Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr.

Michael Malisoff
email: malisoff@lsu.edu.

Stabilization of Nonlinear Time-Varying Systems through a New Prediction Based Approach

Keywords: Delay, robustness, stability, time-varying. F. Mazenc

We propose a prediction based stabilization approach for a general class of nonlinear time-varying systems with pointwise delay in the input. It is based on a recent new prediction strategy, which makes it possible to circumvent the problem of constructing and estimating distributed terms in the expression for the stabilizing control laws. Our result applies in cases where other recent results do not.

I. INTRODUCTION

Constructing delay tolerant feedback controls is a central problem that is of compelling ongoing research interest. Many important systems with input delays are time-varying. For instance, the problem of tracking trajectories leads to challenging problems for time-varying systems with input delays, as explained, e.g., in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF]. See also [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF], and [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] for surveys on delay compensating control.

There are several approaches to designing delay tolerant controls. One involves solving the problem with the input delay set to zero, and then searching for upper bounds on the input delays that the closed loop system can tolerate while still realizing the desired goal. This often involves Lyapunov-Krasovskii functionals (as discussed in [START_REF] Fridman | On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays[END_REF] and [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]), which often lead to satisfactory results when the delay is small; see [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF]. However, many applications have long input delays, such as communication networks or multi-agent problems where the agents are geographically dispersed. In general, stabilization under long input delays needs control designs that use the value of the input delay, and in many cases, distributed delays are used, meaning the control uses all values of the state or input along some interval of past times; see [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], [START_REF] Bekiaris-Liberis | Compensation of time-varying input and state delays for nonlinear systems[END_REF], [START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear timeinvariant systems[END_REF], and [START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF].

In this note, we revisit the problem of applying a prediction based approach to construct globally asymptotically stabilizing control laws for time-varying systems. We adapt the fundamental new approach of [START_REF] Najafi | Closedloop control of dead time systems via sequential sub-predictors[END_REF], which was limited to time-invariant linear systems that satisfy certain matrix inequalities. See also [START_REF] Léchappé | Predictive Control and Estimation of Uncertain Systems with an Input Delay[END_REF], where extensions to linear systems with time-varying delays and approximately known delays are provided. This approach differs from the classical reduction model approach or the prediction based approaches introduced by M. Krstic (as in [START_REF] Bresch-Pietri | Robust compensation of a chattering timevarying input delay[END_REF], [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF], and [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]) which also involve distributed terms.

We use several dynamic extensions, making it possible to obtain a prediction of the state variable without using distributed terms. Many contributions, including [START_REF] Ahmed-Ali | Robust stabilization of nonlinear globally Lipschitz delay systems[END_REF] and [START_REF] Ahmed-Ali | Global exponential sampled-data observers for nonlinear systems with delayed measurements[END_REF], use several dynamic extensions to carry out state prediction, but to the best of our knowledge, they do not apply to the problem we consider here and they use distributed terms. Our prediction stabilization technique applies to nonlinear Lipschitz systems, which is also the case for many prediction ones, e.g., [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF]. We obtain an explicit result, which applies in cases where earlier results such as [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] do not apply. Our work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] covered time-varying linear systems, using distributed terms, and included an application to a linearized pendulum tracking dynamics. To illustrate the value added by our new approach, we apply our new theorem to examples from [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], including a pendulum tracking control that does not use linearization and that does not require any distributed terms.

Prediction without distributed terms has been pursued in several significant works that do not address the challenges that we overcome here. For instance, [START_REF] Ahmed-Ali | Cascade high gain predictors for a class of nonlinear systems[END_REF], [START_REF] Cacace | Memoryless approach to the LQ and LQG problems with variable input delay[END_REF], [START_REF] Cacace | Optimal control of linear systems with large and variable input delays[END_REF], and [START_REF] Zhou | Truncated predictor feedback for linear systems with long time-varying input delays[END_REF] were limited to linear timeinvariant systems ẋ = Ax+Bu, under addition eigenvalue conditions on A and controllability conditions or bounds on the delays, without robustness to uncertainty; [START_REF] Cacace | Stabilization of strict-feedback nonlinear systems with input delay using closed-loop predictors[END_REF] covers strict feedback systems under conditions on coefficient matrices in a new system that is obtained after a diffeomorphic transformation, which we do not require here; [START_REF] Cacace | Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment[END_REF] used partial spectrum assignment to achieve prescribed decay rates for linear time-invariant systems; [START_REF] Cacace | Predictor-based control of linear systems with large and variable measurement delays[END_REF] was also confined to linear time-invariant systems; [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF] covered nonlinear systems under a globally drift-observability condition that we also do not require; and [START_REF] Zhou | Pseudo-predictor feedback stabilization of linear systems with time-varying input delays[END_REF] and [START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF] cover time-varying linear systems and give sufficient conditions for stabilizability under pseudo-predictor feedback using an integral delay system that is not needed here.

