-Affine Systems[END_REF], [START_REF] Rodrigues | Automated control design for a piecewise-affine approximation of a class of nonlinear systems[END_REF].

The problem to find a piecewise-linear model given a nonlinear model has been previously treated ( [START_REF] Sontag | Nonlinear regulation: The piecewise linear approach[END_REF], [START_REF] Hassibi | Quadratic Stabilization and Control of Piecewise-Linear Systems[END_REF], [START_REF] Rantzer | Piecewise linear quadratic optimal control[END_REF], [START_REF] Rodrigues | Observer-Based Control of Piecewise-Affine Systems[END_REF], and some others), in specific biological systems [START_REF] Kolker | A Piecewise-Linear Growth Model: Comparison with Competing Forms in Batch Culture[END_REF]. More recently, an approximation for modeling gene-regulatory networks is presented in [START_REF] Azuma | Lebesgue piecewise affine approximation of nonlinear systems and its application to hybrid system modeling of biosystems[END_REF]. However, these approaches present many parameters that need to be provided by the designer, and finding these parameters is a difficult task, even for simple systems. In this work, a canonical piecewise linear approximation over simplicial partitions is used. It provides a partition of the state space into polytopic cells based on value at vertices [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF], [START_REF] Rodrigues | Automated control design for a piecewise-affine approximation of a class of nonlinear systems[END_REF], [START_REF] Girard | Approximative solutions of odes using piecewise linear vector fields[END_REF]. This choice is motivated by several facts.

First, this class of functions uniformly approximate any continuous nonlinear function defined over a compact domain R n (see [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF]). Moreover, the canonical expression introduced in [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF] uses the minimum and exact number of parameters, and it is the first PWL expression able to represent PWL mappings in arbitrary dimensional domains.

As a consequence of this, an efficient characterization is obtained from the viewpoint of memory storage and numerical evaluation [START_REF] Castro | Wiener-like modelling: A different approach[END_REF]. Second, the approximation can be used in real implementations; the points taken from the nonlinear model may be replaced by points taken from sensors or data directly from the process. Thus, it addresses the problem of finding a PWL approximation of system where a reasonable number of measure samples of the vector field is available (regression set) [START_REF] Storace | Piecewise-linear approximation of nonlinear dynamical systems[END_REF]. Third, this alternative approach deals with an approximation which is easier to handle than the nonlinear model. In fact, it might use many tools developed for hybrid systems -e.g., the MLD model based approach [START_REF] Bemporad | Control of systems integrating logic, dynamics, and constraints[END_REF]-since algorithms for translating MLD systems into PWL systems are available [START_REF] Bemporad | Efficient Conversion of Mixed Logical Dynamical Systems into an Equivalent Piecewise Affine Form[END_REF], [START_REF] Villa | A New Algorithm for Translating MLD Systems into PWA Systems[END_REF]. Finally, this CPWL is used in a model based control, termed probing control in [START_REF] Mojica | Probing control for pwl approximation of nonlinear cellular growth[END_REF], being a first step development a hybrid probing control. The task of the controller is to determine, at every instant, the best control action (the best feed substrate) based on the compilation of the sensor's on-line information (or for the nonlinear model). The fact that the probing strategy for feedback control requires a minimum of process knowledge is exploited in [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF]. This work refers to a probing control as it is presented in [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF] for E. coli. Short pulses to the feed rate are added, and taking into account the system response, the pulse is increased or decreased according with the tuning rule. The probing control strategy avoids acetate accumulation while maintaining a high growth rate [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF], [START_REF] Velut | Bioreactor Control Using a Probing Feeding Strategy and Mid-Ranging Control[END_REF].

The main contribution of Chapter 4 is a hybrid dynamical model using orthonormal high-level canonical piecewise linear functions [START_REF] Mojica | Piecewise-linear approximation of nonlinear cellular growth[END_REF], [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF]. The approximation model is tested by a recently presented control methodology, viz., the probing control strategy, which was developed in [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF] for E. coli cultivations. It is implemented in simulations for this mathematical model [START_REF] Mojica | Probing control for pwl approximation of nonlinear cellular growth[END_REF]. The comparative analysis and error approximation between this new biological model and a nonlinear model developed first in [START_REF] Mojica | Desarrollo y simulación de un modelo para un cultivo celular en bioreactor[END_REF], [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF] are shown. This method is satisfactory for implementation purposes of a hybrid probing control [START_REF] Mojica | Probing control for pwl approximation of nonlinear cellular growth[END_REF].

CHAPTER II

A POLYNOMIAL APPROACH FOR STABILITY

ANALYSIS OF SWITCHED SYSTEMS

The stability analysis of switched non-linear systems, i.e., continuous systems with switching signals under arbitrary switching, is treated in this chapter. Stability analysis under arbitrary switching is a fundamental problem into the analysis and design of switched systems. For this problem, it is necessary that all the subsystems are asymptotically stable. However, in general, the above stability condition is not sufficient to guarantee stability for the switched system under arbitrary switching. It is well known that if there exists a common Lyapunov function for all the subsystems, then the stability of the switched system is guaranteed under arbitrary switching.

In this chapter we present a reformulation of the switched system as a differential continuous system on a constraint manifold. This representation opens several possibilities of analysis and design of switching systems in a consistent way, and also with numerical efficiency [START_REF] Mojica-Nava | Optimal Control of Switched Systems: A Polynomial Approach[END_REF], [START_REF] Mojica-Nava | Stability Analysis of Switched Polynomial Systems Using Dissipation Inequalities[END_REF], which are possible thanks to some tools developed in the last decade for polynomial differential-algebraic equations analysis [START_REF] Ebenbauer | Stability Analysis of Constrained Control Systems: An Alternative Approach[END_REF], [START_REF] Papachristodoulou | Positive Polynomials in Control, ch. Analysis of Non-polynomial Systems Using the Sum of Squares Decomposition[END_REF], [START_REF] Kunkel | Stability Properties of Differential-Algebraic Equations and Spin-Stabilized Descretizations[END_REF].

Using this representation we develop an alternative method to search for a common Lyapunov function for switched systems with an efficient numerical method, using results for stability analysis of polynomial based on dissipativity theory [START_REF] Ebenbauer | Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares[END_REF], [START_REF] Mojica-Nava | Stability Analysis of Switched Polynomial Systems Using Dissipation Inequalities[END_REF]. We propose a methodology to construct common Lyapunov functions for switched nonlinear systems, which provides a less conservative test for proving stability under arbitrary switching. In Section 2.4, we extend the preliminary results to a more general, nonlinear case, and a representative example is presented to show the effectiveness of the methodology by reliable and efficient numerical methods. Basically, this theory is expressed in terms of an inequality involving a generalized system power input, or, supply rate, and a generalized energy function, or storage function [START_REF] Willems | Dissipative Dynamical Systems Part I: General Theory[END_REF]. The interpretation of this storage function establishes the connection between Lyapunov stability and dissipativity. Stability problems can be solved once the dissipativity property is assured, and also once the storage function becomes a Lyapunov function that can be used to construct Lyapunov functions for nonlinear dynamical systems.

Definitions and Preliminaries

Basic Concepts

A switched system is a system that consists of several continuous-time systems with discrete switching events. A switched system may be obtained from a hybrid system by neglecting the details of the discrete behavior and instead considering all possible switching patterns. Switched systems have many application, such as power electric circuits, automotive controllers, chemical processes, etc [START_REF] Liberzon | Switching in Systems and Control[END_REF].

The mathematical model can be described by

ẋ(t) = f σ(t) (x, t), ( 1
)
where the state x ∈ R n , f i : R n × R + → R n are vector fields, and

σ(t) : [0, t f ] → Q ∈ {0, 1, ...,

Stability Analysis under Arbitrary Switching and Dissipativity

The stability problem presents several interesting phenomena. For example, even when all the subsystems are exponentially stable, the switched system may be stable (see Figure 1) or may have divergent trajectories for certain switching signals (see Figure 2). Another scenario is also possible: one may carefully switch between unstable subsystems to make the switched system exponentially stable [START_REF] Liberzon | Switching in Systems and Control[END_REF]. We can see from these examples that the stability of switched systems depends not only on the dynamics of each subsystem but also on the properties of switching signals. Therefore, the stability study of switched systems can be roughly divided into two kinds of problems. One is the stability analysis of switched systems under given switching signals (maybe arbitrary, slow switching, etc.); the other is the synthesis of stabilizing switching signals for a given collection of dynamical systems. We are here dealing with the stability analysis of switched systems under arbitrary switching, i.e., the Switching between stable systems that are becoming unstable switched system state goes to zero asymptotically for any switching sequence. If this holds for any initial conditions, we have global uniform asymptotic stability (GUAS) [START_REF] Liberzon | Switching in Systems and Control[END_REF], [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[END_REF]. For this problem, it is necessary that all the subsystems be asymptotically stable. However, in general, the above subsystem stability assumption is not sufficient to assure stability for the switched systems under arbitrary switching, except for some special cases. On the other hand, if there exists a common Lyapunov function for all the subsystems, then the stability of the switched system is guaranteed under arbitrary switching. This provides us with a possible way to solve this problem, and a lot of efforts have been focused on the common quadratic Lyapunov functions [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results[END_REF].

Common Lyapunov functions

We are interested in obtaining a Lyapunov condition for GUAS. We proceed using the classic Lyapunov formulation. Given a positive definite continuously differentiable function V : R n → R, we say that it is a common Lyapunov function for the family of systems (1) if there exists a positive definite continuous function

W : R n → R such that ∂V ∂x f i (x) ≤ -W (x) ∀x, ∀i ∈ Q.
Theorem 1 If all systems in the family (1) share a radially unbounded common Lyapunov function, then the switched system is GUAS.

Theorem 1 is well known and can be derived in the same way as the standard Lyapunov stability theorem [START_REF] Liberzon | Switching in Systems and Control[END_REF]. The main point is that the rate of decrease of V along solutions is not affected by switching; hence asymptotic stability is uniform with respect to σ.

A converse Lyapunov Theorem

The question now arises whether the existence of a common Lyapunov function is a more severe requirement than GUAS. A negative answer to this question -and a justification for the common Lyapunov function approach-follows from the converse Lyapunov theorem for switched systems [START_REF] Mancilla-Aguilar | A Converse Lyapunov Theorem for Nonlinear Switched Systems[END_REF], [START_REF] Dayawansa | A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching[END_REF], [START_REF] Liberzon | Switching in Systems and Control[END_REF] which claims that the GUAS property of a switched system implies the existence of a common Lyapunov function.

Theorem 2 Assume that the switched system (1) is GUAS, the set {f i (x) : i ∈ Q} is bounded for each x, and the function (x, i) → f i (x) is locally Lipschitz in x uniformly over i. Then all systems in the family (1) share a radially unbounded smooth common Lyapunov function.

There is a useful result which we find convenient to state here as a corollary of Theorem 2. It can be shown that if the switched system is GUAS, then all convex combinations of the individuals subsystems from the family (1) must be globally asymptotically stable. These convex combinations are defined by the vector fields

f p,q,α (x) := αf p (x) + (1 -α)f q (x), p,q ∈ Q, α ∈ [0, 1].
Corollary 3 Under the assumption of Theorem 1, for every α ∈ [0, 1] and all p, q ∈ Q, the system

ẋ = f p,q,α (x)
is globally asymptotically stable.

Dissipativity

A switched system expressed as a polynomial differential-algebraic system allows us to establish an alternative approach for stability analysis. But instead of searching for a common Lyapunov function in order to provide stability under arbitrary switching using traditional techniques (e.g., searching a single Lyapunov function whose derivative along solutions of all systems in the family (1)satisfies suitable inequalities), which usually are very restrictive techniques based on exhaustive algorithms, as it is mentioned in the introduction of this chapter; we look for a Lyapunov function using techniques developed for polynomial continuous systems. It means that we can find a common Lyapunov function using dissipativity inequalities as in [START_REF] Ebenbauer | Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares[END_REF], or study the stability of constrained dynamical systems [START_REF] Prajna | On the Construction of Lyapunov Functions using the Sum of Squares Decomposition[END_REF]. With this reformulation, we are dealing with a polynomial differential system on a manifold. Basically, the stability problem of differential-algebraic systems is related to the problem of stability on manifolds, which are defined by the constraints in the system description. From the concept of dissipativity, it could be inferred that storage functions induced by dissipativity are possible Lyapunov functions that are candidate for stability analysis. This implies that stability and stabilization problems can be solved once the dissipativity property is assured [START_REF] Zhao | Passivity and Stability of Switched Systems: A Multiple Storage Function Method[END_REF]. It is possible to show that if the system is expressed as a purely passive system, the origin is an asymptotically unfluctuating equilibrium point, and the storage function V turns into a Lyapunov function. The functionality of stability analysis using dissipativity is that this property is preserved under interconnection [START_REF] Zefran | A Notion of Passivity for Hybrid Systems[END_REF], [START_REF] Zhao | Passivity and Stability of Switched Systems: A Multiple Storage Function Method[END_REF].

An Equivalent Polynomial Representation

A polynomial expression able to mimic the behavior of a switched system is developed using a new variable s, which works as a parameter. The starting point is to rewrite [START_REF]Special issue on hybrid systems and applications[END_REF] as a continuous non-switched control system in its more general case. The approach followed here has had in spirit some counterpart of 0-1 programs (see for instance [START_REF] Lasserre | An explicit equivalent positive semidefinite program for nonlinear 0-1 programs[END_REF]).

First, we define a drift vector field

F(x) : R n → R n F(x) = [f 0 (x) f 1 (x) ... f q (x)], (2)
where f i (x, u), i ∈ Q, is the function for each subsystem of the switched systems given in [START_REF]Special issue on hybrid systems and applications[END_REF]. Let L be the vector of Lagrange polynomial interpolation quotients [START_REF] Burden | Numerical Analysis[END_REF] defined with the new variable s, i.e.,

L(s) = [l 0 (s) l 1 (s) ... l q (s)] T , (3)
where

l k (s) = q i=0 i =k (s -i) (k -i) . ( 4
)
We define the set

Γ = {s ∈ R |Q(s) = 0}, where Q(s) is the constraint polynomial so that Q(s) = q k=0 (s -k), (5)
which is used to constrain s to take only integer values of the original set Q. Notice that this clearly implies that we cannot find a solution if the starting point does not belong to this set. Finally, the solution of this system may be interpreted as an explicit ordinary differential equation (ODE) on the manifold Γ. A related continuous polynomial system of the switched system (1) is constructed in the following theorem.

Theorem 4 Consider a switched system of the form given in (1) with a drift vector field as is given in [START_REF]Special issue on hybrid systems and applications (2)[END_REF]. Then, there exists a unique polynomial state system with a polynomial state equation p(x, s) of degree q in s, with s ∈ Γ as follows:

ẋ = p(x, s) = F(x)L(s) = q k=0 f k (x)l k (s). ( 6
)
This polynomial system is an equivalent polynomial representation of the switched system [START_REF]Special issue on hybrid systems and applications[END_REF].

Proof. Given a set of q + 1 subsystems f 0 (x), f 1 (x), ..., f q (x), using the definition of the interpolation polynomial in the Lagrange form [START_REF] Burden | Numerical Analysis[END_REF], we obtain a linear combination of the Lagrange basis polynomials as follows:

p(x, j) = f j (x), j = 0, 1, ..., q.
We use the Lagrange quotients that have the properties that l k (s) is a polynomial (with degree q + 1), and that

l k (s) = δ ks ≡ ⎧ ⎪ ⎨ ⎪ ⎩ 1, s = k 0, s = k ,
where δ ks is the Dirac Delta function supported in ks. With this property, the function p(x, s) can be defined as a polynomial in s with degree at most q, and

p(x, j) = q k=0 f k (x)l k (j) = f j (x).
There can be only one solution to the interpolation problem, since the difference of two solutions is a polynomial with degree at most q, and q + 1 zeros. This is only possible if the difference is identically zero, so p(x, s) is the unique polynomial interpolating the given set of subsystems. From the numerator in Equation ( 4), we see that l k (s) is a polynomial of order q having zeros in all subsystems except the k-th ones. The denominator is simply the constant that normalizes its value to 1 at k.

Let q + 1 be the finite number of subsystems of the switched system (1), i.e., f 0 (x), . . . , f q (x). Then, the polynomial state equation p(x, s) is unique because the quotients of the Lagrange polynomial interpolation l 0 , ..., l q are unique. Moreover, the solutions of the algebraic equations Q(s) constrain the values of the variable s to be in the set of finite values of the original set Q. Therefore, for any values of the s ∈ Γ, the polynomial p(x, s) is equivalent to the switched system (1).

For instance, the most simple case arises when q = 1. In this case, the system [START_REF]Special issue on hybrid systems and applications (8)[END_REF] has the same form of the convex combination of two subsystems. When q = 2, the polynomial equivalent representation has the form

ẋ = p(x, s) = 2 k=0 f k (x)l k (s) = 1 2 f 0 (x)(s -1)(s -2) + f 1 (x)(s)(2 -s) + 1 2 f 2 (x)(s)(s -1).
Notice that the trajectories of the original switched system (1) correspond to piecewise constant controls taking values in the set σ ∈ {0, 1, ..., q}.

Results in Stability Analysis for Polynomial Constrained Dynamical Systems

In the previous section, the switched systems are expressed as polynomial differentialalgebraic systems or constrained control systems. With this reformulation, we can apply the approach presented recently for constrained polynomial control systems based on dissipation inequalities [START_REF] Ebenbauer | Stability Analysis of Constrained Control Systems: An Alternative Approach[END_REF], which is in spirit similar to the approach presented in [START_REF] Prajna | On the Construction of Lyapunov Functions using the Sum of Squares Decomposition[END_REF] (however the latter does not consider dissipation inequalities in its analysis). We can show that, with some assumptions, both approaches are equivalent from a computational point of view.

The main idea behind the proposed approach is to include in the system analysis the set of constraints, which are represented in this case by the semi-algebraic set Γ.

Note that the semi-algebraic set Γ is equivalent to taking s as a constrained parameter that takes values on the roots of the polynomial Q(s). We need to check negative semidefinitness of V (x) with respect to the constrained set Γ. We use the idea of penalization used in optimization theory with constraints. For that, we use a function λ(x, s), which can be interpreted as a penalization function or a Lagrange multiplier.

This idea is based on some results presented in [START_REF] Ebenbauer | Stability Analysis of Constrained Control Systems: An Alternative Approach[END_REF] for constrained control systems, where we can use the dissipation inequality concept using storage functions and supply rates [START_REF] Willems | Dissipative Dynamical Systems Part I: General Theory[END_REF]. Therefore, a dissipation inequality has the form V (x) ≤ a(x, ẋ, s), where a(•) is an arbitrary scalar-valued function [START_REF] Ebenbauer | Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares[END_REF]. In the classical point of view, V (x) is considered as the stored energy in the control system, and a(•) as the energy rate supplied into the control system [START_REF] Willems | Dissipative Dynamical Systems Part I: General Theory[END_REF]. Note that the stability of general differentialalgebraic systems has only been recently presented as a dissipation inequality [START_REF] Ebenbauer | Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares[END_REF]. In this approach, we take this idea of constrained stability analysis to deal with singular constrained control systems [START_REF] Ebenbauer | Stability Analysis of Constrained Control Systems: An Alternative Approach[END_REF].