II. DEFINITIONS AND NOTATION

Throughout the paper, we omit arguments of functions when they are clear, and the dimensions of the Euclidean spaces are arbitrary unless otherwise noted. The usual Euclidean norm of vectors, and the induced norm of matrices, are denoted by | • |. We use |φ|I to denote the usual essential supremum of a function φ over any interval I contained in its domain. Consider any system of the form

ẏ(t) = G(t, y(t), U (t -h), µ(t)), (1)
where the state y, the control U , and the unknown measurable locally essentially bounded perturbation µ are valued in any Euclidean spaces R n 1 , R n 2 , and R n 3 , respectively, and h is a constant delay. Due to the delay, its solutions are defined for given initial times t0 ≥ 0, initial functions that are defined on [t0 -h, t0], and choices of the function µ. We assume that (1) is forward complete, meaning all such solutions are uniquely defined on [t0 -h, ∞); see below for assumptions that ensure the forward completeness property. One desirable property for (1) is input-to-state stability (or ISS, which we also use to mean input-to-state stable); see, e.g., [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] for discussions on ISS for systems without delays. The ISS definition for delay systems can be expressed in terms of the standard classes KL and K∞ of comparison functions, defined, e.g., in [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF]Chapt. 4]. By ISS of (1), we mean that there are functions β ∈ KL and γ ∈ K∞ such that for all initial times t0, all initial functions, and all choices of the perturbations µ, the corresponding solutions of (1) all satisfy |y(t)| ≤ β(|y| [t 0 -h,t 0 ] , t -t0) + γ(|µ| [t 0 ,t] ) for all t ≥ t0. This becomes the usual uniform global asymptotic stability property in the special case where there are no perturbations µ in [START_REF] Ahmed-Ali | Cascade high gain predictors for a class of nonlinear systems[END_REF]. We use N to denote the natural numbers {1, 2, . . .}. For any subsets S1 and S2 of Euclidean spaces, a function W : S1 × S2 → R n is called locally Lipschitz in its second variable uniformly in its first variable provided for each constant R > 0, there is a constant LR > 0 such that |W (s1, sa) -W (s1, s b )| ≤ LR|sa -s b | holds for all s1 ∈ S1 and all sa and s b in S2 for which max{|sa|, |s b |} ≤ R.

III. STATEMENT OF MAIN RESULT

Consider the system

ẋ(t) = f (t, x(t), u(t -h), δ(t)), (2)
where the state x, the control u, and the unknown measurable locally essentially bounded perturbation δ are valued in R n , R c , and R d respectively, and h > 0 is a constant delay. We design the control u to prove stability properties for this system when the delay h is known and arbitrarily large. We introduce these two assumptions, the first of which will be used to ensure forward completeness of (2): Assumption 1: The function f is continuous, satisfies f (t, 0, 0, 0) = 0 for all t ≥ 0, and admits a constant k > 0 such that for all t ≥ 0 and U ∈ R c , the inequality

|f (t, z1, U, ∆1) -f (t, z2, U, ∆2)| ≤ k|z1 -z2| + k|∆1 -∆2| (3) holds for all z1 ∈ R n , z2 ∈ R n , ∆1 ∈ R d , and ∆2 ∈ R d .
Assumption 2: There exists a continuous function us : [0, ∞) × R n → R c that is locally Lipschitz in its second variable uniformly in its first variable such that the system

ẋ(t) = f (t, x(t), us(t, x(t) + (t)), δ(t)) (4)
is ISS with respect to the combined disturbance (ε, δ). Before stating our theorem, we first convey the basic ideas of our method, which has two steps. First, we use a closed-loop system, which allows us to replace classical predictors with distributed terms by a new control that is based on a dynamic extension, and that exploits knowledge of future inputs using the delayed feedback property of the input. Second, since the delay h is finite, we use a chain of predictors to apply the method, to cover systems with arbitrarily long delays h. Letting f0(t, x, u) = f (t, x, u, 0), we prove the following (but see Section V for additional assumptions that lead to a standard ISS estimate for our closed loop input delayed system):

Theorem 1: Let (2) satisfy Assumptions 1-2 and m ∈ N satisfy m > 11.4kh.

Choose any constant λa ∈ (0, 1) such that

m > h (2/k)(1 + λa)(4k + λa) 3/2 . ( 6
)
Then there are functions β d ∈ KL and γ d ∈ K∞ such that all solutions x(t) of the system (2), in closed loop with

u(t) = us(t + h, zm(t)), (7)
where zm denotes the last n components of the state of the system

                         ż1(t) = f0 t + h m , z1(t), Φ(t, zm, 1) -(4k + λa)[z1 t -h m -x(t)] ż2(t) = f0 t + 2h m , z2(t), Φ(t, zm, 2) -(4k + λa) z2 t -h m -z1(t) . . . żm(t) = f0 (t + h, zm(t), Φ(t, zm, m)) -(4k + λa) zm t -h m -zm-1(t) (8)
where

Φ(t, zm, i) = us(t + h -h(m -i)/m, zm(t -h(m -i)/m))
for all t ≥ 0 and i ∈ {1, 2, . . . , m}, satisfy

|x(t)| ≤ β d |x| [t0-h,t0+ h m ] + |z| [t0-h,t0+ h m ] , t -t0 + γ d |δ| [t 0 ,t] (9)
for all t0 ≥ h m and all t ≥ t0. Remark 1: Notice that we can write Φ(t, zm, i) = u(t -h(mi)/m) for all t and i, where u is defined by [START_REF] Bresch-Pietri | Robust compensation of a chattering timevarying input delay[END_REF]. A key difference between Theorem 1 and our prior work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] is that Theorem 1 provides a control with no distributed terms that applies to a broad class of time-varying nonlinear systems, while [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] is limited to timevarying linear systems and uses distributed terms. While [START_REF] Mazenc | Local stabilization of nonlinear systems through the reduction model approach[END_REF] applies to more general nonlinear systems that are not necessarily globally Lipschitz, the controls in [START_REF] Mazenc | Local stabilization of nonlinear systems through the reduction model approach[END_REF] use distributed terms and only ensure local stability, which is weaker than the global results we prove here.

Remark 2: Our condition t0 ≥ h/m is used for our Lyapunov-Krasovskii analysis, and is purely technical. It can be relaxed, but for the sake of simplicity and space constraints, we do not modify it.

IV. PROOF OF THEOREM 1

Throughout the proof, all equalities and inequalities are for all t ≥ 0 and along all trajectories of the closed loop system, unless otherwise indicated. Set p = 4k + λa, z0 = x, and ei(t) = zi(t)zi-1 (t + h/m) for all i ∈ {1, ..., m}. Our definition of f0 gives

ė1(t) = -pe1 t -h m + f0 t + h m , z1(t), u t -h(m-1) m -f t + h m , x t + h m , u t -h(m-1) m , δ t + h m . (10)
Similarly, since there are no disturbances δ in (8), we get

ėi(t) = -pei t -h m + f0 t + i h m , zi(t), u t -h(m-i) m -f0 t + i h m , zi-1 t + h m , u t -h(m-i) m + pei-1(t) (11)
when i > 1. We first analyze the e1-subsystem [START_REF] Cacace | Stabilization of strict-feedback nonlinear systems with input delay using closed-loop predictors[END_REF].