The following stability theorem is a particular case of the general result presented in [START_REF] Ebenbauer | Stability Analysis of Constrained Control Systems: An Alternative Approach[END_REF], and it is used to find a common Lyapunov function for the switched system

(1) through the equivalent polynomial representation [START_REF]Special issue on hybrid systems and applications (8)[END_REF].

Theorem 5

The equilibrium point x * = 0 of the equivalent polynomial representation (6) of the switched system ( 1) is stable for any admissible input s(t), if there exist polynomial functions V : R n → R, λ : R n × Γ → R, and a constraint polynomial

Q(s) = 0 such that V (x)
is positive definite in a neighborhood of the origin, and

λ(x, s) ≥ 0 in R n × Γ and the dissipation inequality ∂V ∂x p(x, s) ≤ Q 2 (s)λ(x, s) ( 7 )
is satisfied for some neighborhood of the origin.

Proof. If the dissipation inequality ( 7) is satisfied, then the inequality can be integrated in the interval [0, T )

V (0) -V (T ) ≥ - T 0 Q 2 (s)λ(x, s) dt V (0) -V (T ) ≥ 0.
We have used the fact that Q(s) = 0, and s ∈ Γ. This implies that (∂V /∂x)p(x, s) ≤ 0, for all t ≥ 0, in some neighborhood of the origin. We consider also that Q 2 (s) is positive or zero for all the values of s, in order to check semi-definiteness of V (x), and hence to satisfy the inequality above, we make λ(x, s) positive. It follows from this Lyapunov inequality and the continuity of the trajectories x(t), that V is not increasing and therefore the equilibrium point x = 0 of the system ( 6) is stable. Due to the equivalence presented in Theorem 1, the equilibrium point x * = 0 is also an equilibrium point for the switched system (1) for any admissible s ∈ Γ.

Notice that the Lyapunov function V (x) used in Theorem 2 only depends on the state, i.e., it is a common Lyapunov function under arbitrary switching [START_REF] Liberzon | Switching in Systems and Control[END_REF].

Remark 6 If we are interested in establishing asymptotic stability instead of stability, then ( 7) must be satisfied strictly for all nonzero x in some x-neighborhood, i.e.,

∂V ∂x p(x, s) < Q 2 (s)λ(x, s).
In general, it is very difficult to search for a Lyapunov function V (x) and a function λ(x, s) for practical problems. However, recently established methods based on semidefinite programming and sum of squares decomposition allow us to verify Lyapunov inequalities of the form [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF] very efficiently in the case where Q(s), V (x), and λ(x, s) are assumed to be polynomials [START_REF] Ebenbauer | Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares[END_REF]. Certainly, in our case all of these functions are of polynomial nature. It is impossible to search over all functions V (x), λ(x, s).

In this approach, it is assumed that V (x) and λ(x, s) are polynomials up to certain degrees. Now, we can define the dissipation inequalities for the polynomial representation of the switched system. Since we are studying global uniform asymptotically stable systems (GUAS), it means that we are searching for a common Lyapunov function regardless of the switching sequence. Therefore, if we try to prove global stability of the system (6), the following polynomial inequalities must be satisfied, V (x) > 0, and

∂V ∂x p(x, s) ≤ Q 2 (s)λ(x, s),
for all x ∈ R n and s ∈ R. Note that if V (x) is polynomial and positive definite, it implies that it is radially unbounded. To verify such polynomial inequalities is an NP-hard computational problem [START_REF] Ebenbauer | Analysis and Design of Polynomial Control Systems Using Dissipation Inequalities and Sum of Squares[END_REF]. However, with the help of the sum of squares decomposition, it is possible to verify such polynomial inequalities very efficiently. On the other hand, this problem coincides with the problem of searching for a common Lyapunov function for the vector field

F(x) = [f 0 (x) f 1 (x) ... f q (x)].
For illustration and clarity of exposition, consider the case when q = 1. The dissipation inequality [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF] becomes

∂V ∂x (f 0 (x)(1 -s) + f 1 (x)s) ≤ (s(s -1)) 2 λ(x, s).
Before we state further results, we need to introduce some basic concepts of sum of squares decomposition. A more detailed description can be found in [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF] and references therein.

The Sum of Squares Decomposition

In what follows, we present some basic concepts of the sum of squares decomposition technique to be used in the system analysis. The sum of squares decomposition is a method to check if a polynomial can be decomposed into a sum of squared polynomials.

Definition 7 [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF] For x ∈ R n , a multivariate polynomial p(x) is sum of squares (SOS), if there exist some polynomials r i (x), i = 1, ..., M, such that

p(x) = M i=1 r 2 i (x). (8)
It is clear that p(x) being SOS naturally implies p(x) ≥ 0, for all x ∈ R n . An equivalent characterization of SOS polynomials is given in the following proposition taken from [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF].

Proposition 8 [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF] A polynomial p(x) of degree 2d is an SOS if and only if there exists a positive semidefinite matrix Q and a vector of monomials Z(x) containing monomials in x of degree ≤ d such that

p(x) = Z(x) T QZ(x).
Since we have that p(x, s) is a polynomial vector field, and that we are searching for V (x) that is also a polynomial in x, to solve the testing conditions inequality (7), we can restrict our attention to cases in which the conditions admit SOS decompositions.

The only apparent difficulty is the restriction of V (x) to be positive definite, not just positive semidefinite. To deal with this problem we can use the following proposition taken from [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF].

Proposition 9 [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF] Given a polynomial V (x) of degree 2d, let ϕ(x)

= n i=1 d j=1 i,j x 2j i such that, d j=1 i,j > γ ∀i = 1, ..., n, (9)
with γ a positive number, and i,j ≥ 0 for all i and j. Then the condition that

V (x) -ϕ(x) is SOS (10)
guarantees the positive definiteness of V (x).

Using these ideas, we can rewrite inequality [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF], and a relaxation of Theorem 2 is stated in the following proposition.

Proposition 10 For the equivalent polynomial representation system [START_REF]Special issue on hybrid systems and applications (8)[END_REF], if there exist polynomial functions V (x), λ(x, s), and a positive definite function ϕ(x) of the form given in Proposition 6 such that

V (x) -ϕ(x) is SOS -∂V ∂x p(x, s) + Q 2 (s)λ(x, s) is SOS, ( 11
)
then the polynomials V (x), λ(x, s), and the positive definite function ϕ(x) can be computed using SOSTOOLS [START_REF] Papachristodoulou | Introducing SOS-TOOLS: A General Purpose Sum of Squares Programming Solver[END_REF].

The proof follows the same reasoning as in [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF]. Therefore, Proposition 7 shows that with the polynomial equivalent representation in (5), we can obtain a common Lyapunov function using numerical tools. This Lyapunov function will be used to prove stability of the switched system (1).

Remark 11 Note that in Equation ( 11) the polynomials are sum of squares in terms of x and s.

Numerical Example of a Polynomial Switched System

We present an illustrative example of a switched nonlinear system reformulated by Theorem 1 as an ordinary differential equation on a manifold. With this example we illustrate an efficient computational treatment to study stability analysis of switched systems using Theorem 2. Consider the set of systems described by the drift vector field

F(x) = [f 0 (x) f 1 (x)],
with

f 0 (x) = ⎡ ⎢ ⎣ -β 0 x 1 + x 2 1 + x 2 2 -α 0 x 3 1 -β0x 2 + 2x 1 x 2 -α 0 x 3 2 ⎤ ⎥ ⎦ ,
and

f 1 (x) = ⎡ ⎢ ⎣ -β 1 x 1 + x 2 1 + x 2 2 -α 1 x 3 1 -β1x 2 + 2x 1 x 2 -α 1 x 3 2 ⎤ ⎥ ⎦ ,
This system is considered as a homogeneous switched system presented for stability analysis in [START_REF] Li | Stability of Switched Polynomial Systems[END_REF]. In order to prove stability under arbitrary switching, we use the polynomial equivalent representation obtained using Theorem 1

ẋ(t) = ⎛ ⎜ ⎝ f 0 (x) + ( βx 1 + ᾱx 3 1 )s f 0 (x) + ( βx 2 + ᾱx 3 2 )s ⎞ ⎟ ⎠ , s ∈ Γ = {s ∈ R |Q(s) = s(s -1) = 0 }, with β = (β 1 -β0) and ᾱ = (α 1 -α 0 ).
In [START_REF] Li | Stability of Switched Polynomial Systems[END_REF] it is shown that β i > 2 and α i > 4

for i = 1, 2 in order to obtain a set of stable subsystems. We then set β 0 = 10 and α 0 = 5 and, to keep β > 2 and ᾱ > 4, we set β 1 = 13 and α 1 = 10. We have obtained a representation of the original system with a polynomial form, so that we can use Proposition 10 to analyze stability. First, we search for a Lyapunov function of the polynomial form V (x) = i,j a i,j x i x j . We have tried a function of degree 2 and 4, the latter corresponding to the function that we are looking for. With a degree of 2d = 4, and n = 2, we use a function ϕ(x)

= 11 x 2 1 + 12 x 4 1 + • • • + 22 x 4 2
, where the ij are the unknowns to be found by the tool, with a γ = 0.1, which implies ij ≥ γ.

For the penalty function, we have assumed a polynomial function of the same degree of the Lyapunov candidate function V (x), but considering also the s variable, i.e.,

λ(x, s) = b 11 x 2 1 + b 12 x 4 1 + b 21 x 2 2 + • • • + b 31 s 2 + b 32 s 4 .
The coefficients b ij are again the unknown variables to be found. Using these polynomials and Equation [START_REF] Attia | Sub optimal control of switched nonlinear systems under location and switching constraints[END_REF], we obtain, using the MATLAB toolbox SOSTOOLS [START_REF] Papachristodoulou | Introducing SOS-TOOLS: A General Purpose Sum of Squares Programming Solver[END_REF], a Lyapunov function of fourth degree, i.e.,

V (x) = 0.3x 2 1 + 0.3944x 2 2 + 0.11 • 10 -3 x 4 1 + 0.11 • 10 -3 x 4 2 + 0.7 • 10 -3 x 1 x 2
, which, through Theorem 2, proves that (0, 0) of the homogeneous nonlinear switched system, reformulated as a polynomial DAE [START_REF]Special issue on hybrid systems and applications (8)[END_REF], is a stable equilibrium point. Note that there is not a specific procedure to set the value of the degree of V (x) and the minimum value for γ (it should be noticed also that these functions are not unique). These parameters are chosen through different attempts. We start trying with a degree d = 1 and a small value of γ, and then we increase the degree until the properties of the Proposition 7 are met, and hence, we obtain a Lyapunov function. It may be interesting to develop an automatic pre-treatment algorithm to choose these values.

A Generalization for Nonlinear Switched Systems

In the previous sections we have focused our attention on switched systems of polynomial form, i.e., each subsystem is modeled by a polynomial system. In this section we will extend the results to a more general class of switched systems, those modeled by elementary and nested elementary functions. This class of functions is related with explicit symbolic derivatives, such as exponential, logarithm, power-law, trigonometric, and hyperbolic functions. For this aim, we transform, using a recasting process, the system obtained by the equivalent representation in a system with polynomial form, and then we use the results of Section 4 for stability analysis.

The Recasting Process for Stability Analysis

We use a recasting process introduced in [START_REF] Savageau | Recasting Nonlinear Differential Equations as S-systems: A Canonical Nonlinear Form[END_REF], and later used for stability analysis of nonlinear systems in [START_REF] Papachristodoulou | Positive Polynomials in Control, ch. Analysis of Non-polynomial Systems Using the Sum of Squares Decomposition[END_REF]. It is a procedure that goes through several steps until the system has the expected form. The algorithm is as follows:

• Step 0. Equivalent Representation: We consider the equivalent representation for the switched system obtained in Theorem 1, and we name it as the original system (i.e., before the recasting process), with ξ = (ξ 1 , . . . , ξ n ) as the state of the original system.

• Step 1. Original State Equations: The original system is described by

ξi = j a j k p ijk (ξ, s), i = 1, . . . , n. ( 12
)
Here a j s are real numbers, and the factors p ijk are elementary functions, or nested elementary functions of elementary functions.

• Step 2. Decomposition of Non-Polynomial Functions: Let x i = ξ i , for i = 1, . . . , n. For each p ijk (ξ, s) in Equation ( 12) that is not already a power-law function, replace it with a new variable x n+1 . This variable simplifies the differential equation to sums and products of power-law functions. An additional differential equation is generated for each new variable, using the chain rule of differentiation.

• Step 3. Recasting Process: When the recasting process leads to some constraints in the new variables, we have to introduce an n-dimensional manifold on which the solutions to the original differential equation lie. The particular choice of initial conditions defines the reference manifold.

• Step 4. The Polynomial Form: If the set of equations is in polynomial form, then the recasting process is complete. If not, repeat steps 2-3 until to obtain a system of equations with a rational or polynomial form.

Remark 12 Notice that the constraints introduced by the definition of new variables,

and their initial conditions, restrict the system behavior to a manifold of the same dimension of the original problem.

As a result of the recasting process we have obtained new variables, which are considered. Suppose that for a switched system consisting of subsystems of non-polynomial form, we apply the equivalent representation and obtain a system, ξ = p(ξ, s).

The recasted system obtained using the procedure presented above is written as

ẋo = p o (x o , x r , s), ẋr = p r (x o , x r , s), ( 13
)
where

x o = (x 1 , ..., x n ) = ξ
Γ r = {(x o , x r , s) ∈ R n+m+1 | g k (x o , x r , s) = 0, for all k = 1, ..., m} (14)
be the set of constraints from the recasting process. The following proposition is an extension of the stability Theorem 2, and can be used to prove that the origin of a nonlinear switched system is a stable equilibrium point.

Proposition 13

The equilibrium point x * = 0 of the equivalent polynomial representation obtained after the recasting process of the nonlinear switched system ( 1) is stable for any admissible input s(t) if there exist polynomial functions V : R n+m → R, λ o : R n+m × Γ → R, λ r : R n+m × Γ r → R, and constraint polynomials Q(s) = 0, and

g k (x o , x r , s) = 0, with k = 1, ..., m, such that V (x o , x r
) is positive definite in a neighborhood of the origin, and

λ o (x o , x r , s) ≥ 0 in R n+m ×Γ, λ k (x o , x r , s) ≥ 0 in R n+m ×Γ r , k = 1, ..

., m and the dissipation inequality

∂V ∂xo p o (x o , x r , s) + ∂V ∂xr p r (x o , x r , s) ≤ Q 2 (s)λ o (x o , x r , s) + m k=1 g k (x o , x r , s)λ k (x o , x r , s) ( 15
)
is satisfied for some neighborhood of the origin.

The above proposition establishes non-negativity conditions, which can be relaxed to appropriate sum of squares conditions (see Proposition 7), so that we can use the methods based on semidefinite programming and sum of squares decomposition to verify Lyapunov inequalities efficiently. We extend the Proposition 7 as follows.

Proposition 14 For the equivalent polynomial representation system [START_REF] Azuma | Lebesgue piecewise affine approximation of nonlinear systems and its application to hybrid system modeling of biosystems[END_REF], if there

exist polynomial functions V (x o , x r ), λ o (x o , x r , s), λ k (x o , x r , s), for k = 1, ...,
V (x o , x r ) -ϕ(x o , x r ) is SOS -( ∂V ∂xo p o (x o , x r , s) + ∂V ∂xr p r (x o , x r , s)) + Q 2 (s)λ o (x o , x r , s) + m k=1 g k (x o , x r , s)λ k (x o , x r , s) is SOS, ( 16
)
then the polynomials V (x o , x r ), λ o (x o , x r , s), λ k (x o , x r , s), for k = 1, .
.., m, and the positive definite function ϕ(x o , x r ) can be computed using SOSTOOLS [START_REF] Papachristodoulou | Introducing SOS-TOOLS: A General Purpose Sum of Squares Programming Solver[END_REF].

Example of a Non-Polynomial Switched System

In this example we are dealing with a two-dimensional model of a pendulum, where the acceleration of its pivot is assumed to be the control input. Swinging up and stabilization of the pendulum is usually solved by switching between different laws.

We use a damping-pumping strategy as it is proposed in [START_REF] Åström | A Family of Smooth Controllers for Swinging Up a Pendulum[END_REF]. The normalized model of the pendulum is given by

ẋ1 = x 2 ẋ2 = sin x 1 -(2 sin x 1 + F x 2 cos x 1 ) cos x 1 , (17)
where x 1 is the angular position with respect to the origin at the upright position, and x 2 is the velocity. Considering stabilization conditions, it is shown that we can set F as a gain of -1 in some regions, and a gain of 1 in some others, so that the system minimizes the energy consumption all the time. Therefore, we can obtain a switched system depending of the gain F . For F = -1 we set f 0 (x) and for F = 1 we set f 1 (x).

We then use Theorem 1 to obtain an equivalent continuous representation of the switched model related to [START_REF] Bemporad | The explicit solution of model predictive control via multiparametric quadratic programming[END_REF] and we begin the recasting process, ξ(t) = f 0 (ξ)(1 -

s) + f 1 (ξ)s with s ∈ Ω 1 = {s ∈ R |Q(s) = s(s -1) = 0 }. We obtain the following equivalent continuous system, ξ1 = ξ 2 ξ2 = sin ξ 1 -2 sin ξ 1 cos ξ 1 + (1 -2s)ξ 2 cos 2 ξ 1 . ( 18
)
Now, following the recasting process, it is clear that ( 18) is in the same form as [START_REF] Axelsson | Optimal mode-switching for hybrid systems with unknown initial state[END_REF], but in this case the elementary functions are trigonometric functions. Let us follow Step 2 to Step 4 in the recasting process. As a result, we obtain a new set of differential equations given by

ẋ1 = x 2 ẋ2 = x 3 -2x 3 x 4 + (1 -2s)x 2 x 2 4 ẋ3 = x 2 x 4 ẋ4 = -x 2 x 3 . ( 19
)
As we know by Step 3 in the recasting process, when we introduce the new variables

x 3 and x 4 , a set of constraints arise. For this case, we have that the manifold on which the solutions to the original system (18) lie is given by Ω 2 = {x 2 3 + x 2 4 -1 = 0}.

The resulting system is in a polynomial form so that Proposition 9 can be used to prove stability. Due to the form of the original system, we expect that the Lyapunov function has some trigonometric terms. We are searching for a Lyapunov function

of the form V (x) = a 1 x 2 1 + a 2 x 2 2 + a 3 x 2 3 + a 4 x 2 4
+ a 5 . These coefficients must satisfy a 4 + a 5 = 0 for V (x) to be equal to zero at (0, 0). Which is equivalent to

V (ξ) = a 1 ξ 2 1 + a 2 ξ 2 2 + a 3 sin 2 (ξ 1 ) + a 4 sin 2 (ξ 1 -1)
in the original variables. The problem now is to search for those a i s coefficients. In order to guarantee that V (x) is positive definite, the polynomial function ϕ(x o , x r ) is chosen as

ϕ(x o , x r ) = 1 x1 2 + 2 x 2 2 + 3 x 2 3 + 4 (1 -x 3 )
where s are positive constants. In this particular case, we set all i ≥ 1 for i = 1, ..., 4.

In the same way, we define the functions λ o and λ 1 to be monomials of degree two.

CHAPTER III ON OPTIMAL CONTROL OF SWITCHED SYSTEMS USING A POLYNOMIAL APPROACH

We propose an alternative approach for solving effectively the optimal control problem for an autonomous, nonlinear switched system. We are considering a set of several continuous-time subsystems with a discrete switching law. The switching law consists of the switching times and mode transitions. The essence of this method is the transformation of a nonlinear, non-convex optimal control problem into an equivalent optimal control problem with linear and convex structure, a formulation more appropriate to be solved by high performance numerical computing. Therefore, using the Generalized Maximum Principle (GMP), we propose to convexify the state and the control variables, using the method of moments in the polynomial expression to deal with this problem.

At a first stage, we focus our analysis on vector fields and running costs that are of polynomial form. However, it is well known that the functions so-called nested elementary functions can be recasted exactly in a polynomial systems with a larger state dimension [START_REF] Savageau | Recasting Nonlinear Differential Equations as S-systems: A Canonical Nonlinear Form[END_REF]. Therefore, we use the fact that all system data are polynomial after the recasting process, to apply the theory of moments as it has been mentioned above.

Definitions and Preliminaries

Switched Systems and Its Optimal Control Problem

The switched system adopted in this chapter has a general mathematical model described by

ẋ(t) = f σ(t) (t, x(t), u(t)), (20)
where x(t) is the state,

f i : R + × R n × R m → R n are vector fields, x(t 0 ) = x 0 , are fixed initial values, u(t) ∈ U ⊂ R m
is the exogenous input constrained to the convex and compact set U, and

σ : [t 0 , t f ] → Q ∈ {0, 1, 2, ..., q}
is a piecewise constant function of time, with t 0 and t f as the initial and final times respectively. Every mode of operation corresponds to a specific subsystem ẋ(t) = f i (t, x(t), u(t)), for some i ∈ Q, and the switching signal σ determines which subsystem is followed at each point of time, into the interval [t 0 , t f ]. The control inputs, σ and u, are both measurable functions. In addition, we consider a non-Zeno behavior, i.e., we exclude an infinite switching accumulation points in time. Finally, we assume that the state does not have jump discontinuities. Moreover, for the interval [t 0 , t f ], the control functions must be chosen such that the initial and final conditions are satisfied.

Definition 15 A control for the switched system in ( 20) is a triplet consisting of (a) a finite sequence of modes, (b) a finite sequence of switching times such that t 0 < t 1 < • • • < t q = t f , (c) a sequence of exogenous control inputs, each control input function being associated with a mode.

Let us define the optimization functional in bolza form to be minimized as

J = ϕ(x(t f )) + t f t 0 L σ(t) (t, x(t), u(t))dt, ( 21
)
where ϕ(x(t f )) is a real-valued function, and the running switched costs L σ(t) :

R + × R n × R m → R are continuously differentiable, for each σ ∈ Q.

Switched Optimal Control Problem (SOCP)

We want to solve the switched optimal control problem, which can be state in a general form as follows:

Definition 16 Given the switched system in [START_REF] Berg | Moment Problems and Polynomial Approximation[END_REF] and a Bolza cost functional J as in [START_REF] Blondel | Survey on the State of Systems and Control[END_REF], the switched optimal control problem (SOCP) is given by min σ(t),u(t)

J(t 0 , t f , x(t 0 ), x(t f ), x(t), σ(t), u(t)) ( 22
)
subject to the states x(•) satisfies Equation [START_REF] Berg | Moment Problems and Polynomial Approximation[END_REF].

The SOCP can have the usual variations of fixed or free initial or terminal state, free terminal time, etc. In [START_REF] Garavello | Hybrid Necessary Principle[END_REF] it is noted that, in this setting, it is not appropriate first to choose a sequence of controls and then determine the trajectory associated to it, because the sequence could not be admissible a priori (in the sense that there could exist no trajectory corresponding to it). This is due to the fact that in every mode i ∈ Q, it is possible to use only a subset of U, depending on the switching strategy.

Maximum Principle and Necessary Conditions

The Maximum Principle gives a necessary condition for a trajectory x(•) to be a solution of the switched optimal control problem (SOCP). The set of variations involves trajectories having the same history of the candidate's optimal one, which is having the same switching strategy (see [START_REF] Sussmann | A maximum principle for hybrid optimal control problems[END_REF], [START_REF] Garavello | Hybrid Necessary Principle[END_REF]). Variations of the classical Maximum Principle for general hybrid systems have been presented previously in ( [START_REF] Branicky | A unified framework for hybrid control: Model and optimal control theory[END_REF], [START_REF] Sussmann | A maximum principle for hybrid optimal control problems[END_REF],

[109], [START_REF] Riedinger | An Optimal Control Approach for Hybrid Systems[END_REF], [START_REF] Garavello | Hybrid Necessary Principle[END_REF], [START_REF] Bengea | Optimal control of switching systems[END_REF], [START_REF] Shaikh | On the Hybrid Optimal Control Problem: Theory and Algorithms[END_REF]). In particular in [START_REF] Sussmann | A maximum principle for hybrid optimal control problems[END_REF], the principle is presented as an abstract mathematical statement that can be rendered specific in various ways, giving rise to different versions of it. We will state in the next section a specific version concerning our approach. However, we need to introduce first a brief summary of the Maximum Principle for continuous systems in its basic form.

The Maximum Principle

Consider a dynamical system defined by a differential equation on time the interval

[t 0 , t f ], i.e., ẋ(t) = f (t, x(t), u(t)), ( 23
)
x(t 0 ) = x 0 ,
where x(t) ∈ R n , the control input u(t) ∈ U ⊂ R m , and the vector field

f : R × R n × R m → R n .
Then, the problem can be stated as follows: choose the initial conditions

x 0 and the control input u(t) so that the functional

J = ϕ(t f , x(t f )) + t f t 0 L(t, x(t), u(t))dt ( 24
)
is minimized. In order to solve the problem, the following assumptions are made:

(A 1 )
The control domain U is bounded.

(A 2 ) The vector fields f (t, x, u) and L(t, x, u) are continuous and continuously differentiable functions with respect to the state variables and the time variable.

(A 3 ) ϕ is continuous and continuously differentiable.

Define the Hamiltonian function as

H(t, λ(t), x(t), u(t)) = L((t, x(t), u(t)) + λ T f (t, x(t), u(t)) ( 25
)
and the Hamiltonian system as

ẋ = ∂H ∂λ (t, λ(t), x(t), u(t)) λ = ∂H ∂x (t, λ(t), x(t), u(t)), λ(t f ) = -∂ x φ(t f , x(t f )), (26)
Theorem 17 (Necessary Conditions) [START_REF] Pedregal | Introduction to Optimization[END_REF] If (x * (t), u * (t)) is the pair of the corresponding trajectory for the control problem and an admissible optimal control with the assumptions (A 1 ) -(A 3 ), then there must exist a function λ(t) on [t 0 , t f ], such that λ * (x * (t), u * (t)) satisfies the Hamiltonian system ( 25)-( 26) almost everywhere as well as the following maximum condition:

H(t, λ * (t), x * (t), u * (t)) = sup u∈U H(t, λ * (t), x * (t), u(t)).
The question now is whether necessary conditions of optimality are sufficient. In general, the maximum principle is not a sufficient condition for global optimality of (x * , u * ). Nevertheless, the maximum principle becomes a sufficient condition for optimality of (x * , u * ) under some additional assumptions about the control system [START_REF] Branicky | A unified framework for hybrid control: Model and optimal control theory[END_REF] and the functional [START_REF] Branicky | Algorithms for optimal hybrid control[END_REF].

Theorem 18 (Sufficiency of Optimality Conditions) [START_REF] Pedregal | Introduction to Optimization[END_REF] Assume that L(t, x, u) and the subset U are convex in (x, u), and that f (t, x, u) is linear in (x, u).

If the triplet (λ, x, u) satisfies the Hamiltonian system [START_REF] Burden | Numerical Analysis[END_REF] and the maximum condition, then the pair (x * , u * ) is a global optimal solution of the corresponding optimal control problem.

Quite often, optimal solutions for control problems cannot be found, either because there are too many (or not so many) variables involved, so that it is almost impossible to handle them all by hand, or else because optimality conditions cannot be solved explicitly or it is really cumbersome and tedious to find explicit formulas [START_REF] Pedregal | Introduction to Optimization[END_REF]. In this context, efficient computational tools appear as an important alternative to solve nonlinear optimal control problems.

Relaxation and Young Measures

We present the main results concerned with the analysis of optimal control problems governed by ordinary differential equations by means of Young measures. We show a rather general existence theorem for generalized solutions of control problems in the form of Young measures, and first order necessary conditions of optimality that this generalized solutions must verify. These conditions come in the form of a generalized Maximum Principle, which, after all, imposes constraints on the support of optimal Young measures. We describe the relaxation of optimal control problems, with some hypotheses concerning the different ingredients of the problem, so that, regardless of convexity assumptions, existence of optimal solutions can be achieved. Nonexistence (under the coercivity assumption) is always related to an oscillatory behavior which in form is induced by a lack of convexity. A classical relaxation theorem establishes, under some technical assumptions, that the infimum of any functional does not change when we replace the integrand by its convexification. The relaxed version of the problem we are using is formulated in terms of Young measures associated with sequences of admissible controls [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF].

Consider minimizing the functional defined previously in Equation ( 24) with initial time t 0 = 0, and final time fixed t f = T . Where (a) the control is assumed measurable and takes values in a given closed set U (not assumed bounded for the moment);

(b) the state of the system governed by the equation of state ( 23) is assumed to be measurable on the variable t, continuous in (x, u), continuously differentiable with respect to x, and satisfies a uniform Lipschitz condition with respect to x

|f (t, x 1 , u) -f (t, x 2 , u)| ≤ ε|x 1 -x 2 |, ε > 0,
so that problem [START_REF] Branicky | A unified framework for hybrid control: Model and optimal control theory[END_REF] always has a unique solution;

(c) the running cost L is assumed to be continuous on the pair (x, u), differentiable with respect to x, measurable on t, also to satisfy the coercivity requirement

c(|u| p -1) ≤ L(t, x, u), p > 1, c > 0; and (27)
(d) the function ϕ is assumed to be continuous and differentiable.

Let us further set

h(x, u) = sup{|f (t, x, u)| : 0 < t < T } and L U (t, x, u) = ⎧ ⎪ ⎨ ⎪ ⎩ L(t, x, u), u ∈ U +∞, else.
Notice that the function h is continuous with respect to x, even Lipschitz continuous.

We postulate as well the behavior

lim |u| →∞ h(x, u) (|u| p ) = 0, (28)
for each x ∈ R n . Let U be the set of admissible controls

U = {u : u ∈ L p (0, T ), u(t) ∈ U}.
The target space for functions in L p (0, T ) is assumed throughout to be R m and thus will not be indicated explicitly. The set of Young measures associated to sequences

in U is [78] U = ν = {ν t } t∈(0,T ) : supp(ν t ) ⊂ U, a.e., t ∈ (0, T ), T 0 U |η| p dν t (η)dt < ∞ .
where ν is a probability measure supported in U. The extended functional J defined on U is given by

J (ν) = ϕ(x(T )) + T 0 U L(t, x(t), η)dν t (η)dt,
where x(t) is the solution of

ẋ(t) = U f (t, x(t), η)dν t (η), x(0) = x 0 .
This initial value problem is well-posed because this function satisfies the Lipschitz condition in x, necessary to ensure a unique solution (cf. [START_REF] Muñoz | A Refinement on Existence Results in Nonconvex Optimal Control[END_REF], [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF]).

Theorem [START_REF] Bengea | Optimal control of switching systems[END_REF] If in addition to the coercivity condition [START_REF] Colaneri | Stabilization of Continuoustime Switched Systems[END_REF] and the behavior indicated in [START_REF] Curto | The Truncated Complex K-moment Problem[END_REF] (no convexity in L U is assumed), we have the upper bound

L(t, x, u) ≤ k(x)(1 + |u| p )
where

k ∈ L ∞ loc (R n ), then inf{J(u) : u ∈ U} = min{ J : ν ∈ U}.
Notice the use of min in the generalized problem to emphasize the existence of a solution. We state the generalized maximum principle as in [START_REF] Young | Lectures on the Calculus of Variations and Optimal Control Theory[END_REF], [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF], [START_REF] Muñoz | A Refinement on Existence Results in Nonconvex Optimal Control[END_REF]. Before stating the generalized maximum principle, we introduce the generalized Hamiltonian by

H(t, x, μ, λ) = H(t, x, •, λ), μ ,
where

H(t, x, u, λ) = L(t, x, u) + λf (t, x, u),
and μ is a probability measure supported in U such that

U |η| p dμ(η) < ∞.
Theorem 20 If ν = {ν t } t∈(0,T ) is a minimizer for J in U, and the assumptions (a)-(d) above hold, then there exists a function λ(t), t ∈ (0, T ), such that for a.e.

t ∈ (0, T ), λ(t) = - U ∂H ∂x (t, x, η, λ)dν t (η), λ(T ) = ∇ϕ(x(T )),
with the generalized Hamiltonian condition

U H(t, x, η, λ)dν t (η) = inf{H(t, x, η, λ) : η ∈ U },
where the initial value problem is

ẋ(t) = U f (t, x, η)dν t (η), x(0) = x 0 .
Notice that ∂H/∂x is a continuous function of u, because both L and f are continuous functions of (x, u), so that the integral that appears in λ above is well-defined. The conclusion of this theorem is a direct generalization of the classical Pontryagin principle, and its proof does not involve any particular difficulty. There are some other ways of obtaining this type of optimality conditions based on sliding mode variation of relaxed controls, as it is mentioned in [START_REF] Muñoz | A Refinement on Existence Results in Nonconvex Optimal Control[END_REF].

An Equivalent Polynomial Optimal Control Problem

Equivalent Representations

The starting point is to rewrite (20) as a continuous non-switched control system as it has been shown in Chapter 2 for the stability analysis. The polynomial expression in the control variable able to mimic the behavior of the switched system is developed using a variable s, which works as a control variable.

Let Lagrange polynomial interpolation quotients [START_REF] Burden | Numerical Analysis[END_REF] be defined with the new variable s, i.e.,

l k (s) = q i=0 i =k (s -i) (k -i) . ( 29
)
We define the set of polynomial constraints which is used to constrain s to take only integer values of the original set Q as,

Ω = {s ∈ R |Q(s) = 0},
where Q(s) is the constraint polynomial defined by

Q(s) = q k=0 (s -k). ( 30
)
The solution of this system may be interpreted as an explicit ODE on the manifold Ω.

A related continuous polynomial system of the switched system (20) is constructed in the following theorem.

Theorem 21 Consider a switched system of the form given in [START_REF] Berg | Moment Problems and Polynomial Approximation[END_REF]. Then, there exists a unique continuous state system with polynomial dependence in the control variable s, p(x, s) of degree q in s, with s ∈ Ω as follows:

ẋ = p(x, s) = q k=0 f k (x)l k (s). ( 31
)
This polynomial system is an equivalent polynomial representation of the switched system [START_REF] Berg | Moment Problems and Polynomial Approximation[END_REF].

Proof. The proof of this proposition is presented in Theorem 4 of Chapter 2.

Note that the trajectories of the original switched system (20) correspond to piecewise constant controls taking values in the set σ ∈ {0, 1, ..., q}.

Similarly, we define a polynomial equivalent representation for the running cost L σ(t) using the Lagrange's quotients as follows,

Proposition 22 Consider a switched running cost of the form given in [START_REF] Blondel | Survey on the State of Systems and Control[END_REF]. There exists a unique polynomial running cost equation L(x, s) of degree q in s, with s ∈ Ω as follows:

L(x, s) = q k=0 L k (x)l k (s). ( 32
)
with l k (s) defined in [START_REF] Das | Optimally switched linear systems[END_REF]. This polynomial system is an equivalent polynomial representation of the switched running cost in [START_REF] Blondel | Survey on the State of Systems and Control[END_REF].

Proof. The proof of this proposition uses the same ideas as Theorem 4 of Chapter 2.

Equivalent Optimal Control Problem

Based on the reformulation presented in the previous section, we define an optimal control problem based on these equivalent polynomial representations. Consider the equivalent optimal control problem (EOCP):

The functional using Equation ( 32) is defined by

J = ϕ(x(T )) + T 0 L(x, s)dt, subject to the system ẋ(t) = p(x, s) = q k=0 f k (x)l k (s) , with x ∈ R n , s ∈ Ω, x(0) = x 0 ,
where l k (s), Ω, and L are defined as above. Note that this control problem is a continuous polynomial system with the input constrained by a polynomial Q(s). Notice also that we are using s as a control variable. In order to develop a methodology based on Young measures and the theory of moments, we make the following assumption, which will be omitted in the next section for a generalization of this approach.

Assumption 23 All functions in the drift vector field

F(x) = [f 0 (x)f 1 (x) • • • f q (x)],
and running cost functions L 0 (x), ..., L q (x), are polynomials.

We rewrite the EOCP using this polynomial dependence.

Let R[x, s] = [x 1 , ..., x n , s]
denote the ring of polynomials in the variables x and s, its bases being ordered lexicographically, as it is shown in the Appendix 1:

1, x 1 , x 2 , . . . , x n , x 2 1 , x 1 x 2 , . . . , x 1 x n , x 2 2 , • • • , x 2 x 3 , . . . , x 2 n , . . . , x r 1 , . . . , x r n , s, s 2 , • • • , s q ,
for the vector space of real-valued polynomials of degree at most r in x, and the scalar polynomial variable s of degree at most q. Then, an r-degree polynomial running cost L(x, s) : R n+1 → R is written as

L(x, s) = η,k a ηk x η s k ,
where η is the biggest degree of the polynomial L in the x variable, and a polynomial map p(x, s) :

R n+1 → R n p(x, s) = γ,k p γk x γ s k ,
where γ is the biggest degree of the polynomial p in the x variable, and some coefficients a kη and p kγ , that depend on L(x, s) and p(x, s), respectively. We define β = max{η, γ} as the biggest degree of the polynomials in x, and use a canonical basis on that, putting zeros where necessary. Then, we use a unified exponent and introduce the Hamiltonian as

H(x, λ, s) = L(x, s) + λ T p(x, s) = k,β a kβ x β s k + k,β λ T p kβ x β s k , ( 33
)
where λ is the vector of co-states. With this polynomial Hamiltonian, we can establish a polynomial Hamiltonian problem as follows:

The function of co-state λ(t) can be expressed as polynomial in the form:

(a) The co-state equation

λ(t) = - ∂H ∂x = - k,β a kβ ∇ x x β s k - k,β p kβ ∇ x x β λs k , ( 34
)
with

λ(T ) = ∇ x ϕ(x(T )).

(b)

The minimum condition over s

min s H = min s k,β a kβ x β s k + k,β λ T p kβ x β s k = min s k,β α kβ (λ)x β s k . ( 35
)
Notice that due to the linear relation of λ T , we can obtain a function α βk (λ), which is linear and depends on the constant coefficients a βk and p βk .

(c) The state equation

ẋ(t) = k,γ p kγ x γ s k . ( 36
)
Thanks to this polynomial form of the Hamiltonian system, we can use some concepts introduced in [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF] for relaxation of functions and use the generalized maximum principle presented in Theorem 20. This relaxation is mainly based on parametrized measures, as it is shown in the next section.