The Fundamental Theorem of Calculus gives

ė1(t) = -pe1 (t) + p t t-h m ė1( )d +f0 t + h m , z1(t), u t -h(m-1) m -f t + h m , x t + h m , u t -h(m-1) m , δ t + h m . (12)
Let ν(e1) = 1 2 |e1| 2 . Then Assumption 1 and our formula for f0 give

ν(t) ≤ -p|e1(t)| 2 + p t t-h m e1(t) ė1( )d + |e1(t)|k z1(t) -x t + h m + δ t + h m = (k -p)|e1(t)| 2 + p t t-h m e1(t) ė1( )d + k|e1(t)| δ t + h m . (13)
From Young's inequality, it follows that

ν(t) ≤ (k -p)|e1(t)| 2 + k|e1(t)| δ t + h m + p t t-h m m 2h |e1(t)| 2 + h 2m | ė1( )| 2 d = k -1 2 p |e1(t)| 2 + ph 2m t t-h m | ė1( )| 2 d + k|e1(t)| δ t + h m . (14)
Next observe that [START_REF] Cacace | Stabilization of strict-feedback nonlinear systems with input delay using closed-loop predictors[END_REF] gives the following for all ≥ 0:

| ė1( )| ≤ p e1 -h m + f0 + h m , z1( ), u -h(m-1) m -f + h m , x + h m , u -h(m-1) m , δ + h m ≤ p e1 -h m + k|e1( )| + k|δ + h m |, by Assumption 1.
Hence, Young's Inequality gives this for all ≥ 0:

| ė1( )| 2 ≤ 2p 2 |e1( -h/m)| 2 + 2k 2 |e1( )| 2 + |δ( + h/m)| 2 +2|e1( )||δ( + h/m)|) ≤ 2p 2 |e1( -h/m)| 2 + 2k 2 (1 + λa/4) |e1( )| 2 +(1 + 4/λa)|δ( + h/m)| 2 .
Therefore, [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF] gives the following for all t ≥ h m :

ν(t) ≤ k -1 2 p |e1(t)| 2 + p 3 h m t-h/m t-2h/m |e1( )| 2 d + phk 2 m (1 + λa/4) t t-j/m |e1( )| 2 d + phk 2 m 1 + 4 λa t+h/m t |δ( )| 2 d + k|e1(t)||δ(t + h/m)|. (15)
Since Young's Inequality also gives

k|e1(t)||δ(t + h/m)| ≤ λa 2 |e1(t)| 2 + k 2 2λa |δ(t + h/m)| 2 , (16)
we can use [START_REF] Khalil | Nonlinear Systems, Third Edition[END_REF] to upper bound the last term in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] and use our choice p = 4k + λa and finally our choice of ν to get

ν(t) ≤ -k|e1(t)| 2 + (4k + λa) 3 h m t-h/m t-2h/m |e1( )| 2 d +(4k + λa)k 2 (1 + λa/4) h m t t-h/m |e1( )| 2 d + M |δ| 2 [t,t+h/m] ≤ -2kν(e1(t)) + 2h m (4k + λa) 3 t t-2h/m ν(e1( ))d + M |δ| 2 [t,t+h/m] , (17)
where

M = k 2 /(2λa) + (phk 2 /m) (1 + 4/λa) (h/m). Let µ(e1,t) = ν(e1(t)) + 2h(1+λa)(4k+λa) 3 m t t-2 h m t s ν(e1( ))d ds.
Here and in the sequel, ei,t is the restriction of ei to [t -2h/m, t] for all t ≥ t0 and i ∈ {1, 2, . . . , m}. Then, for all t ≥ h/m,

d dt µ(e1,t) ≤ -2kν(e1(t)) + M |δ| 2 [t,t+h/m] + 2h m (4k + λa) 3 t t-2 h m ν(e1( ))d + 2h(1+λa) m ν(e1(t)) -2h(1+λa) m (4k + λa) 3 t t-2 h m ν(e1( ))d . (18)
Consequently,

d dt µ(e1,t) ≤ 2k -1 + 2h 2 m 2 k (4k + λa) 3 (1 + λa) ν(e1(t)) -2h m λa(4k + λa) 3 t t-2 h m ν(e1( ))d + M |δ| 2 [t,t+h/m] . (19)
Therefore, our lower bound on m from (6) combined with µ(e1,t) ≤ ν(e1(t))

+ 4h 2 m 2 (1 + λa)(4k + λa) 3 t t-2h/m ν(e1( ))d provides a constant 0 > 0 such that, for all t ≥ h/m, d dt µ(e1,t) ≤ -0 µ(e1,t) + t t-h/m |e1( )| 2 d + M |δ| 2 [t,t+h/m]
holds along all trajectories of the e1 dynamics.