Relaxation of the Equivalent Optimal Polynomial Problem

In the previous section we introduced the basic concepts on relaxation and Young measures, which are used in this section to obtain, in regard of the special structure, a polynomial form of the optimal control problem, and hence, a relaxation of this equivalent optimal control problem as it is in Equations ( 34)- [START_REF] Galbraith | Optimal control of hybrid systems with an infinite set of discrete states[END_REF]. Due to the polynomial dependence on x and s, we are concerned with moments of such probability measures. We obtain a convexification of the state x and the control s by using moment variables, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high performance numerical computing. To solve non-convex polynomial programs as [START_REF] Fotiou | Parametric optimization and optimal control using algebraic geometry methods[END_REF] subject to ( 34) and ( 36), we use the convex hull of the graph of the polynomial [START_REF] Fotiou | Parametric optimization and optimal control using algebraic geometry methods[END_REF], once it has been provided the coercivity requirement, which is assumed if α βk > 0, with β and k be even.

Let Ω be the set of admissible controls s(t) up to time T , Ω = {s(t) ∈ R|Q(s) = 0}, with Q(s) as is defined in Equation [START_REF] Dayawansa | A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching[END_REF]. The set of Young measures associated to admissible state-control in S is

Ω = μ = {μ t } t∈(0,T ) : supp(μ t ) ⊂ Ω, a.e., t ∈ (0, T ), T 0 S |η| p dμ t (η)dt < ∞ ,
where μ is a probability measure supported in Ω. The extended functional J(x, s) defined on Ω is now given by

J(x, s) = ϕ(x(T )) + T 0 Ω L(t, x(t), η)dμ t (η)dt, where x(t) is the solution of ẋ(t) = Ω p(x, η)dμ t (η), x(0) = x 0 .
This initial value problem is well-posed because this function satisfies the Lipschitz condition in x, necessary to ensure a unique solution [START_REF] Muñoz | A Refinement on Existence Results in Nonconvex Optimal Control[END_REF], [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF]. We can now restate the generalized maximum principle presented in Theorem 20 as a generalized maximum principle as follows:

Theorem 24 (Generalized Maximum Principle -GMP)

If λ = {λ t } t∈(0,T ) is a minimizer for J in Ω, and the assumptions (a)-(d) in Section 3.1.3 hold, then there exists a function λ(t) ∈ R n such that for a.e. t ∈ (0, T ),

λ(t) = - Ω ∂H ∂x (t, x, η, λ)dμ t (η), λ(T ) = ∇ϕ(x(T )). ( 37
)
The generalized Hamiltonian minimum condition is now written as

Ω H(t, x, η, λ)dμ t (η) = inf{H(t, x, η, λ) : η ∈ Ω}, ( 38
)
and the initial value problem is

ẋ(t) = Ω p(x, η)dμ t (η), x(0) = x 0 . ( 39
)
With this generalized maximum principle, we have obtained an infinite dimensional linear program. Note that ( 35) is feasible whenever there exists an admissible control. This linear program is a rephrasing of the polynomial Hamiltonian system [START_REF] Fotiou | Nonlinear parametric optimization using cylindrical algebraic decomposition[END_REF][START_REF] Fotiou | Parametric optimization and optimal control using algebraic geometry methods[END_REF][START_REF] Galbraith | Optimal control of hybrid systems with an infinite set of discrete states[END_REF] in terms of the Young measures of its trajectories (x, s). In order to obtain a semidefinite program (SDP), which is a relaxation of the GMP, we use the fact that all functions of the EOCP are polynomials. Indeed, this is precisely the relaxation in moments of the global optimization of the polynomial Hamiltonian H(x, λ, s), when the state and the variable s are transformed into a vector m. Thus, every minimizer of the convex formulation (37) attains the minimum value of the equivalent polynomial optimal control problem (34-36); the minimizers therefore attain the minimum value of the switched optimal control problem [START_REF] Blondel | Survey on the State of Systems and Control[END_REF]. Since this method is a relaxation, H always produces a lower bound for the optimal value H * .

If H(x, s)dμ(η), [START_REF] Henrion | Nonlinear Optimal Control Synthesis via Occupation Measures[END_REF] in order to find the global minima of the objective function H in Ω. The relaxed problem [START_REF] Henrion | Nonlinear Optimal Control Synthesis via Occupation Measures[END_REF] contains information about all the global minima of the function H in S. The solution of this problem is the family of all probability measures supported in arg min(H) (see [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], [START_REF] Meziat | An alternative approach for non-linear optimal control problems based on the method of moments[END_REF]). However, it cannot be solved easily in practice (consider, for instance, the difficulty of describing all possible convex combinations of points in S). The linear program ( 40) is infinite dimensional, and thus not tractable as it stands. Therefore, we present a relaxation scheme that provides a sequence of semidefinite programs, or linear matrix inequality relaxations, each with finitely many constraints and variables.

As we know, the function H is a polynomial, and hence it can be expressed as a linear combination of simpler functions. In this case, the simpler functions are the algebraic system of integer exponents. With these considerations, we can deal with polynomial optimization problems using the method of moments.

The method of moments is a general method for treating non-convex optimization problems. It takes a proper formulation in probability measures of a non-convex optimization problem (in Appendix 1 we present the main ideas behind this method).

Therefore, when the problem can be stated in terms of polynomial expressions, we can transform the measures into algebraic moments to obtain a new convex program defined in a new set of variables that represent the moments of every measure [START_REF] Meziat | The method of moments in global optimization[END_REF].

We can express the linear combination as:

H = β,k α βk ψ βk , β ∈ N r , k ∈ N q
where the function basis {ψ i } is the algebraic system ψ β,k = x β s k . Then, we are dealing with the algebraic system in the form

H(x, λ, s) = β,k α βk (λ)x β s k . ( 41
)
We show that it is possible to determine the global minima for this algebraic polynomials. For that, consider the following optimization problem min

x,s∈S

H(x, s, λ)

We apply the convexification of this function, obtaining thus the envelope of the function H:

min x,s∈S H(x, λ, s) → min μ S H(x, λ, s)dμ(x, s) = min μ I, min μ I = min μ S β,k α βk x β s k dμ(x, s), min μ I = min μ β,k α βk S x β s k dμ(x, s) K-truncated moments , min m I = min m β,k β,k α βk m β,k . ( 42
)
In a similar way, we obtain the convexification of the state

ṁβ0 (t) = S β,k p βk x β s k dμ(x, s) = β,k p kβ m β,k (t), (43)
We have that S ⊂ R n+1 is a semialgebraic set. However, we recall the fact that the vector variable m should denote moments of a measure μ with support contained in S.

We invoke recent results of real algebraic geometry on the representation of positive polynomials on a compact set, and obtain necessary and sufficient conditions on the variables m βk to denote, indeed, moments of a measure μ with appropriate support.

Therefore, a sequence m has a representing measure μ supported on S only if these moments are restricted to be the entries on a positive semidefinite moment matrix M n (m), with m 0 = 1, and a localizing matrix defined as follows (cf. [START_REF] Curto | The Truncated Complex K-moment Problem[END_REF], [START_REF] Lasserre | An explicit equivalent positive semidefinite program for nonlinear 0-1 programs[END_REF], [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF]).

(More details about moment and localizing matrices can be found in the Appendix 1 and references therein.)

Definition 25 Moment matrix: For a given real sequence m = {m γ } γ∈N n ×N q of real numbers, the moment matrix M r (m) of order r associated with m, has its rows and columns indexed in the canonical basis {x β , s k }, and is defined by

M r (m)(γ, α) = m γ+α , γ,α ∈ N n × N q , |γ|, |α| ≤ r, ( 44
)
where |γ| := j γ j .

M r (m) is symmetric nonnegative (denote M r (m) 0, for every r). We define the localizing matrix M r (θm) whose positivity is directly related to the existence of a representing measure for m with support in K = {(x, s) ∈ R[x, s] : θ(x, s) ≥ 0}.

Definition 26

Localizing matrix: For a given polynomial θ ∈ R[x, s], written as

θ(x, s) = β,k θ β,k x β s k ,
we define the localizing matrix M r (θm) associated with m, θ, and with rows and columns also indexed in the canonical basis of R[x, s], by

M r (θm)(γ, α) = β,k θ β,k m (θ,k)+γ+α , γ, α ∈ N n × N q , |γ|, |α| ≤ r. ( 45
)
M r (θm) is also symmetric nonnegative (denote M r (θm) 0 for every r). The Kmoment problem identifies those sequences m that are moments-sequences of a measure with support contained in the semialgebraic set S.

The important property of all the above conditions is that when it is stated for all polynomials of degree less than r, they translate into linear matrix inequalities

SDP Relaxation of the Optimal Control Problem

We present the semidefinite relaxation of the optimal control problem obtained from the polynomial EOCP using the theory of moments. Consider the polynomial Hamiltonian system defined by [START_REF] Fotiou | Parametric optimization and optimal control using algebraic geometry methods[END_REF] and subject to [START_REF] Galbraith | Optimal control of hybrid systems with an infinite set of discrete states[END_REF] and [START_REF] Fotiou | Nonlinear parametric optimization using cylindrical algebraic decomposition[END_REF], where we have the set S = {R n × Ω} of admissible control values. Recall that originally we have defined the set Ω as the constraint polynomial Ω = {s ∈ R | Q(s) = 0}, with Q(s) defined as in Equation [START_REF] Dayawansa | A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching[END_REF]. Now, to be coherent with the definitions of localizing matrix and the representation results of the Appendix 1, we treat the polynomial Q(s) as two opposite inequalities, Q 1 (s) = Q(s) ≥ 0 and Q 2 (s) = -Q(s) ≥ 0, and we redefine the compact set Ω to be

Ω = {Q i ≥ 0, i = 1, 2}.
Define the space of moments as

Γ = m = {m βk } : m βk = S x β s k dμ(x, s), μ ∈ P (S), • • • • • • M i (m) 0, M i-d i (Q 1 m) 0, M i-d i (Q 2 m) 0} ,
where μ is a probability measure supported in P (S), M i is a moment matrix associated to the vector of moments m, M i-d i (Q 1 m) and M i-d i (Q 2 m) are localizing matrices related to the vector of moments constrained to the set Ω, and

d i = deg(Q 1 )/2 .
We easily see from [START_REF] Dayawansa | A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching[END_REF] that deg(Q 1 ) = deg(Q 2 ) = q + 1, where q + 1 is the number of modes of the switched system, so that d i = (q + 1)/2 . Since the mapping μ ∈ P (S) → Γ is linear, we conclude that Γ is a convex set of vectors [START_REF] Pedregal | Existence Results for Optimal Control Problems with Some Special Nonlinear Dependence on State and Control[END_REF].

We can take advantage of the moment structure of the Hamiltonian and the state equation to rewrite the relaxed formulation obtained in Theorem 24 as a SDP. For

i ≥ max[deg(H), max i deg(Q i )
] consider the positive semidefinite programs (LP i ):

Semidefinite programs-LP i :

LP i : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Minimize the Hamiltonian defined in moments

H * i (m) = min m βk ∈Γ β,k (a βk + λ T p βk )m β,
M i (m) 0, M i-d i (Q 1 m) 0, M i-d i (Q 2 m) 0.
Note that a sequence m = {m βk } indexed in the basis of R[x, s] is given. We denote {m β0 } β∈N n the marginal with respect to the variable x, and {m 0k } k∈N q the marginal with respect to the control variable s. These sequences are indexed in the canonical basis of R[x] and R[s] respectively. Also note that the optimum H * i is a lower-bound on the global optimum H * of the original problem [START_REF] Julián | High-level canonical piecewise linear representation using a simplicial partition[END_REF], since any feasible solution (x, s) yields a feasible solution m of LP i through Equation ( 42). Moreover,

H * i ≤ H * i ,
when i ≥ i . We refer to problem LP i as the semidefinite program relaxation of order i of [START_REF] Julián | High-level canonical piecewise linear representation using a simplicial partition[END_REF]. If any feasible point of the relaxation of order i is bounded, then H * i → H * as i → ∞. The LMI constraints of LP i state necessary conditions for m to be the vector of moments up to order 2i, of some probability measure μ with support constrained in S. This implies that inf LP i ≤ H * , as the vector of moments of the Dirac measure at a feasible point of ( 41), is feasible for LP i . Since the convex relaxation of the polynomial optimal control problem has convex structure in the state x, and in the control variable s, one may suppose, under mild assumptions, that the problem has a minimizer m * . Hence, we can obtain minimizers of the polynomial problem, and then obtain minimizers of the switched optimal problem [START_REF] Blondel | Survey on the State of Systems and Control[END_REF]. We can now state an important result in the following theorem:

Theorem 27 Consider the problem defined in ( 34)-( 36) and let deg(Q 1 ) = q + 1.

Then for every i ≥ n + q + 1.

(a) LP i is solvable with H * = min LP i , and to every optimal solution (x * , s * ) of ( 34)-( 36) corresponds the optimal solution

m * β,0 = (x * 1 , ..., x * n , (x * 1 ) 2 , x * 1 x * 2 , ..., (x * 1 ) 2i ) ( 46
)
and

m * 0,k = (s * , ..., (s * ) 2i ) ( 47
)
of LP i ;

(b) every optimal solution m * of LP i is the finite vector of moments of a probability measure finitely supported on v optimal solutions of ( 34)-( 36), with

v = rankM i (m) = rankM n (m).
In many cases, low order relaxation (i.e., with i << n) will provide the optimal value H * . We provide a criterion, based on the work presented in [START_REF] Lasserre | An explicit equivalent positive semidefinite program for nonlinear 0-1 programs[END_REF], to detect whether some relaxation LP i achieves the optimal value H * . One way is to determine by inspection whether an optimal solution m of LP i is a moment vector. This will be the case if, for instance, rankM r (m) = 1. However, if Equations ( 34)-( 36) have multiple optimal solutions, it can happen that m is a convex combination of moments of Dirac measures supported on the optimal solutions, which in general is not easy to detect. The next criterion allows us to test if the relaxation LP i achieves the optimal value H * [START_REF] Lasserre | An explicit equivalent positive semidefinite program for nonlinear 0-1 programs[END_REF].

Theorem 28 Consider the problem defined in ( 34)- [START_REF] Galbraith | Optimal control of hybrid systems with an infinite set of discrete states[END_REF] and let m * be an optimal solution of LP i with i < n + q + 1. If

rankM i-q (m * ) = rankM i-q-1 (m * ),
then min LP i = H * and m * is the vector of moments of a probability measure supported

on v = rankM i (m * ) = rankM i-q-1 (m *
) optimal solutions of ( 34)- [START_REF] Galbraith | Optimal control of hybrid systems with an infinite set of discrete states[END_REF].

Considering the result presented in Theorem 27, we can set the correspondence between the minimizer of ( 35) and the minimizers of LP i , and then set a correspondence with the minimizers of the original optimal switched problem [START_REF] Blondel | Survey on the State of Systems and Control[END_REF].

Theorem 29 If the moment sequence m * = {m * β,k } is the unique minimizer of the semidefinite program LP i , then the problem ( 34)-( 36) admits a unique minimizer

(x * , s * ) = m * β1 , ( 48
)
where we can state, using the equivalent representation, that the autonomous optimal switching is given by

σ * (t) = s * (t) = m * 0,1 (t), ( 49
)
and the optimal trajectory is given by the first n-terms of the moments sequence

x * (t) = m * β,0 (t). ( 50
)
This result is particularly convenient to obtain an algorithm for switching control law, as it is presented in the next section.

Switched Optimization Algorithm

In this section we sketch the algorithm we use to numerically solve the optimal control problem formulated in Definition 16. The algorithm is mainly based on the interrelation of the following:

(a) The equivalent optimal control problem EOCP

The EOCP is formulated in Section 3.1.1, where it is used the equivalent representation of the switched system and the running cost to obtain a polynomial Hamiltonian system.

(b) The relaxation of the EOCP

The relaxation allows us to obtain a generalized maximum principle which transforms the problem in a suitable form for applying the theory of moments. We therefore obtain an equivalent linear convex formulation.

(c) SDP relaxation algorithm

With the theory of moments we obtain a semidefinite program, which can be solved efficiently by a numerical algorithm; we can then apply Theorem 27 and Theorem 29 to obtain an optimal switching law.

We propose the following control algorithm:

Algorithm SOCP

We start by partitioning the time interval [t 0 , t f ] into N subintervals with points

t 0 < t 1 < • • • < t N = t f .
Step 1. Obtain the equivalent representation of the optimal switched problem using Theorem 21 and Proposition 22.

Step 2. Apply the SDP relaxation over the equivalent polynomial optimal control problem, obtaining the new set of moment variables and the LP i programs.

Step 3. Set the initial conditions:

• m 00 = 1;

• the switching signal m 0k = 0 for all 1 ≤ k ≤ q (the switching system starts with the subsystem f 0 at t 0 , i.e., σ(0) = 0);

• the state variables in the moment variables: m β0 = x(0) = x 0 .

Step 4. Solve the LP i program for the corresponding point of time and calculate the optimal value of H * using Theorem 28. In order to solve the LP i program we use an indirect shooting method.

Step 5. Calculate the moments corresponding to the next point of time, use Theorem 29 to set the switching signal and the state variables for the corresponding point of time.

Step 6. If terminal conditions are not fulfilled, go to Step 4.

In the next section we present a numerical example to show the results presented in this section.

Numerical Example: Artstein's Circle

We present an illustrative example of a switched nonlinear optimal control problem reformulated by Theorem 1 and Proposition 2 as a polynomial optimal control problem and solved by the Algorithm SOCP proposed using the theory of moments. We illustrate an efficient computational treatment to study the optimal control problem of a switched system reformulated as a polynomial expression. Examine the so-called Artstein's circle, considered as a polynomial switched system presented for stability analysis in [START_REF] Prieur | A Robust Globally Asymptotically Stabilizing Feedback: The Example of the Artstein's Circle[END_REF], [START_REF] Colaneri | Stabilization of Continuoustime Switched Systems[END_REF]. It is established that this system is asymptotically controllable and asymptotically stabilizable. It is described by the equation

f (x) = ⎡ ⎢ ⎣ (-x 2 1 + x 2 2 )u -2x 1 x 2 u ⎤ ⎥ ⎦ , ( 51
)
where u < 0 generates clockwise movement, u > 0 generates counterclockwise movement, and u = 0 induces an equilibrium point. For this system, a continuous feedback law that asymptotically stabilizes the origin does not exist. One characteristic of this problem is that if the state is initially on the circle

x 2 1 + (x 2 -c) 2 = c 2 ,
then any resulting trajectory remains on that circle regardless of the input. We impose some state constraints {x ∈ R 2 , x 1 ≤ c} with c ∈ (0, 1). If we set c = 0.577, and

x 1 = c = 0.577, we obtain x 2 = 1.1143. In Figure 3 a phase plane for two different initial conditions is shown. The system shows a stable behavior for both cases. We propose to use the SOCP to stabilize the origin of system (51) respecting the input and state constraints. Consider the set of systems described by the drift vector field

F(x) = [f 0 (x) f 1 (x)],
with

f 0 (x) = ⎡ ⎢ ⎣ -x 2 1 + x 2 2 -2x 1 x 2 ⎤ ⎥ ⎦ and f 1 (x) = ⎡ ⎢ ⎣ x 2 1 -x 2 2 2x 1 x 2 ⎤ ⎥ ⎦ ,
Note that f 0 (x) and f 1 (x) are obtained from ( 51) setting u = 1 and respectively u = -1. We use the polynomial equivalent representation to obtain the polynomial optimal control problem min x,s

t f 0 x T R 0 x(1 -s) + x T R 1 xs dt s.t. ẋ(t) = f 0 (x)(1 -s) + f 1 (x)s x ∈ R n , s ∈ Ω, x(0) = x 0 .
With this polynomial problem we obtain a polynomial Hamiltonian system as in ( 34)- [START_REF] Galbraith | Optimal control of hybrid systems with an infinite set of discrete states[END_REF]. Hence, we are ready to apply the SOCP Algorithm using the SDP relaxation in moments. Consider a regulator problem: We want to stabilize the system minimizing the control energy, in this case, the switching between the subsystems, i.e., σ ∈ Q = {0, 1}. We use matrices R 0 = R 1 = I 2×2 to set the running cost for both subsystems,

L 0 = L 1 = x T I 2×2
x, with initial time t 0 = 0 and final time t f = 10. The degree of the polynomial equivalent system is the biggest degree of the field and the running cost, i.e., r = 2. The number of variables is the number of states plus s, i.e., n = 3 because R 2 × R. We obtain with this data a basis in a lexicographical order, i.e.,

1, x 1 , x 2 , s, x 2 1 , x 1 x 2 , x 1 s, x 2 2 , x 2 s, s 2 .
We recall the fact that moment and localizing matrices have the rows and columns indexed in the previous basis of polynomials.

Define the sets

Ω = {s ∈ R| Q 1 = s(s -1) ≥ 0, Q 2 = s(1 -s) ≥ 0}, S = {Ω × R 3 },
the moment matrix with i ≥ max deg = 2, M 2 (m), and the localizing matrices with

d i = (q + 1)/2 = 1, M 1 (Q 1 m) and M 1 (Q 2 m).
Using the set S and moment and

The Recasting Process

The recasting process, as it is presented in Chapter 2 for stability analysis and presented here for convenience in the reading, is expounded as a procedure with several steps until the system has the expected form. The algorithm is thus:

• Step 0. Equivalent Representation: We consider the equivalent representation for the switched system obtained in Theorem 1, and we name it as the original system (i.e., before the recasting process), with ξ = (ξ 1 , . . . , ξ n ) as the state variables and s as the control variable.

• Step 1. Original State Equations: The original system is described by ξi = j a j k p ijk (ξ, s), i = 1, . . . , n; [START_REF] Li | Control parameterization enhancing transform for optimal control of switched systems[END_REF] nested elementary functions of elementary functions.

• Step 2. Decomposition of Non-Polynomial Functions: Let x i = ξ i , for i = 1, . . . , n. For each p ijk (ξ, s) in equation ( 52) that is not already a power-law function, replace it with a new variable x n+1 . This variable simplifies the differential equation to sums and products of power-law functions. An additional differential equation is generated for each new variable, using the chain rule of differentiation.

• Step 3. Recasting Process: When the recasting process leads to some constraints in the new variables, we have to introduce an n-dimensional manifold on which the solutions to the original differential equation lie. The particular choice of initial conditions defines the reference manifold.

• Step 4. The Polynomial Form: If the set of equations is in polynomial form, then the recasting process is complete. If not, repeat steps 2-3 until a system of equations with a rational or polynomial form is obtained.

Remark 30 Notice that the constraints introduced by the definition of new variables,

and their initial conditions, restrict the system behavior to a manifold of the same dimension of the original problem.

Let us first define the set of polynomial constraints as it has been done in the previous section. We treat the polynomial Q(s) as two opposite inequalities Q 1 (s) = Q(s) ≥ 0, and Q 2 (s) = -Q(s) ≥ 0, and we redefine the compact set Ω to be

Ω 1 = {Q i ≥ 0, i = 1, 2}.
As a result of the recasting process we have obtained new variables, which we take into account. Suppose that, for a switched system consisting of subsystems of nonpolynomial form, we apply the equivalent representation and obtain a system ξ = p(ξ, s).

The recasted system obtained using the procedure presented above is written as

ẋo = p o (x o , x r , s), ẋr = p r (x o , x r , s), ( 53
)
where x o = (x 1 , ..., x n ) = ξ are the state variables of the original system, x r = (x n+1 , ..., x n+m ) are the new variables introduced in the recasting process, p o (x o , x r , s), and p r (x o , x r , s) have polynomial forms. Previously in the recasting process we have also obtained new polynomial constraints. Consider the real-valued polynomial g k (x o , x r , s), with k = 1, ..., m, where m is the number of polynomial constraints generated in the recasting process. Let Ω 2 be the set of polynomial constraints defined by the real-valued polynomial that we obtain from the recasting process. Again, we treat the polynomials g k (x o , x r , s) as two opposite inequalities g k (x o , x r , s) = g k (x o , x r , s) ≥ 0, and g k+m (x o , x r , s) = -g k (x o , x r , s) ≥ 0, so that we define the following set. Let

Ω 2 = {(x o , x r , s) ∈ R n+2m × Ω 1 | g k (x o , x r , s) ≥ 0, g k+m (x o , x r , s) ≥ 0, k = 1, ..., m}
be the set of constraints generated by the recasting process. Having finished the recasting process, we have to redefine the set of constraints of the equivalent polynomial representations. Let

D = {(x, s) ∈ R n+m × Ω 1 × Ω 2 } ( 54
)
be the compact semi-algebraic set, where Ω 1 and Ω 2 are defined as above. Using this polynomial dependence, we rewrite the EOCP.

Let R[x o , x r , s] = [x 1 , ..., x n , x n+1 , ..., x n+m , s] denote the ring of real-valued polynomials in the variables x o , x r , s, with bases ordered lexicographically as it is shown in the previous section, for the vector space of real-valued polynomials of degree at most r in x o , x r , and the scalar polynomial variable s of degree at most q.

Then, an r-degree polynomial running cost L(x o , x r , s) : R n+m+1 → R is written as L(x o , x r , s) = η,k a ηk x η s k and a polynomial map p(x o , x r , s) : R n+m+1 → R n+m+1 , p(x o , x r , s) = γ,k p γk x γ s k for some coefficients a kη and p kγ , that depend on L(x o , x r , s) and p(x o , x r , s), respectively for every η, k ∈ N r × N. We define β = max{η, γ} as the biggest degree of the polynomials, and use a canonical basis on that, putting zeros where it is needed. We then use an unified exponent and introduce the Hamiltonian as in [START_REF] Egerstedt | Optimal control of switching times in switched dynamical systems[END_REF].

SDP Relaxation

Consider the polynomial form of the Hamiltonian. We now use the same ideas of the previous section for relaxation based on Young measures and the theory of moments.

Let D be the set of admissible trajectories

D = {(x, s) : (x(t), s(t)) ∈ Ω 1 × Ω 2 }.
The set of Young measures associated to trajectories in D is

D = μ : supp(μ) ⊂ D, a.e., t ∈ (0, T ), T 0 D x α s ν dμ(x, s)dt < ∞ ,
where μ is a probability measure supported in D. The extended functional J(x, s) defined on D is given by

J(x, s) = ϕ(x(T )) + T 0 D L(x, s)dμ(x, s)dt, where x(t) is the solution of ẋ(t) = D p(x, s)dμ(x, s), x(0) = x 0 .
This initial value problem is well-posed because this function satisfies the Lipschitz condition in x, necessary to ensure a unique solution.

Let m = {m β,k } be a sequence of real numbers, and let the moment and localizing matrices be

M i (m) 0, M i-d i (Q 1 m) 0, M i-d i (Q 2 m) 0,
where M i (m) is a moment matrix associated to the vector of moments m, M i-

d i (Q 1 m),
and M i-d i (Q 2 m) are localizing matrices related to the vector of moments constrained to the set Ω 1 , and

d i = deg(Q 1 /2) .
We easily see from Equation (30) that deg(Q 1 ) = deg(Q 2 ) = q + 1, where q + 1 is the number of modes of the switched system, so that d i = (q + 1)/2 . Let w k = 2v k or w k = 2v k -1 be the degree of the polynomial g k (x o , x r , s) depending on its parity.

Consider the localizing matrices related to the polynomial constraints obtained by the recasting process, for i

≥ max k v k , M i-v k (g k m) 0, ∀k = 1, ..., 2m,
then m has a representing measure with support contained in D. As in the previous section, we define a space of moments as

Λ = m = {m βk } : m βk = D x β s k dμ(x, s), μ ∈ P (D), • • • • • • M i (m) 0, M i-d i (Q 1 m) 0, M i-d i (Q 2 m) 0, M i-v k (g k m) 0} ,
where μ is a probability measure supported in P (D). It was shown in the previous section that we can take advantage of the moment structure of the Hamiltonian and the state equation to rewrite the relaxed formulation obtained in Theorem 24 as a SDP.

For i ≥ max[deg(H), max i deg(Q i ), max k v k ] consider the positive semidefinite pro- grams (GLP i ):
Semidefinite programs-GLP i :

LP G i : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Minimize the Hamiltonian defined in moments

H * i (m) = min m βk ∈Λ β,k (a βk + λ T p βk )m β,k , subject to the adjoint equation λ = -β,k ∂ ∂m β,0 (a βk + λ T p βk )m β,k , λ(T ) = ∇ m β,0 ϕ(T ), the state equation in moment variables ṁβ0 = β,k p βk m βk , m β0 = x(0) = x 0 ,
the corresponding moment and localizing matrices

M i (m) 0, M i-d i (Q 1 m) 0, M i-d i (Q 2 m) 0,
and the localizing matrices related to the polynomials obtained in the recasting process

M i-v k (g k m) 0, ∀k = 1, ..., 2m.
The GLP i programs can be used in a slightly variant of the Algorithm SOCP, taking into account the same considerations stated in the previous section for the LP i programs. Basically, in Step 4 of the Algorithm SOCP we replace LP i by GLP i programs.

In the next section we present a numerical example to illustrate the effectiveness of the approach presented in this section.

Numerical Example: Swinging up a Pendulum

In this example we are dealing with a two-dimensional model of the pendulum, and thus, the acceleration of its pivot is assumed to be the control input. The swinging up and stabilization of the pendulum is usually solved by switching between different laws. We use a damping-pumping strategy as it is proposed in [START_REF] Åström | A Family of Smooth Controllers for Swinging Up a Pendulum[END_REF]. The normalized model of the pendulum, when the control input is the acceleration of the pivot, is given by

ẋ1 = x 2 , ẋ2 = sin x 1 -(2 sin x 1 + F x 2 cos x 1 ) cos x 1 , (55)
of i = 2, we obtain a suboptimal value of H * = 2, 7745, the moment sequence, which allows us to calculate the switching signal, the suboptimal trajectories, and the co-states (see Figure 6). The system response shows that the trajectories reach an equilibrium point (0, 0).

CELLULAR GROWTH CONTROL

In this chapter the peculiar features of mammalian cells growth in fed-batch operating condition are addressed. The task of the controller is to determine, in every instant, the best feed substrate, using the compilation of information online from the sensor. The determination of an optimal strategy of feed substrate using the nonlinear modeling, even if the kinetics are known, is not a straightforward matter and is often further complicated by the presence of constraints imposed on the state variables [START_REF] Bastin | Nonlinear and Adaptive Control in Biotechnology: A Tutorial[END_REF]. This chapter is related with a modeling class of hybrid systems, piecewise-linear (PWL) systems. The PWL approximation, i.e., systems which are linear or affine on each of the components of a polyhedral partition of the state space [START_REF] Sontag | Nonlinear regulation: The piecewise linear approach[END_REF], have shown advantages of implementation, performance analysis, and calculations [START_REF] Hassibi | Quadratic Stabilization and Control of Piecewise-Linear Systems[END_REF], [START_REF] Rantzer | Piecewise linear quadratic optimal control[END_REF], [START_REF] Rodrigues | Observer-Based Control of Piecewise-Affine Systems[END_REF], [START_REF] Rodrigues | Automated control design for a piecewise-affine approximation of a class of nonlinear systems[END_REF]. In this work a canonical piecewise linear approximation is used over simplicial partitions. It provides a state space partition into polytopic cells based on value at vertices [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF], [START_REF] Rodrigues | Automated control design for a piecewise-affine approximation of a class of nonlinear systems[END_REF], [START_REF] Girard | Approximative solutions of odes using piecewise linear vector fields[END_REF]. This choice is motivated by several facts. First, this class of functions uniformly approximate any continuous nonlinear function defined over a compact domain R n (see [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF]). Moreover, the canonical expression introduced in [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF] uses the minimum, exact number of parameters, and it is the first PWL expression able to represent PWL mappings in arbitrary dimensional domains. As a consequence of this, an efficient characterization is obtained from the viewpoint of memory storage and numerical evaluation [START_REF] Castro | Wiener-like modelling: A different approach[END_REF]. Second, the approximation can be used in real implementations. The points from the nonlinear model may be replaced by points from sensors or data directly from the process; it thus addresses the problem of finding a PWL approximation of a system where a reasonable number of measure samples of the vector field is available (regression set) [START_REF] Storace | Piecewise-linear approximation of nonlinear dynamical systems[END_REF]. Third, this alternative approach deals with an approximation which is easier to handle than the nonlinear model. In fact, it can use many tools developed for hybrid systems, e.g., the MLD model based approach [START_REF] Bemporad | Control of systems integrating logic, dynamics, and constraints[END_REF], since algorithms for translating MLD systems into PWL systems are available [START_REF] Bemporad | Efficient Conversion of Mixed Logical Dynamical Systems into an Equivalent Piecewise Affine Form[END_REF], [START_REF] Villa | A New Algorithm for Translating MLD Systems into PWA Systems[END_REF]. Finally, this CPWL is used in a model based control termed, probing control in [START_REF] Mojica | Probing control for pwl approximation of nonlinear cellular growth[END_REF], and so it is a first step in the development of a hybrid probing control. This work refers to a probing control as it is presented in [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF] for E. coli. Short pulses to the feed rate are added, and taking into account the system response, the pulse is increased or decreased according with the tuning rule. The probing control strategy avoids acetate accumulation while maintaining a high growth rate [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF], [START_REF] Velut | Bioreactor Control Using a Probing Feeding Strategy and Mid-Ranging Control[END_REF].

The approximation model is tested by the probing control strategy. It is implemented in simulations for this mathematical model. The comparative analysis and error approximation between this new biological model and a nonlinear model developed first in [START_REF] Mojica | Desarrollo y simulación de un modelo para un cultivo celular en bioreactor[END_REF], [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF] are shown. This method is satisfactory for implementation purposes of a hybrid probing control.

Process Description

The mammalian cells of Baby Hamster Kidney are used in the production of the vaccine against the foot-and-mouth disease. They are cultivated in a bioreactor located in the biotechnology laboratory Limor de Colombia S.A, in Bogotá, Colombia. It has a capacity of 2500 liters, operating in fed-batch mode, and is temperature and agitation speed controlled. Several experiments are carried out, allowing to obtain information of several cellular cultures. Then, the data are used for parameter adjustment of the mathematical nonlinear model [START_REF] Mojica | Desarrollo y simulación de un modelo para un cultivo celular en bioreactor[END_REF]. In the nonlinear model approximation, four state variables are used in order to analyze the basic cellular behavior, and the capability of the CPWL model to approximate better this behavior, compared with a simpler approximation of PWL based on a partition form of the state space.

Nonlinear Model

The biological system class of nonlinear dynamics may be described using the following model:

⎧ ⎪ ⎨ ⎪ ⎩ • x(t) = f (x(t)) + B(x(t))u(t) y(t) = Cx(t) , ( 58
)
where x(t) ∈ R n is a vector of state at time t, f : R n → R n are nonlinear vectorvalued functions, B is a state-dependent n × m input matrix, u : R → R m is an input signal, C is an n × k output matrix, and y : R → R k is the output signal. The control variable u denotes the dilution rate, which can be shown in equation [START_REF] Mancilla-Aguilar | A Converse Lyapunov Theorem for Nonlinear Switched Systems[END_REF] to have a direct relation with state variables; it therefore has a highly nonlinear behavior.

We present a description with four state variables: cellular concentration, glucose concentration, dissolved oxygen concentration, and a waste product, i.e., acetate.

The latter is the most relevant waste component produced in the cellular growth process. In addition, this model facilitates the reactor start-up and steady-state operation conditions, since the presence of less-desired steady states at the same inlet conditions makes this bioreactor a challenging problem for control design. It can be shown in the simulation section that the transient analysis evidences this behavior.

As it has been shown in [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF], the BHK cells have several growth phases:

i. Latency, immediately after inoculation.

ii. Acceleration state, when cells begin the growth process.

iii. Exponential state, when the cells reach a constant growth with the major growth rate.

iv. Deceleration state, when cells reduce the constant growth for absence of substrate and accumulation of toxic substrates.

v. Stationary state, when the cellular growth reaches a constant value, since the substrate is over or is in a low constant value.

For further details about the nonlinear model, refer to [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF]. The nonlinear model is

as follows ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ẋ1 = μ m x 2 x 2 2 k i +x 2 +ks x 3 x 3 +ka x 1 - k dm k ds (μm-k dm x 4 )(k ds +x 2 ) x 1 -x 1 u, ẋ2 = -k 1 μ m x 2 x 2 2 k i +x 2 +ks x 3 x 3 +ka x 1 + (S 1i -x 2 )u, ẋ3 = k 3 μ m x 2 x 2 2 k i +x 2 +ks x 3 x 3 +ka x 1 -x 3 u, ẋ4 = Kla(C * -x 4 ) -k 1 μ m x 2 x 2 2 k i +x 2 +ks x 3 x 3 +ka x 1 -x 4 u, (59)
where x 1 (g/L) is the cellular concentration; x 2 (g/L) is the glucose concentration;

μ m (1/h) is the maximum cellular growth rate; k s (g/L) is the glucose saturation parameter; x 3 (g/L) is the lactate concentration; x 4 (g/L) is the dissolved oxygen concentration; k a = 0.06 (g/L) is the lactate saturation parameter; k 1,2,3 are yield coefficients, 1.46, 0.9, and 0.06 respectively; S 1i = 1.5 (g/L) is the initial glucose concentration in the feeding medium; k dm = 0.012 (1/h) is the maximum cellular death rate; and k ds = 0.08 is the death saturation parameter. The volumetric oxygen transfer coefficient, Kla is a function of stirred speed but is also affected by the air flow rate and factors like viscosity and foaming. C * (g/L) is the dissolved oxygen concentration in equilibrium with the oxygen in gas bubble. All the nonlinear model parameters are taken from the real data through several experiments and are presented in [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF].

Figure 7 shows the schematic of the bioreactor in a fed-batch operation mode with dissolved oxygen concentration as input variable and feed rate as control variable.

The Biological CPWL model

The approximation presented here deals with the PWL approximation technique developed in the past few years in [START_REF] Julián | High-level canonical piecewise linear representation using a simplicial partition[END_REF]. To guarantee that the dynamical behavior of the PWL-approximate flow will be faithful to that of the original system for any values of some significant parameters, the input variable u is used as a parameter, and is simulated the PWL-flow. Figure 8 shows the schematic of the Bioreactor with CPWL model with the control loop and variables involved in this approach. First, some definitions about CPWL (Canonical Piecewise-linear) are given.