Similarly, since there are no δ's in the z system, [START_REF] Cacace | Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment[END_REF] and the relation 2rs ≤ λar 2 /4 + 4s 2 /λa for all r ≥ 0 and s ≥ 0 give

| ėi( )| 2 ≤ 2p 2 |ei( -h/m)| 2 + 2 (k|ei( )| + p|ei-1( )|) 2 ≤ 2p 2 |ei( -h/m)| 2 + 2k 2 (1 + λa/4)|ei( )| 2 + 2p 2 (1 + 4/λa)|ei-1( )| 2 (20)
for any i ∈ {2, 3, . . . , m}, which implies that the function

µ(ei,t) = 1 2 |ei(t)| 2 + 2h m (1 + λa)(4k + λa) 3 t t-2 h m t s ν(ei( ))d ds (21)
satisfies the following for all t ≥ h/m:

d dt µ(ei,t) ≤ -0 μ(ei,t) + p|ei(t)||ei-1(t)| + p 3 h m (1 + 4/λa) t t-h/m |ei-1( )| 2 d ≤ -0 μ(ei,t) + 0 4 |ei(t)| 2 + p 2 0 |ei-1(t)| 2 + p 3 h m (1 + 4/λa) t t-h/m |ei-1( )| 2 d ≤ -0 2 μ(ei,t) + p 2 0 |ei-1(t)| 2 + p 3 h m (1 + 4/λa) t t-h/m |ei-1( )| 2 d , (22)
where μ(ei,t) = µ(ei,t)

+ t t-h/m |ei( )| 2 d . (23)
Therefore, the function

µ (et) = m i=1 4 0 1 + p 2 0 + 1 + 4 λa p 3 h 2m m-i µ(ei,t)
where et = (e1,t, e2,t, . . . , em,t) admits a constant ¯ > 0 such that

d dt µ (et) ≤ -¯ µ (et) + M 4 0 1 + p 2 0 + 1 + 4 λa p 3 h 2m m-1 |δ| 2 [t,t+h/m] (24)
holds for all t ≥ h/m along all trajectories of the closed loop system. For instance, when m > 2, we get

d dt µ(em,t) + 4 0 1 + p 2 0 + 1 + 4 λa p 3 h 2m µ(em-1,t) ≤ -0 2 μ(em,t) + p 2 0 |em-1(t)| 2 + 1 + 4 λa p 3 h m t t-h/m |em-1( )| 2 d -0 2 μ(em-1,t) 4 0 1 + p 2 0 + 1 + 4 λa p 3 h 2m + 4 0 1 + p 2 0 + 1 + 4 λa p 3 h 2m p 2 0 |em-2(t)| 2 + 1 + 4 λa p 3 h m t t-h/m |em-2( )| 2 d ≤ -0 2 μ(em,t) -0 2 μ(em-1,t) 4 0 + 4 0 1 + p 2 0 + 1 + 4 λa p 3 h 2m p 2 0 |em-2(t)| 2 + 1 + 4 λa p 3 h m t t-h/m |em-2( )| 2 d , (25)
by using the formula for μ(em-1,t) to cancel the terms in curly braces in [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF]. Then [START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear timeinvariant systems[END_REF] follows from a similar induction argument. Take any t0 ∈ [h/m, ∞). Using [START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear timeinvariant systems[END_REF], an integrating factor, and the quadratic upper and lower bounds on ν, and setting e = (e1, e2, . . . , em), we can find βa ∈ KL and γa ∈ K∞ such that

|e(t)| ≤ βa(|e| [t 0 -h,t 0 ] , t -t0 + h) + γa(|δ| [t 0 ,t+h/m] ) (26)
holds along all solutions of the closed loop system when t0 ≥ h/m. Next observe that zm(t) = em(t) + zm-1 (t + h/m), zm-1(t) = em-1(t) + zm-2 (t + h/m), ..., and z1(t) = e1(t) + x t + h m all hold. Consequently, we can prove (e.g., by induction on m) that

zm(t) = m-1 l=0 e m-l t + l h m + x(t + h) . (27)
Therefore, (2) in closed loop with the control [START_REF] Bresch-Pietri | Robust compensation of a chattering timevarying input delay[END_REF] gives