Orthonormal Canonical Piecewise Linear Functions

The orthonormal definition of the PWL functions given before in [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF] is used in this work to represent the nonlinear static mapping of cellular growth process in bioreactor [START_REF] Margaliot | Stability analysis of switched systems using variational principles: An introduction[END_REF]. A brief description of this representation and its most important characteristics are given as following. For further details refer to [START_REF] Julián | High-level canonical piecewise linear representation using a simplicial partition[END_REF], [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF]. For this approximation, given a nonlinear system as in (3), the following steps are necessary to obtain a CPWL approximation:

Step 1. Order all vertices of the grid partition.

Step 2. Group the vertices into simplicial cells.

Step 3. Find CPWL basis functions.

Step 4. Find the CPWL model approximation.

Figure 8 shows a schematic of this piecewise approximation. In order to perform these steps, the following definitions and methods are presented: Definition 31 . Let x 0 , x 1 , ..., x n be n + 1 points in the n-dimensional space. A simplex (or polytope) (x 0 , x 1 , ..., x n ) is defined by where 0 ≤ μ i ≤ 1, i ∈ {0, 1, ..., n} and n i=0 μ i =1. A simplex is said to be proper if and only if it cannot be contained in a (n -1) dimensional hyperplane.

(x 0 , x 1 , ..., x n ) = x : x = n i=0 μ i x n ,
The representation proposed in [START_REF] Julián | High-level canonical piecewise linear representation using a simplicial partition[END_REF] requires the definition of a rectangular compact domain of the form

S = {x ∈ R m : 0 ≤ x i ≤ n i δ k , i = 1, 2, ..., m} , ( 60
)
where δ is the grid size and n i ∈ Z + (the set of positive integers). This domain is Figure 9 shows the basic concepts of the simplicial partition and the grid size for an illustrative two-dimensional system.

There are many possible choices for the PWL basis functions, each of which is made up of N (linearly independent) functions belonging to PWL Ndimensional linear space. Thus, any basis can be expressed as a linear combination of the elements of the β basis, which is defined by recursively applying the following generating function γ(u, v) [START_REF] Julián | High-level canonical piecewise linear representation using a simplicial partition[END_REF], [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF]:

γ(x 1 , x 2 ) = {||x 1 | + x 2 | -||x 2 | -x 1 | + |x 1 | + |x 2 | -|x 2 -x 1 |}/4. ( 61
)
π k,j k (x) =x k -j k δ k , k = 1, 2, ..., m, j k = 0, 1, ..., n k -1.
As a result, the basis can be expressed in vector form, ordered according to its nesting level as

Λ = Λ 0 T , Λ 1 T , ..., Λ m T , ( 63
)
where Λ i is the vector containing the generating functions defined in [START_REF] Julián | High-level canonical piecewise linear representation using a simplicial partition[END_REF] with i nesting levels. Accordingly, any f p ∈ PWL[S H ] can be written as

f p (x) = c T Λ(x), (64)
where c = c 0 T , c 1 T , ..., c m T T , and every vector c i is a parameter vector associated with the vector function Λ i .

In order to obtain an orthonormal basis, it is necessary to define an inner product on PWL[S H ]. The new basis elements are a linear combination of (64), i.e., Υ(x) =T Λ(x), and the matrix T may be obtained by two different methodologies.

In this work we use the one built using the Gran-Schmidt procedure as given in [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF].

To find the required approximation, we use a routine of [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF] that finds a vector of where ∀x 1 , x 0 ∈ S and L ≥ 0 is the Lipschitz constant. Then, f CP W L satisfies

f CP W L -f (x) ≤ δL n i=0 e i , ∀x ∈ S,
where e i is the error of each simplicial partition. It should be noticed that a choice of δ such that δL n i=0 e i = e, or equivalently

δ = e L n i=0 e i ,
guarantees that the approximation error is bounded as follows:

f CP W L -f (x) ≤ e.

Cellular Growth CPWL Model

The cellular growth approximation scheme is based on the orthonormal canonical piecewise linear functions. It means the approximation of the right-hand side nonlinear function to a CPWL function for each differential equation. The PWL approximation based on a nonlinear model is obtained over the domain values of subdivisions, it is found that the last values perform the system well, from both qualitative and quantitative points of view. It is noticed that the algorithm that uses orthonormal basis allows a larger number of inputs due to the ability of handling sparse matrices. Thus, it is possible to detect the simplices that contribute to the approximation, and it gives a measure of the nonlinearity of the function. It is possible to take smaller values of m in order to reduce the number of parameters, but the quality of the approximation dereases.

S = y ∈ R 3 : a 1 ≤ x 1 ≤ b 1 , a 2 ≤ x 2 ≤ b 2 , a 3 ≤ x 3 ≤ b 3 , a 4 ≤ x 4 ≤ b 4 , a 5 ≤ u ≤ b 5 } ,

Simulation Results

Model Simulation Results: Nonlinear vs CPWL

In order to analyze the system behavior with different input values, corresponding to the dilution rate u, the original nonlinear system with the CPWL approximation for u = 0 is simulated. It is the batch operation mode, i.e., the natural system behavior.

The simulation provides a plausible explanation of the different steady states. First, as in the previous section, the CPWL model is simulated without input variable, dilution rate, in order to obtain a batch mode behavior. All the simulations are made with initial value x 0 = (0.12, 4.2, 2.6, 0.0). Figure 10 shows a comparison between the nonlinear system and the CPWL approximation obtained by numerically integrating the dynamical system that has a dilution rate u = 0 on the right-hand side of f CP W L .

In this case, it can be shown that the behavior of the four state variables that use the CPWL model is qualitatively similar to the original system. For simulation purposes, the dilution rate is increased to the best experimental value obtained form the data of many experiments [START_REF] Mojica | Identificación y Control de un Proceso de Crecimiento Celular en Bioreactor[END_REF]. The CPWL shows a very good agreement in this case. The dilution rate was then increased to reach a high dilution rate, where normally the system is not working, and the cellular concentration is too low. It is of particular interest to show that the obtained model is able to indicate the multiple steady states. Figure 11 shows the comparison between the nonlinear system and the CPWL approximation for these cases, the dilution rate is u = 0.031/h.

Transient Analysis of Cells Concentration

In this section, the transient analysis for different initial conditions is considered at the same dilution rate, and several steady states are found. In order to further investigation of these steady states, the dilution rate is established at u = 0.036/h, and several initial conditions, with cellular and lactate initial conditions left unmodified and varying glucose and glutamine initial concentrations ( see Table 1).

Three steady states can be clearly observed (see Figure 12). Steady state 1 results from low levels of S 1 and S 2 being maintained in a middle value. In a low value, the curves 1 and 4 are shown. Steady state 2 results from a middle value of S 1 and high value for S 2 , as also S 2 middle and S 1 high (see curves 2 and 3). Finally, the steady state 3 results from high levels of S 1 and S 2 (see curve 5). The capability of the CPWL model to show the three steady states can thus be noticed. High concentration of glucose and glutamine in the initial conditions produces high concentration of lactate generating the cellular death (see Figure 12).

Probing Feed Controller

This section briefly describes the control technique developed in [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF] for E. coli cultivations. The key idea is to exploit the characteristic saturation in the oxygen supply system that occurs at acetate formation, when the glucose uptake rate exceeds the glucose uptake rate critic. This is achieved by superimposing a short pulse on the feed visible, the feed rate is increased at the end it. When overfeeding is detected (no pulse response) the feed rate is decreased. This control has some optimal characteristics, in the sense that it gives the maximum growth rate without overflow metabolism, thus minimizing the cultivation time. A simple flow diagram of the control algorithm is shown in Figure 13.

Feedback Algorithm

A simple algorithm that can be interpreted as a proportional incremental controller is used to adjust the feed rate F . At each cycle of the algorithm a pulse is given and, depending on the response, the feed rate is adjusted according to the flow. A reaction to a pulse is said to occur if the amplitude of the response exceeds a critic oxygen reactive during the pulse [START_REF] Åkesson | Probing Control of Fed-Batch Cultures: Analysis and Tuning[END_REF]. In this specific case, a proportional probing feed controller is used. The controller changes depending if it is in a pulse up or a pulse down. In the first case the increase in the feed F is decided by

dF (t) dt = k C pulse (t) -C pref C * -x 4 F
where C pulse (t) is the pulse response, C pref is the desired pulse response, and k the controller gain. In the case of down pulses the feed is given by

dF (t) dt = k C pulse (t) -C pref C * -x 4 -γ p F,
where γ p is the pulse amplitude. The pulse height, F pulse , must give an oxygen response that exceeds the reaction levels, but it must also be ensured that the dissolved oxygen level does not reach zero during the pulse.

F pulse = γ p F γ p ≈ 4C reac /(C * -C sp ),

CONCLUSIONS AND FUTURE WORK

In this chapter we will overview the main contributions of this dissertations and discuss briefly future research directions.

Summary of Contributions

• In Chapter 2 we developed a new method for stability analysis of switched systems based on a polynomial approach. First, we transformed the original problem into a polynomial system, which is able to mimic the switching behavior but with a continuous differential-algebraic nonlinear representation. From a theoretical point of view, we showed that the representation of the original switched problem into a continuous polynomial system allowed us to use the dissipation inequality for polynomial systems. With this method and from a theoretical point of view, we provided an alternative way to search for a common Lyapunov function for switched systems.

• In Chapter 3 we considered a new method for solving the optimal control problem of nonlinear switched systems based on a polynomial approach. First, we transformed the original problem into a polynomial system, which is able to mimic the switching behavior with a continuous polynomial representation. After that, we transformed the polynomial problem into a relaxed convex problem, using the method of moments. From a theoretical point of view, we provided necessary and sufficient conditions for the existence of minimizer, by using particular features of the relaxed, convex formulation. Even in the absence of classical minimizers of the switched system, the solution of its relaxed formulation provided minimizers. However, in some cases some functions of the system were not in a polynomial form, to solve this issue, we applied the recasting process to obtain a complete polynomial system, and thus extended the results to a more general nonlinear switched systems.

• In Chapter 4 we presented some simulation results on a piecewise linear approximation based on orthonormal CPWL functions of nonlinear cellular growth. It was also proved that this structure allows us to approximate the dynamical behavior for different initial conditions in the transient analysis. Moreover, the nonlinear characteristic presented over cellular growth on three steady states was shown for the CPWL model that captures the partially substitutable and partially complementary nature of the two substrates, glucose and glutamine. This model is useful for a nonlinear control, in particular to hybrid control. It is possible to take a nonlinear system to obtain a PWL approximation from it, and then to apply the hybrid control.

One interesting point over this CPWL model is the fact that some points of the nonlinear system model are used. In a real implementation these points being instead of taken from nonlinear systems, could be taken from the sensors or data from the process.

Future Research Directions

In this section we will outline several future research directions that are related to our work. Further directions for the optimal control can be focused on the development of a computational tool to solve the convex relaxed problem in general cases, i.e., nonlinear vector fields, and to prove the computational efficiency of the proposed method. Moreover, an extension of this approach using SDP relaxation should be deployed for subsystems modeled by differential-algebraic equations [START_REF] Kunkel | Stability Properties of Differential-Algebraic Equations and Spin-Stabilized Descretizations[END_REF], [START_REF] Roubi'cek | Optimal control of causal differentialalgebraic systems[END_REF]. On the other hand, we have several tools for switched systems using this polynomial representation, different from the sum of squares decomposition. Some of them open several possibilities for system analysis, such as controllability and observability.

In the area of model and control of bioreactors using hybrid systems, future work should focused on developing an optimal control for this class of hybrid system, so as to implement it on cellular cultures based on the relaxed approach developed in Chapter 3. This model, even though it has been utilized in BHK experiments, seems to be generally applicable to mammalian systems other than BHK, such as Hybridoma and Chinese Hamster Ovary (CHO) cells that display similar behavior [START_REF] Namjoshi | Unveiling steady-state multiplicity in hybridoma cultures: The cybernetic approach[END_REF]. The CPWL model has been tested with the probing proportional controller based on the dissolved oxygen signal. Having preformed optimally, we can therefore extend this result to a more general biological process.

A.1.2 Measures

Formally, a measure is a function defined on a σ-algebra B over a set Ω such that the following properties are satisfied: Definition 41 Let Ω be a set, B an algebra on Ω. A function μ:

Ω → R + ∪ {∞} is called a measure if 1.
The empty set has measure zero. μ(A) ≥ 0 for any A ∈ B and μ(∅) = 0;

Countable additivity or σ-additivity. If

(A i ) i≥1 is a disjoint family of sets in B (A i ∩ A j = ∅ for any i = j) such that ∞ i=1 A i ∈ B, then μ ∞ i=1 A i = ∞ i=1 μ(A i ).

A.1.2.1 Properties

Some elementary properties of a measure can be derived from the definition of a countably additive measure.

(c) If A, B ∈ B and μ(A ∩ B) < ∞ then μ(A ∪ B) = μ(A) + μ(B) -μ(A ∩ B). (d) If (A i ) i≥1 ⊂ B, not necessarily disjointed, such that ∞ i=1 A i ∈ B, then μ ∞ i=1 A i ≤ ∞ i=1 μ(A i ).
Theorem 42 Let B be an algebra, and

(A i ) i≥1 ⊂ B a monotonically increasing se- quence of sets (A i ⊂ A i+1 ) such that i≥1 ∈ B. Then μ ∞ i=1 A i = lim n→∞ μ(A n ). f -1 ([a, b]) ∈ X for all a < b.
It can be shown that this is equivalent to requiring that the pre-image of any Borel subset of R be in X. We will make this assumption from now on. The set of measurable functions is closed under algebraic operations, but more importantly the class is closed under various kinds of pointwise sequential limits.

We build up an integral

Ω f dμ = Ω f (x)μ(dx)
for measurable real-valued functions f defined on Ω in stages.

The Lebesgue integral has the following properties: Monotone convergence theorem: Suppose (f n ) is a sequence of real, nonnegative measurable functions with limit f such that

f n (x) ≤ f n+1 (x) ∀k ∈ N, ∀x ∈ Ω. Then A f dμ = lim n→∞ A f n dμ, A ∈ Ω.
N.B: The value of any of the integrals is allowed to be infinite.

Fatou's lemma: If (f n ) is a sequence of non-negative measurable functions defined a.e., and

f (x) = lim n→∞ inf f n (x) then A f dμ ≤ lim n→∞ inf A f n dμ, A ∈ Ω Theorem 46 Let A ∈ B, (f n ) be a sequence of non-negative measurable functions and f (x) = ∞ n=1 f n (x), x ∈ A. Then A f dμ = ∞ n=1 A f n dμ.
f n dμ = b a f (t)dt.
In other words, the Lebesgue integral of f is equal to the Riemann integral of f .

A.2.0.2 Expectation

Definition 50 If f : Ω → R is a random variable on (Ω, B, μ), then the mean or expectation of f is defined by

E(f ) = Ω f dμ
It has all usual properties of the integrals. Recall that f is a measurable function.

In analogy with discrete values and the classical probability theory, the expectation is

E(X) = k j=1 x j p j ,
where X is a discrete variable that can take values x 1 , x 2 , ..., x k with probabilities p 1 , p 2 , ..., p k .

A.2.1 Some Facts About Young Measures

This section is a summary of a few basic, general, important facts to be used in providing results for Chapter 3. Our fundamental reference for this material is [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF] and [START_REF] Muñoz | A Refinement on Existence Results in Nonconvex Optimal Control[END_REF]. The first one is a basic existence theorem for Young measures, these are parameterized measures that are associated with certain subsequences of a given bounded sequence of measurable functions. The second one relates to the fact that a lack of oscillations is reflected on the triviality of the Young measure.

A basic definition concerning normed spaces or L p spaces, which are in fact vector spaces since they are stable under the vector space operations.

Definition 51

The space of sequences l p , p ≥ 1, consists of sequences

(x n ) satisfying ∞ n=1 |x n | p < ∞; the norm in l p is x n p = ( ∞ n=1 |x n | p )
ψ(x) = R m ψ(x, λ)dν x (λ).
Notice that there is always a Young measure associated to a bounded sequence {u j } in L p (Ω) for p > 1.

Proposition 53 Let z j = (x j , u j ) : Ω → R n × R m be a bounded sequence in L p such that {x j } converges strongly to x in L p . If μ = {μ t } t∈Ω is the parametrized measure associated to {z j }, then μ t = δ x(t) ⊗ ν t a.e. t ∈ Ω, where {ν t } t∈Ω is the parametrized measure corresponding to {u j }.

The following fact is a remarkable, convenient property.

Lemma 54 If ν = {ν x } x∈Ω is a family of probability measures supported in R m such that Ω R m |λ| p dν x (λ)dx < ∞, p > 1,
then there exists a bounded sequence in L p (Ω), {u j }, whose corresponding Young measure is ν and such that {|u j | p } is equiintegrable. If moreover supp(ν t ) ⊂ K for a.e. x ∈ Ω, where K ⊂ R m is a convex set, then each u j can be chosen taking values in K.

|P (x)| 2 dμ(x) > 0, for every complex-valued polynomial P (x), unless P vanishes on the support of μ.

This gives rise to matrix conditions, necessary on any sequence of moments, namely that certain Hankel matrices are positive semi definite.

A general description of the problem of moments is as follows.

Given a set of functions h 1 , ..., h k defined in Ω ⊂ R n , and a sequence of values (m k ), the problem of moments consists in determining a positive measure μ such that

m i = Ω h i (x)dμ(x) ∀i = 1, ..., k,
whenever that is possible. Thus, the problem of moments also includes the search for requirements in order to characterize the sequence (m k ) as a set of moments.

Depending on the function basis h 1 , ..., h k and the set Ω, the problem of moments can take different forms. Usually, we refer to the function basis as an algebraic system, i.e. h i = x i .

A.3 Basics of Convex Optimization

In convex optimization we find a fusion of three different domains: Convex analysis, optimization, and numerical computation.

A.3.1 Convex Sets

We call a point of the form θ

1 x 1 + • • • + θ k x k , where θ 1 + • • • + θ k = 1 and θ i ≥ 0, i = 1, • • • , k, a convex combination of the points x 1 + • • • + x k .
It can be shown that a set is convex if and only if it contains every convex combination of its points.

Definition 56

The convex hull of a set C, denoted coC, is the set of all convex combinations of points in C:

(c) more generally, f (E(x)) ≤ E(f (x)).
Another convexity preserving operation is that of minimizing over some variables.

Specially, if h(x, y) is convex in x and y, then

f (x) = inf y h(x, y)
is convex in x. This is because the operation above corresponds to projection of the epigraph, (x, y, a) → (x, t).

Definition 59

The convex hull or envelope of a function f : R n → R is defined as

h(x) = cof (x) = inf{a |(x, a) ∈ coepif (x) }.
Geometrically, the epigraph of h(x) is the convex hull of the epigraph of f . h(x) is the largest convex underestimator of f .

A.4 Optimization over Polynomials Using the Method of Moments

The method of moments is a general method for treating non-convex optimization

where the objective function f (x) is a linear combination of simple functions.

One approach to this problem comes from convex analysis, since we can use the convex envelope of the function f in order to locate its global minima. As we have shown, every convex combination of points in Ω can be described as a discrete probability distribution μ supported in Ω, such that every integral

m i 2n i=0 c i m i , ( 68
)
and one works with the finite sequence m of the moments of μ, up to order r, instead of μ itself. Of course, not every sequence m has a representing measure μ; that is, given an arbitrary finite sequence m, there might not be any probability measure μ, all of whose moments up to order r coincide with m α scalars. Consider the function polynomial f . It can thus be expressed as a linear combination of simple functions.

In this case the simple functions are the algebraic system of integer exponents. Let

x α = x α 1 1 x α 2 2 • • • x αn n
is a monomial of degree r and p α is a coefficient. A polynomial p of degree r can thus be identified with the sequence of its coefficients (p α ) |α|≤r in the canonical basis of monomial {x α : |α| ≤ r}, and the space of polynomial of degree r can be viewed as a vector space, which we name P r . If necessary, a polynomial of degree r can be viewed as a polynomial of higher degree r , by setting the coefficients of monomials of degree higher than r to zero. Given two index strings, α = (α 1 , ..., α s ) and β = (β 1 , ..., β t ), we define their concatenation as α • β = (α 1 , ..., α s , β 1 , ..., β t ). It thus follows that

x α x β = x α•β .
In order to study moment problems on a semialgebraic set of R n , we consider

) 70
where |α| := j α j .

For illustration, consider the two-dimensional case. The moment matrix M r (m) is the block matrix {M i,j (m)} 0≤i,j≤2r defined by

M i,j (m) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ m i+j,0 m i+j-1,1 • • • m i,j m i+j-1,1 m i+j-2,2 • • • m i-1,j+1 • • • • • • • • • • • • m j,i m i+j-1,1 • • • m 0,i+j ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
, where m i,j represents the (i + j)-order moment x i y j dμ(x, y) for some probability measure μ. For instance, consider the particular case when n = 2 and r = 2. We obtain This means that M r (θm) 0 whenever μ m has its support contained in the set K.

M 2 (m) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 | m 1,0 m 0,1 | m 2,
The K-moment problem identifies those sequences m that are moments-sequences of a measure with support contained in the semialgebraic set K.

To illustrate how to construct a localizing matrix, consider a moment matrix

M 1 (m) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 m 1,0 m 0,1 m 1,0 m 2,0 m 1,1 m 0,1 m 1,1 m 0,2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦
, and a polynomial θ(x) = ax 2 1x 2 2 . We obtain then the following localizing matrix: We should briefly outline now the idea developed for the optimization in polynomials with Ω as an arbitrary subset of R n . One first reduces the optimization polynomial problem to the equivalent convex optimization (67) on the space of probability measures μ with support contained in Ω. We have the following proposition summarizing the main result for constrained optimization in polynomials.

M 1 (m) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ a -m 2,0 -m 0,2 am
Proposition [START_REF] Mojica | Desarrollo y simulación de un modelo para un cultivo celular en bioreactor[END_REF] The problems [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF] and [START_REF] Mojica | Piecewise-linear approximation of nonlinear cellular growth[END_REF] are equivalent, that is, (a) inf(66) = inf(67);

(b) If x * is a global minimizer of [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF], then μ * := δ x * is a global minimizer of ( 67);

(c) Assuming [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF] has a global minimizer, then, for every optimal solution μ * of (67), f (x) = min(66), μ * -almost everywhere (μ *a.e); and (d) If x * is the unique global minimizer of [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF], then μ * := δ x * is the unique global minimizer of [START_REF] Mojica | Piecewise-linear approximation of nonlinear cellular growth[END_REF].

A.4.1 Convergent Semi-definite Relaxations

Consider the constrained optimization problem in [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF], where the g i (x) are all real- where d i = deg(g k )/2 . Note that the optimum f i is a lower-bound on the global optimum f * of the original problem [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF], since any feasible solution x of (66) yields a feasible solution m of SDP i through [START_REF] Mojica | Probing control for pwl approximation of nonlinear cellular growth[END_REF]. Moreover, f i ≤ f i when i ≥ i . We refer to problem SDP i as the semidefinite program relaxation of order i of (66). If any feasible point of the relaxation of order i is bounded, then f i → f * as i → ∞. The LMI constrains of SDP i state necessary conditions for m to be the vector of moments up to order 2i, of some probability measure μ m with support contained in Ω. This clearly implies that inf SDP i ≤ f * , as the vector of moments of the Dirac measure at a feasible point of (66) is feasible for SDP i . It has been established that there are no more than 2 n -1 variables. We can now state an important results in the following theorem.

Theorem 66 Consider the problem defined in [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF] and let v = max k=1,...,m v k . Then for every i ≥ n + v:

(a) SDP i is solvable with f * = min SDP i , and to every optimal solution x * of [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF] corresponds the optimal solution m * = (x * 1 , ..., x * n , ..., (x * 1 ) 2i , ...,

(x * n ) 2i ) ( 72
)
of SDP i ; and

(b) every optimal solution m * of SDP i is the (finite) vector of moments of a probability measure finitely supported on t optimal solutions of ( 66), with t = rankM i (m) = rankM n (m).

In many cases, low order relaxations (that is with i n) will provide the optimal value f * . Therefore, one would want to have a test to detect whether some relaxation SDP i achieves the optimal value f * . One way to achieve that is to determine by inspection whether an optimal solution m of SDP i is a moment vector. This will be the case if, for instance, rankM r (m) = 1. However, in the case in which (66) has multiple optimal solutions, it can happen that m is a convex combination of moments of Dirac measures supported on the optimal solutions, which in general is not easy to detect. We next provide a criterion to test whether the SDP relaxation SDP i indeed achieves the optimal value f * .

Theorem 67 Consider the problem defined in [START_REF] Mojica | Canonical piecewise-linear approximation of nonlinear cellular growth[END_REF] and let v = max k=1,...,m v k . Let m * be an optimal solution of SDP i with i < n + v. If

rankM i-v+1 (m * ) = rankM i-v (m * ),
then min SDP i = f * and m * is the vector of moments of a probability measure supported on t = rankM i (m * ) = rankM i-v (m * ) optimal solutions of (66).

Une

  Approche polynomiale pour l'analyse et la commande optimale des systèmes non-linéaires à commutation Résumé : Dans cette thèse nous étudions comment la géométrie semi-algébrique convexe et l'optimisation polynôme globale peuvent être employées pour analyser et concevoir les systèmes non linéaires à commutations. Pour traiter l'analyse de stabilité des systèmes non-linéaires à commutations on montre que la transformation du problème original à commutations vers un système polynômial continu nous permet d'employer l'inégalité de dissipation pour les systèmes polynômiaux. Avec cette méthode et d'un point de vue théorique, nous fournissons une manière alternative de rechercher une fonction commune de Lyapunov pour les systèmes non linéaires à commutations. L'idée principale derrière l'approche proposée est d'inclure dans l'analyse fonctionnelle les contraintes cachées. Nous devons vérifier le définition semi-négative de dV /dt en ce qui concerne l'ensemble de contraintes. Pour cela, nous employons l'idée de la pénalisation utilisée dans la théorie d'optimisation avec contraintes. Une fonction λ(x, s) est introduite et elle peut être interprétée comme fonction de pénalisation ou multiplicateur de Lagrange. Cette idée est basée sur des résultats pour les systémes de commande contraints, oú nous pouvons employer le concept d'inégalité de dissipation utilisant des fonctions de stockage et des taux d'approvisionnement. Pour cela, nous employons l'idée de la pénalisation utilisée dans la théorie d'optimisation avec des contraintes. Ainsi nous étendons alors les résultats á une classe plus générale des systémes commutés, ceux modélisés par des fonctions élémentaires. Cette classe de fonctions provient des dérivés symboliques explicites, telles que l'exponentielle, le logarithme, les fonctions trigonométriques, et les fonctions hyperboliques. Pour ce faire, nous transformons, en utilisant un processus de réécriture le systéme obtenu iii par la représentation équivalente dans un systéme sous la forme polynômiale et puis, nous employons les résultats de la section précédente pour l'analyse de la stabilité. En plus de l'analyse de stabilité, des problèmes de commande optimale pour les systèmes non-linéaires commutés sont également étudiés. Nous proposons une approche alternative pour résoudre le problème de commande optimale pour un système non linéaire autonome à commutations, basé sur le principe de maximum généralisé (GMP). L'essentiel de cette méthode est la transformation d'un problème de commande optimale non-linéaire et non-convexe, c'est-á-dire, le système commuté, en un problème de commande optimale équivalent avec la structure linéaire et convexe, qui permet d'obtenir une formulation convexe équivalente plus appropriée pour être résolue par un calcul numérique plus efficace. En conséquence, nous proposons de convexifier les variables d'état et de commande au moyen de la méthode des moments afin d'obtenir des programmes SDP. Une généralisation pour résoudre le problème de commande optimale des systèmes commutés non-linéaires est étudiée à partir du processus réécrit. En conclusion, nous étudions l'application industrielle obtenue par une approximation linéaire par morceaux de la croissance cellulaire non-linéaire en utilisant des fonctions canoniques linéaires orthonormales. Elle est commandée par une stratégie de ¡¡probing control¿¿. Nous traitons les cellules mammifères BHK (rein de bébé hamster) dans un bio-réacteur. Les résultats de simulation prouvent que cette approximation linéaire par morceaux est bien adaptée pour modeléliser une telle dynamique non-linéaire. Mots clés : Optimisation convexe, commande optimale, systèmes commutés, analyse de stabilité A Polynomial Approach for Analysis and Optimal Control of Switched Nonlinear Systems Abstract: In this dissertation we investigate how convex semialgebraic geometry and global polynomial optimization can be used to analyze and to design switched nonlinear systems. To deal with stability analysis of switched nonlinear systems it is shown that the representation of the original switched problem into a continuous polynomial system allows us to use the dissipation inequality for polynomial systems.

2 Figure 1 :

 21 Figure 1: Switching between stable systems that are becoming stable

2 Figure 2 :

 22 Figure 2: Switching between stable systems that are becoming unstable

(

  LMI) conditions on m, via moment and localizing matrices associated with m and S. It is shown that the global minima of the optimization problem are equivalent to the minima of the programming problem[START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF]. It can be shown that the optimization problem given in (42) is a semidefinite program (SDP) because of the symmetry on the moments matrices. Consequently, we have now the necessary elements to state a linear convex program related with the original problem[START_REF] Boccadoro | Beyond the construction of optimal switching surfaces for autonomous hybrid systems[END_REF].

  k subject to the adjoint equation λ =β,k ∂ ∂m β,0 (a βk + λ T p βk )m β,k , λ(T ) = ∇ m β,0 ϕ(T ), and the state equation in moment variables ṁβ0 = β,k p βk m βk , m β0 = x(0) = x 0 , and the corresponding moment and localizing matrices related to the space of moments Γ

Figure 3 :

 3 Figure 3: Phase plane for two different initial conditions

2 Figure 4 : 2 Figure 5 :

 2425 Figure 4: States, co-states, and switching signal for the Arstein's circle example

2 Figure 6 :

 26 Figure 6: States and switching signal for the pendulum example

Figure 7 :

 7 Figure 7: A cell bioreactor in a feed-batch operation mode

Figure 8 :

 8 Figure 8: Schematic of the Bioreactor with CPWL model

  then subdivided using a simplicial boundary configuration H. A simplicial boundary configuration is characterized by the fact that it produces a division of the domain into simplices. If S is defined as in (60), the space of all continuous PWL mappings defined over the domain S partitioned with a simplicial boundary configuration H is denoted by PWL[S H ]. At this point it is convenient to introduce the set V S of all simplex vertices contained in S. Depending on the number of coordinates different from zero, these vertices are organized into classes; for instance, all the vertices in S that have k (k ≤ m) coordinates different from zero are called class k vertices. The set V S plays an important role, as the function values at the vertices of V S constitute all the information that is needed to fully characterize any PWL function f p : S -→ R 1 .

with a 1

 1 = a 2 = a 3 = a 4 = a 5 = 0.0, b 1 = 4, b 2 = 4.5, b 3 = 2.6, b 4 = 1.8, b 5 = 0.3. The domain S is partitioned by performing m 1 , m 2 , m 3 , m 4 subdivisions along the state components x 1 , x 2 , x 3 , x 4 respectively, and m 5 subdivisions along the parameter component u as the control variable. The coefficients are derived from a set of samples of f corresponding to a regular grid of n 1 × n 2 × n 3 × n 4 × n 5 points over the domain S. In particular, the following PWL approximation f CP W L of f is considered: m 1 = m 2 = m 3 = m 4 = 6, m 5 = 3, n 1 = n 2 = 15, n 3 = n 4 = n 5 = 10. Using several

Figure 10 :Figure 11 :

 1011 Figure 10: Comparison of the system response computed with the nonlinear and CPWL models to a dilution rate u = 0.0

Figure 12 :

 12 Figure 12: Transient response of cells and glucose concentration for u=0.036 for different start-up conditions

  probi ng control duri ng fed-batch si mul ati on Feed rate [L/h]

Figure 14 :

 14 Figure 14: System response for probing control in a part of the fed-batch cultivation

  (a) If A, B ∈ B and B ⊂ A then μ(B) ≤ μ(A). (b) If (A, B ∈ B and B ⊂ A and μ(B) < ∞ then μ(A\B) = μ(A)μ(B).

Linearity:

  If f and g are Lebesgue integrable functions, and a and b are real numbers, then af + bg is Lebesgue integrable and (af + bg)dμ = a f dμ + b gdμ. Monotonicity: If f ≤ g, then f dμ ≤ gdμ.

Definition 60 (

 60 Convex Relaxation) Let f : S ⇒ R where S ⊂ R n is a nonempty convex set. Then, a convex function h : S ⇒ R is a convex relaxation of f if h(x) ≤ f (x)∀x ∈ S. Definition 61 (Convex Envelope) Let f : S ⇒ R, where S ⊂ R n is a nonempty convex set. The convex envelope of f over S (denoted f S ) is a convex relaxation such that, for any other convex relaxation h of f on S, we have f S ≥ h(x), ∀x ∈ S. The convex envelope is the tightest possible convex relaxation of a nonconvex function. The convex envelopes of many functions are known. However, in general, finding the convex envelope of an arbitrary function is as hard as finding the global minimum. On the other hand, a number of polynomial algorithms exist for constructing convex relaxations of quite general classes of functions. minimize c T x + d subject to Gx h Ax = b, where G ∈ R m×n and A ∈ R p×nSecond-order Cone ProgramA problem that subsumes both linear and quadratic programming is the secondorder cone program (SOCP):minimize f T x subject to A i x + b i 2 c T i x + d i , i = 1, ..., m F x = g, where x ∈ R n is the optimization variable, A i ∈ R n i ×n , and F ∈ R p×n .A problem which subsumes linear, quadratic and second-order cone programming is called semidefinite program (SDP), and has the formminimize c T x subject to x 1 F 1 + • • • + x n F n + G 0 Ax = b,where G, F i ∈ S k , and A ∈ R p×n . The inequality here is a linear matrix inequality.As shown earlier, since SOCP constraints can be written as LMIs, SDPs subsume SOCPs, and hence LPs as well. (If there are multiple LMIs they can be stacked into one large block diagonal LMI, where the blocks are the individual LMIs.)

  problems. It takes a proper formulation in probability measures of a non-convex optimization problem. In this way, when the problem can be stated in terms of polynomial expressions, we can transform the measures into algebraic moments to obtain a new convex program defined in a new set of variables that present the moments of every measure. In global optimization the main objective is to find the global minima of a function f defined on a subset Ω of the Euclidean space R n . In other words, we are interested in solving a mathematical program given in the general form min x∈Ω f (x),

Ωf

  (x)dμ(x) represents one point over the convex envelope of the function f . For this reason, we study the relaxed problem min μ Ω f (x)dμ(x) (67) in order to find the global minima of the objective function f in Ω. The relaxed problem (67) contains information about all the global minima of the function f in Ω. However, it cannot be solved easily in practice: Consider, for instance, the difficulty of describing all possible convex combinations of points in Ω. It can be shown how to transform problem (67) in order to make it more treatable. Since f (x)is a polynomial of degree r, the criterion f dμ involves only the moments of μ up to order m and, in addition, is linear in the moment variables. We therefore next replace μ with the finite sequence m = {m α } of all its moments, up to order r;

R

  n [x] = R[x 1 , ..., x n ] denote the space of real polynomials in n real variables, and let R n r [x] denote the polynomials of degree r at most, then d(r) = dim R n r [x] = ( n+r r ) is its dimension. For x = (x 1 , ..., x n ) ∈ R n , a monomial can then be associated to a string α = (α 1 , ..., αn) of integers α i ∈ {1, ..., n}. Then, an r-degree polynomial p(x) : R n → R n can be expressed as p(x) = |α|≤r p α x α , where

Definition 63 1 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 , x 2 3 .

 631223 the following notions of the truncated moment problem. Given a real sequence m = {m α } α∈Z n ,|α|≤N , the truncated moment problem for m concern conditions for the existence of positive Borel measure μ on R n satisfyingm α = x α dμ(x)(≡ x α 1 1 • • • x αn n dμ(x 1 , ..., x n )) (|α| ≤ N). (69)A measure μ satisfying (69) is a representing measure for m; if, moreover, K ⊆ R n is closed and suppμ ⊆ K, then μ is a K-representing measure for m. We next introduce the definitions of moment matrix and localizing matrix. Let N = 2r. In this case m corresponds to a real moment matrix M r = M n r (m) defined as follows: Moment matrix: Let P r denote the basis of monomials inR n r [x],ordered lexicographically, e.g., for n = 3, r = 2, this ordering is 1,x 1 , x 2 , x 3 ,x2 The size of M r is dim R n r [x] = ( r+n n ), with rows and columns indexed as {T i } i∈Z n ,|i|≤r , following the same lexicographic order as above.The entry of M r in row Ti , column T j is m i+j , i, j ∈ Z n , |i| + |j| ≤ 2r. Note that for n = 1, M n r (m) is the Hankel matrix (m i+j ) associated with the classical Hamburger moment problem (K = R). Another way of constructing M r (m) is as follows. For a given real sequence m = {m α } α∈N n ×N q of real numbers, the moment matrix M r (m) of order r associated with m has its rows and columns indexed in the canonical basis {x α }, and is defined by M r (m)(α, γ) = m α+γ , γ, α ∈ N n , |γ|, |α| ≤ r, (

0 m 1 | m 3 ,0 m 2 , 1 | m 4 ,0 m 3 ,1 m 2, 2 m 1 , 1 |.Definition 64

 13214321164 0 | m 2,0 m 1,1 | m 3,0 m 2,1 m 1,2 m 0,1 | m 1,1 m 0,2 | m 2,1 m 1,2 m 0,3 m 2,1 m 1,2 | m 3,1 m 2,2 m 1,3 m 0,2 | m 1,2 m 0,3 | m 2,2 m 1,3 m 0,4We next define the localizing matrix M r (θm) whose positivity is directly related to the existence of a representing measure for m with support inK = {x ∈ R[x] : θ(x) ≥ 0}.Localizing matrix: For a given polynomial θ ∈ R[x], written as θ(x) = β θ β x β , we define the localizing matrix M r (θm) associated with m, θ, and with rows and columns also indexed in the canonical basis of R[x], as M r (θm)(γ) = β θ β m (θ)+γ , γ ∈ N n , |γ| ≤ r. (71)

  1,0m 3,0m 1,2 am 0,1m 2,1m 0,3 am 1,0m 3,0m 1,2 am 2,0m 4,0m 2,2 am 1,1m 3,1m 1,3 am 0,1m 2,1m 0,3 am 1,1m 3,1m 1,3 am 0,2m 2,2m 0,4

valued polynomials, i = 1 ,

 1 ..., m. Let Ω = {x ∈ R n | g i (x) ≥ 0, i = 1, ..., m} be the feasible set. Depending on its parity, let w k = 2v k or w k = 2v k -1 be the degree of the polynomial g k (x), k = 1, ..., m. When needed below, for i ≥ max k w k , the vectors g k ∈ R d(w k ) are extended to vectors of R d(w k ) by completing them with zeros. For i ≥ max[deg(f ), max i deg(g i )], consider the positive semidefinite programSDP i ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ f i = min m α f α m α s.t. M i (m) 0, M i-d i (g k m) 0, ∀k = 1, ..., m,

TABLE OF CONTENTS

 OF DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introductory Remarks and Motivation . . . . . . . . . . . . . . . .