ẋ(t) = f t, x(t), us t, m-1 l=0 e m-l t + l h m -h + x(t) , δ(t) . (28)
From Assumption 2 and the fact that γ(r +s) ≤ γ(2r)+γ(2s) holds for all r ≥ 0 and s ≥ 0 and any γ ∈ K∞, we can then find functions β b ∈ KL and γ b ∈ K∞ such that for all t ≥ t0 and t0 ≥ h/m,

|x(t)| ≤ β b (|x| [t 0 -h,t 0 ] , t -t0) + γ b (2m|e| [t-h,t-h/m] ) + γ b (2|δ| [t 0 ,t] ). (29)
Consider any t ≥ t0 and s ∈ [t -h, t -h/m], and these two cases.

(i) s ∈ [t0 -h, t0]. Then |e(s)| ≤ |e| [t 0 -h,t 0 ] . (ii) s / ∈ [t0 -h, t0]. Then, s ≥ t0. From (26), we obtain |e(s)| ≤ βa(|e| [t 0 -h,t 0 ] , s -t0 + h) + γa(|δ| [t 0 ,s+h/m] ). ( 30
)
We deduce that for all s ∈ [t -h, t -h/m] the inequality

|e(s)| ≤ βc(|e| [t 0 -h,t 0 ] , s -t0 + h) + γa(|δ| [t 0 ,s+h/m] ) (31)
holds with βc(q, r) = βa(q, r) + qe -r+h . This gives

|e| [t-h,t-h/m] ≤ βc(|e| [t 0 -h,t 0 ] , t -t0) + γa(|δ| [t 0 ,t]
). This inequality in combination with [START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF] gives

|x(t)| ≤ β b (|x| [t 0 -h,t 0 ] , t -t0) +γ b 2mβc(|e| [t 0 -h,t 0 ] , t -t0) +2mγa(|δ| [t 0 ,t] ) + γ b (2|δ| [t 0 ,t] ) ≤ β b (|x| [t 0 -h,t 0 ] , t -t0) +γ b 4mβc(|e| [t 0 -h,t 0 ] , t -t0) +γ b 4mγa(|δ| [t 0 ,t] ) + γ b (2|δ| [t 0 ,t] ). (32)
Also, our formulas for the ei's give

|e| [t 0 -h,t 0 ] ≤ 2m|z| [t 0 -h,t 0 +h/m] +|x| [t 0 -h,t 0 +h/m]
, where z = (z1, z2, . . . , zm). Hence, we can choose β d (q, r) = β b (q, r) + γ b (4mβc(2mq, r)) and γ d (r) = γ b (4mγa(r)) + γ b (2r).

V. REMARKS ON ASSUMPTIONS AND ISS

The conclusion of Theorem 1 is weaker than ISS because the right side of ( 9) involves x and z values on [t0 -h, t0 + h/m]. We can strengthen its conclusion to an ISS estimate provided we (a) fix any function ᾱ ∈ K∞ and only consider initial functions that satisfy

|zi| [t 0 -h,t 0 ] ≤ ᾱ(|x| [t 0 -h,t 0 ]
) for all i ∈ {1, 2, . . . , m} and (b) assume that there are constants L1 > 0 and L2 > 0 such that

|f (t, x, U, 0)| ≤ L1(|x| + |U |) and |us(t, x)| ≤ L2|x| (33)
hold for all t ≥ 0, x ∈ R n , and U ∈ R c , as follows.

Assumption 1 combined with the first bound in (33

) implies that |f (t, x, U, ∆)| ≤ L1(|x| + |U |) + k|∆| holds for all t ≥ 0, x ∈ R n , U ∈ R c ,
and ∆ ∈ R d . Hence, our choice of the control in Theorem 1 implies that along all trajectories of the closed loop system, we get

| ẋ(t)| ≤ L1 |x(t)| + L2|zm(t -h)| + k|δ(t)| and | żi(t)| ≤ L1|zi(t)| + (4k + λa) zi t -h m + |zi-1(t)| + L1 L2 zm t -h(m-i) m
for all t ∈ [t0, t0 + h/m] and i ∈ {1, 2, . . . , m}. Hence,

|(x(t), z(t))| ≤ (m + 1)|(x(t0), z(t0))| + L1 t t 0 |x( )|d + h m L1 L2|zm|[t 0 -h,t 0 ] + hk m |δ| [t 0 ,t] + m L1 t t 0 |z( )|d + m L1 L2 h m |zm| [t 0 -h,t 0 ] + t t 0 |zm( )|d + m(4k + λa) |z| [t 0 -h,t 0 ] + t t 0 |z( )|d ≤ L3|(x, z)| [t 0 -h,t 0 ] + L4 t t 0 |(x( ), z( ))|d + hk m |δ| [t 0 ,t] for all t ∈ [t0, t0 + h/m], where L3 = m + 1 + (h/m) L1 L2(1 + m) + (4k + λa)m and L4 = L1(1 + m(1 + L2)) + m(4k + λa). Hence, Gronwall's Inequality gives |(x(t), z(t))| ≤ L3|(x, z)| [t 0 -h,t 0 ] + hk m |δ| [t 0 ,t] e L4 h/m ≤ L3|x|[t 0 -h,t 0 ] + m L3 ᾱ |x| [t 0 -h,t 0 ] + hk m |δ| [t 0 ,t] e L4 h/m for all t ∈ [t0 -h, t0 + h/m].
Therefore, in terms of the functions β d ∈ KL and γ d ∈ K∞ from our proof of Theorem 1, the choices βe(q, r) = β d 4 L3(q + m ᾱ(q))e L4 h/m , r and

γe(r) = β d 4hk m e L4 h/m r, 0 + γ d (r) (34)
are of class KL and K∞ respectively and give the final ISS estimate

|x(t)| ≤ βe |x| [t 0 -h,t 0 ] , t -t0 + γe(|δ| [t 0 ,t] ) for all t ≥ t0, using the relation max{|x(t)|, |z(t)|} ≤ |(x(t), z(t))|.

VI. EXAMPLES AND SIMULATIONS

We illustrate the value added by Theorem 1 by revisiting two examples from [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF]. Our treatment of the examples differs from [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] because (i) the controls from [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] have distributed terms and (ii) the work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] only applied to linear time-varying systems and so only applied to a linearized pendulum, unlike our treatment below that covers the time-varying nonlinear pendulum dynamics.

A. Scalar Example

The one dimensional example from [23, Section VII.A] is

ẋ(t) = sin(t/9) + sin(11πt) x(t) + u(t -h) + δ(t) (35)
with the disturbance δ. It is beyond the scope of standard reduction model technique or prediction results since the vector fields are time-varying and not periodic. However, [23, Theorem 1] provides functions β ∈ KL and γ ∈ K∞ such that along all trajectories of (35), in closed loop with the control u that is defined by

u(t) = K(t) x(t) + t t-h λ(t, r + h)u(r)dr , (36)
with the choices λ(t, ) = e -9 cos(t/9)-1 11π cos(11πt)+9 cos( /9)+ 1 11π cos(11π ) and K(t) = e 9 cos(t/9)+ 1 11π cos(11πt)-9 cos( t+h 9 )-1 11π cos(11π(t+h))

× -1 -sin(t/9) -sin(11πt) , (37)
and for all choices of the constant delay h > 0, we have

|x(t)| + |u| [t-h,t] ≤ β |x(t0)| + |u| [t 0 -h,t 0 ] , t -t0 + γ(|δ| [t 0 ,t] ) (38)
for all initial times t0 ≥ 0 and all t ≥ t0. By contrast, Theorem 1 above provides a control without distributed terms, as follows.