1.2 Contributions, Literature Review, and Outline . . . . . . . . . . . . 1.2.1 For Stability Analysis . . . . . . . . . . . . . . . . . . . . . 1.2.2 For the Optimal Control Problem . . . . . . . . . . . . . . . 1.2.3 For the Piecewise Linear Model and Control of a Bioreactor II CHAPTER I INTRODUCTION

  are the state variables of the original system, x r = (x n+1 , ..., x n+m ) are the new variables introduced in the recasting process, p o (x o , x r , s) and p r (x o , x r , s) have polynomial forms. Previously, in the recasting process we have also obtained new polynomial constraints. Consider the real-valued polyno-

mial g k (x o , x r , s), with k = 1, ..., m, where m is the number of polynomial constraints generated in the recasting process. Let

  In other words, we are interested in solving a mathematical

	program given in the general form
	min s∈Ω	H(x, s),
	where the objective function H(x, s) is a linear combination of simpler functions.
	One approach to tackle this problem comes from convex analysis, since we can use
	the convex envelope of the function H in order to locate its global minima. As we
	have shown, every convex combination of points in Ω can be described as a discrete
	probability distribution μ supported in Ω, such that every integral

we consider the global minimization of H as a global optimization problem, it is well known that the main objective is to find the global minima of a function H defined on a subset S. Ω H(x, s)dμ(η) represents one point over the convex envelope of the function H. For this reason, we study the relaxed problem min μ∈P (Ω) Ω

  The first element of the basis is a constant term γ 0 (1). The remaining elements are formed by the composition of the function γ k (•, ..., •), k = 1, 2, ..., m, with the linear functions

	γ(x 1 , x 2 ), and in general
			γ k (x 1 , ..., x k ) = γ(x 1 , γ k-1 (x 2 , ..., x k ))	(62)
	are defined. A distinctive property of (62) is that it possesses k nesting of absolute
	value functions, and accordingly, it is said to have nesting level (n.l) equal to k.
	x
	0	1	2	3	4	5	6	7	8	x 1
			Figure 9: Simplicial partition and grid size
							78

Table 1 :

 1 Start-up Conditions for Transient Analysis

		S 1 (Glucose g/L) S 2 (Glutamine g/L)
	1	1 . 2	2 . 6
	2	4 . 2	4 . 2
	3	7 . 2	2 . 6
	4	4 . 2	0 . 6
	5	7 . 2	4 . 2

  1/p . Theorem 52 Let Ω ⊂ R n be a measurable set and let u j : Ω → R m be measurable = {ν x } x∈Ω (the associated parametrized measure), depending measurably on x, with this property: Whenever the sequence {ψ(x, u j (x))} is weakly convergent in L 1 (Ω) for any Carathéodory function ψ(x, λ) : Ω × R m → R

	functions such that
	sup	ψ(|u j |)dx < ∞,
	j	Ω
	where ψ : [0, ∞) → [0, ∞] is a continuous, nondecreasing function such that
	lim t →∞ ψ(t) = ∞. There exists a subsequence, not relabeled, and a family of probabil-
	ity measures, ν

*

, which implies that this function is measurable in x, continuous in t, and has a bounded Lebesgue integral, then the weak limit is the (measurable) function

ACKNOWLEDGEMENTS

First and foremost, I would like to express deep gratitude towards my advisors, Alain Gauthier, Naly Rakoto, Jean-Jacques Loiseau, and Nicanor Quijano. Alain, for his guidance and support all these years; I will be forever thankful for this opportunity.

Naly, for providing a unique working environment that I mostly enjoyed and also for his encouragement during the time at EMN. Jean-Jacques, for letting me be part of the research group ACSED under his direction, and also for his many interesting advices. At last but not the least, my greatest appreciation goes to my closest advisor, Nicanor. His advice and friendship lead me through the life of a graduate student; Nick made my Uniandes experience terrific, creating numerous opportunities for me to learn and grow as a researcher. I also thank the other members of my doctoral committee: Peter Caines, Didier Henrion and Pierre Riedinger, for their encouragement, generosity, and insightful discussions. Furthermore, I am grateful to Rene Meziat for our long conversations, many of which found their way into the pages of this thesis.

To all my friends at Uniandes: Andrés Pantoja, Iván muñoz, and the GIAP group gang, thank you for the Café and Coffee evenings-you have been great. Furthermore, I thank my closest friends at EMN, Yonatan and Chio, for their noble hearts. I also want to mention Roberto Bustamante and Ángela Cadena, professors at Uniandes, for their good humor and inspiration.

Finally, I am forever grateful to my parents, Alirio and Cecilia, and the rest of my family for their constant encouragement, confidence, friendship, and unconditional love for all these years. And also, I want to express my endless love to Carito, "mi mitis", who even though was for almost a year 8,960 km away, always waited for me.

V = 0.892 + 0.82x 2 2 -2.596x 4 + 2.596x 2 3 + 1.704x 2 4 , which in the original variables is V = 0.892 + 0.82ξ 2 2 -2.596 cos(ξ 1 ) + 1.704 sin 2 (ξ 1 ) + 1.704 cos 2 (ξ 1 ).

Therefore, using Proposition 11, we can show that the origin is asymptotically stable.

localizing matrices, we set the problem in moment variables obtaining the positive semidefinite programs (LP i ). Solving the (LP i ) programs in time, with a relaxation order i = 2 and the algorithm SOCP, we obtain an optimal value of H * = -8.547, and the moment sequence which allows us to calculate the switching signal and the trajectories. Figure 4 shows the trajectories, the co-state, and the switching signal obtained for a relaxation of order i = 2 solution of the corresponding optimal control problem. In Figure 5 a phase plane of the system trajectories with the switching signal is depicted. The points where the two subsystems switch between them are clear, and eventually the switched system reach the stable point (0, 0). This numerical example allow us to confirm that the first moment, i.e., m 001 , is equivalent to the polynomial variable s, and therefore with the switching signal σ(t) as it is stated in Theorem 27.

Extension Results to More General Nonlinear Optimal Control Problems

In this section we will omit the Assumption 23, and let the state equations and the running costs be of a more general class. Namely, we deal with elementary and nested elementary functions. This class of functions is related with explicit symbolic derivatives, for instance, exponential, logarithmic, power, trigonometric, and hyperbolic functions, among others. We have mentioned that a very large class of non-polynomial nonlinearities can be converted into polynomial systems. This process, introduced in [START_REF] Savageau | Recasting Nonlinear Differential Equations as S-systems: A Canonical Nonlinear Form[END_REF], and later used for stability analysis in [START_REF] Papachristodoulou | Positive Polynomials in Control, ch. Analysis of Non-polynomial Systems Using the Sum of Squares Decomposition[END_REF] and in Chapter 2, is based on the recasting process of elementary and nested elementary functions. In what follows we will describe the recasting process and how can it be used in optimal control problems of switched systems.

where x 1 is the angular position with respect to the origin, at the upright position, and x 2 is the velocity. Considering stabilization conditions, it has been shown that we have to take F negative in some regions, and positive in some others, so that the system minimizes the energy consumption all the time. Therefore, this is equivalent to a switched system, where f 0 is defined by equation ( 55) when F is negative, and

First, we use Theorem 1 to obtain an equivalent continuous representation of the switched model related to [START_REF] Lin | Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results[END_REF] for the pendulum, ξ(t

We obtain the following equivalent continuous system:

Now, following the recasting process, it is clear that ( 56) is in the same form as [START_REF] Li | Control parameterization enhancing transform for optimal control of switched systems[END_REF], but in this case the elementary functions are trigonometric functions. Let us follow Step 2 to Step 4 in the recasting process. As a result, we obtain a new set of differential equations given by

As we know by Step 3 of the recasting process, a set of constraints arises when we introduce the new variables x 3 and x 4 . For this case, we have that the manifold on which the solutions to the original system (56) lie is given by Ω

Using this reformulation we recast the nonlinear optimal control problem in an equivalent polynomial problem. Then, this reformulation allows us to apply the positive semidefinite relaxation using the theory of moments. We want to design a control law in such a way that the closed-loop energy presents a minimum at the desired position, and the energy controller is globally defined. Since the chosen target energy has other minima different from the desired equilibrium, a combination of energy dissipation (damping) and injection (pumping) is needed in order to globally stabilize the origin.

Suppose that we start the pendulum at the position x 0 = (-π, 0.5). Previously we have obtained a recasted system for the pendulum in equation ( 52), constrained by the set Ω 2 = {x 2 3 + x 2 4 -1 = 0}. We define the set for the Young measures D. In this case, the constrained sets Ω 1 and Ω 2 are redefined as

we therefore obtain the constrained set

The degree of the equivalent polynomial system is the highest degree of the field plus the running cost, i.e., r = 2. The number of variables is the number of states obtained in the recasting process plus s, i.e., n = 5. Hence, we obtain with this data the basis in a lexicographical order, i.e., 1,

x 2 s, x 2 3 , x 3 x 4 , x 3 s, x 2 4 , x 4 s, s 2 , which leads to the moment and localizing matrices. We recall that moment and localizing matrices have the rows and columns indexed in the previous basis of polynomials. We obtain a Hamiltonian in moment sequences and co-states H(m, λ). We obtain a relaxed Maximum Principle (RMP) conditions for the moment sequence m = {m β,k } to be optimal, and we obtain the positive semidefinite programs GLP i . With sets defined above we obtain the moment matrix with i ≥ max deg = 2, M 2 (m), and the localizing matrices, with

, and M 1 (g 2 m). Using the set D and moment and localizing matrices we set the problem in moment variables obtaining the positive semidefinite programs (GLP i ). Solving the (GLP i ) programs in time, with a relaxation order parameters c that is the solution of the least square problem min x |Ax -b| 2 , where A = Υ T (X), X is the input matrix, and b is the output to be approximated in sparse format. In accordance with [START_REF] Julián | Orthonormal high level canonical pwl functions with applications to model reduction[END_REF], the CPWL approximation of the nonlinear function g is defined as the function

Equation ( 65) presents the CPWL approximation for the nonlinear system.

Analysis of CPWL Approximation: Error Estimation

In this section, the error estimation between the nonlinear system and the CPWL approximation ( 65) is presented. It can be noticed that it is only necessary to know the values of f at the vertices because, as stated above, this is all the information needed to uniquely define a function belonging to PWL[S H ].

If the function f is continuous, the following results quantify the precision of the approximation:

Lemma 32 If f is continuous in S, which is the union of nonoverlapping simplices, then

Moreover, if the function is assumed to be Lipschitz continuous in S, a useful relationship between the approximation error and the grid size δ can be obtained.

Lemma 33

Let f : S → R 1 be a function satisfying where C reac is the reaction amplitude, and which should be chosen large enough for the algorithm to be unaffected by the background variability in C pulse . A value in 3 -5%, is a reasonable default choice.

Figure 14 shows a simulation result with a proportional controller, according to the description above. The oxygen concentration, the variable of the process, and the cell concentration are shown. The figure thus depicts the exponential cell growth by simulation.

MATHEMATICAL BACKGROUND

In this appendix we will present some of the mathematical ideas, concepts, and definitions that are employed in this thesis. They are based mainly on [START_REF] Ben-Tal | Lectures on Modern Convex Optimization[END_REF], [START_REF] Berg | Moment Problems and Polynomial Approximation[END_REF], [START_REF] Curto | The Truncated Complex K-moment Problem[END_REF], [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], [START_REF] Lasserre | An explicit equivalent positive semidefinite program for nonlinear 0-1 programs[END_REF], [START_REF] Meziat | The method of moments in global optimization[END_REF], [START_REF] Rao | Measure Theory and Integration[END_REF], [START_REF] Pedregal | Parametrized Measures and Variational Principles[END_REF], [START_REF] Muñoz | A Refinement on Existence Results in Nonconvex Optimal Control[END_REF]. This appendix is a brief compilation of notions of measure theory and integration, probability theory, convex optimization, and optimization over polynomials using the theory of moments.

A.1 Brief Introduction to Measure Theory and Integration

Measure theory provides a way to extend our notions of length, area, volume, etc. to a much larger class of sets (but not all of its applications have to do with physical sizes).

Informally, given some base set, a measure is any consistent assignment of sizes to the subsets of the base set. Depending on the application, the size of a subset may be interpreted as its physical size, the amount of something that lies within the subset, or the probability that some random process will yield a result within the subset. The main use of measures is to define general concepts of integration over domains with a more complex structure than the intervals of the real line. Such integrals are used extensively in probability theory and in much of mathematical analysis. In order to be able to make use of measures and integrals, we need to know that the class of measurable sets is closed under certain types of operations.

A.1.1 Systems of Sets

Our universe is denoted by Ω, i.e., all the sets we shall consider are subsets of Ω.

Recall some standard notation. 2 Ω everywhere denotes the set of all subsets of a given set Ω. The complement (in Ω) of a set A is denoted by A c . By A B the symmetric difference of A and B is denoted, i.e. A B = (A\B) ∩ (B\A).

Definition 34 A ring of sets is a non-empty subset in 2 Ω which is closed with respect to the operations ∩ and \.

Definition 35 A semi-ring is a collection of sets A ⊂ 2 Ω with the following properties:

2. For every A, B ∈ A there exists a finite disjoint collection (C j ) j = 1, 2, ..., n of In other words, it is the intersection of all σ-algebras of sets containing U. Some important measures. We are mainly interested in three important measures.

(a) The Lebesgue measure on R is the measure on a σ-algebra containing the inter-

(b) The probability measure. μ(Ω) = 1, i.e. takes the value 1 on the whole space.

(c) The Dirac measure μ A . The measure of a set is 1 if it contains the point A, and 0 otherwise.

A.1.2.2 The Lebesgue Measure

Bounded Sets of R. Let B be the algebra of all finite unions of semi-segments (semi-intervals) on R 1 , i.e. all sets of the form

The Lebesgue measure μ is a mapping μ : B → R defined by:

A.1.2.3 The Probability Measure

The probability measure is a function from B σ-algebra to the real numbers that assigns to each event a probability between 0 and 1. It must satisfy μ(Ω) = 1.

Because the probability measure is a function on B and not on Ω, the set of events is not required to be the complete 2 Ω of the sample space; that is, not every set of outcomes is necessarily an event.

A.1.2.4 The Dirac Measure

A Dirac measure is a measure δ x on a set Ω (with any σ-algebra of subsets of Ω) that gives the singleton set {x} the measure 1, for a chosen element x ∈ Ω : δ x ({x}) = 1.

In general, the measure is defined by

The Dirac measure is a probability measure, and it represents the almost sure outcome x in the sample space Ω. Dirac measures are the extreme points (in a convex set S, if a point in S which does not lie in any open line segment joining two points of S, intuitively, is a "corner" of S) of the convex set of probability measure on Ω.

A.1.3 Integration

Let Ω be a set, B is a σ-algebra of subsets of Ω and μ is a measure on it.

Definition 45 A triple (Ω, B, μ) is called a measure space.

The most important example of a measure space is the Lebesgue measure space.

A.1.3.1 Lebesgue Integration

The integral of a function f between limits a and b can be interpreted as the area under the graph of f . This is easy to understand for familiar functions such as

A.2 Some Results on Probability Theory

The definition of the probability space is the foundation of probability theory. It was introduced by Kolmogorov in the 1930s.

(c) μ is countably additive:

The sample space Ω is a nonempty set whose elements are known as outcomes or states of nature and are often given the symbol ω. The set of all possible outcomes of an experiment is known as the sample space of the experiment.

B is a σ-algebra of subsets of Ω. Its elements are called events, which are sets of outcomes for which one can ask a probability. The complement of any event is an event, and the union of any (finite or countable infinite) sequence of events is an event. Usually, the events are the Lebesgue-measurable or Borel-measurable sets of real numbers.

The probability measure μ is a function from B to the real numbers that assigns to each event a probability between 0 and 1. Because μ is a function defined on B and not on Ω, the set of events is not required to be the complete power set of the sample space; that is, not every set of outcomes is necessarily an event. Measure theory gives us a way to deal simultaneously with continuous, discrete, and mixed distributions in a unified way.

One more lemma: Existence It was realized that the problem of moments is closely connected to Hilbert spaces and spectral theory. In more concrete terms, there is a condition on a positive measure μ, namely that

The convex hull coC is always convex, and is the smallest convex set that contains C.

The idea of a convex combination can be generalized to include infinite sums, integrals, and, in the most general form, probability distributions.

Suppose

and that

More generally, suppose p : R n → R satisfies p(x) ≥ 0, for all x ∈ C, and

if the integral exists. In the most general form, suppose C ⊂ R n is convex and x is a random vector with x ∈ C with probability one. Then the expectation E(x) ∈ C. Indeed, this form includes all the others as special cases. For example, suppose the random variable x only takes on the two values x 1 and x 2 , with prob(x = x 1 ) = θ and prob

and we are back to a simplex convex combination of two points.

A.3.2 Convex Functions Definition 57 A function

Definition 58 The epigraph of a function

From the basic definition of convexity, it follows that f is a convex function if, and only if its epigraph epif is a convex set.

The convexity of a differentiable function f : R n → R can also be characterized by conditions on its gradient ∇f and Hessian ∇ 2 f . Recall that, in general, the gradient yields a first order Taylor approximation at x 0 .

The first-order condition: f is convex if and only if for all x, x 0 ∈ domf ,

i.e., the first order approximation of f is a global underestimator.

Remember also that the Hessian of f , ∇ 2 f , yields a second order Taylor series expansion around x 0 . We have the necessary and sufficient second-order condition: a twice differentiable function f is convex if and only if, for all x ∈ domf, ∇ 2 f (x) 0, i.e., its Hessian is positive semidefinite on its domain.

Elementary properties helpful in verifying convexity:

(a) nonnegative sums of convex functions are convex;

(b) nonnegative infinite sums, integrals: p(y) ≥ 0, g(x, y) convex in x ⇒ p(y)g(x, y)dy convex; and (c) expected value: f (x, u) convex in x ⇒ g(x) = E u f (x, u) convex. In this way, we know that convex optimization problems have three crucial properties that makes them fundamentally more tractable than generic nonconvex optimization problems:

Jensen's inequality: If

(a) no local minima: any local optimum is necessarily a global optimum;

(b) exact infeasibility detection: using duality theory, hence algorithms are easy to initialize;

(c) efficient numerical solution methods.

There are several canonical optimization problem formulations, for which extremely efficient solution codes are available. Thus, if a real problem can be cast into one of these forms, then it can be considered as essentially solved.

The most important category of these canonical problems is known as Conic programming, and it is called conic because the inequalities are specified in terms of affine functions and generalized inequalities. Geometrically, the inequalities are feasible if the range of the affine mapping intersects the cone of the inequality.

Linear Program

A general linear program (LP) has the form