The assumptions of Theorem 1 are satisfied with the choices k = 2, us(t, x) = -[1 + sin(t/9) + sin(11πt)]x, h = 1, m = 23 and λa = 0.0225, and without any restrictions on the sup norm of the disturbance or on the initial functions, and the corresponding control [START_REF] Bresch-Pietri | Robust compensation of a chattering timevarying input delay[END_REF] has no distributed terms. We undertook three simulations of the closed loop system (35) with the preceding choices and t0 = 0, using NDSolve in the program Mathematica; see [START_REF]The world's definitive system for modern technical computing[END_REF] and Remark 2. In our first two simulations, we chose the initial functions for x and the zi's all equaling 1 on [-1, 0]. We chose the zero perturbation δ(t) = 0 in the first simulation, and our second simulation used δ(t) = e -t sin(t). In our third simulation, we chose the same values as in our first two simulations, except with all of the initial functions equaling 7 on [-1, 0], instead of 1, and with the perturbation δ(t) = 0.1 sin(t). We plotted the corresponding simulations for x(t) in Fig. 1 below. They illustrate convergence towards 0, with overshoots depending on the perturbation values, and therefore help illustrate Theorem 1.

B. Pendulum

We next illustrate our theory using the model

ṙ1(t) = r2(t) ṙ2(t) = -g l sin(r1(t)) + 1 Ml 2 v(t -h) (39)
of the simple pendulum, where g = 9.8 m/s is the gravity constant, l is the pendulum length in meters, M is the pendulum mass, v is the input, and h is the constant delay. As in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], we wish to track a C 1 reference trajectory (r1,s(t), r2,s(t)) such that ṙ1,s(t) = r2,s(t).

The error variables ri = ri -ri,s(t) for i = 1, 2 and the feedback produce the tracking dynamics

u(t -h) = 1 Ml 2 v(t -h) -ṙ2,s(t) -g l sin(r1,s(t)) (40)
     ṙ1(t) = r2(t) ṙ2(t) = g l [sin(r1,s(t)) -sin(r1(t) + r1,s(t))] + u(t -h). (41)
The work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] showed that for the case where r1,s(t) = ωt where ω > 0 is a large enough constant and h = 1, the linearization

ẋ1(t) = x2(t) ẋ2(t) = -g l cos(ωt)x1(t) + u(t -h), (42)
of (41) at 0 has the globally exponentially stabilizing distributed control u defined by u(t) = -0.6x1(t) -0.4x2(t) -t t-1 0.6(t -r -1) + 0.4 u(r)dr. However, [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] does not cover the original nonlinear pendulum dynamics (41) and it imposes conditions on ω, and [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] cannot be applied to (42) without imposing an upper bound on h. By contrast, (41) is covered by Theorem 1 without any bounds on the delay or on ω, and our theorem provides a controller with no distributed terms.

To apply Theorem 1 to (41), we chose

M = h = 1, l = 10, us(t, x) = -0.98(sin(ωt) -sin(x1 + ωt)) -x1 -x2, (43)
and x = (x1, x2) = (r1, r2). For simplicity, we only consider the case where ω = 1, but analogous reasoning applies for any positive constants ω, h, M, and l. The requirements of our theorem are then satisfied with k = 1, m = 16, and λa = 0.35. We simulated the corresponding perturbed pendulum dynamics

ẋ1(t) = x2(t) ẋ2(t) = g l [sin(t) -sin(x1(t) + t)] + u(t -h) + δ(t) (44)
with the control from our theorem, again using the command NDSolve in the Mathematica program, with different choices of δ, which represents control uncertainty. Recalling our change of feedback with u = us, it follows that the original controller becomes v(t) = 98 sin(zm1(t) + t + 1) -100(zm1(t) + zm2(t)), (45)

where zm = (zm1, zm2) is the bottom system in the dynamic control corresponding to m = 16. In our first simulation, we took the initial functions for all components of x and the zi's from our dynamic controller to be identically 1, again with the initial time t0 = 0, and with the zero perturbation δ = 0. In Fig. 2, we plotted the corresponding paths for the tracking error components x1(t) and x2(t) and the control v(t -1). The plots show rapid convergence of the tracking error towards 0, and the convergence of the control values, and therefore help demonstrate our theory. In Fig. 3, we show the solutions from the first simulation in red, and in blue we show the corresponding uncontrolled trajectories for x1 and x2 that we obtained by simulating (44) with u = δ = 0 in (44), for comparison. When u = 0, the solution x1(t) obtained from Mathematica for the unperturbed δ = 0 case satisfies limt→∞ x1(t) = ∞, which is another motivation for our new control design.

Then we repeated the simulations with the same choices as our first pendulum simulations, except with the initial functions for all components of x and the zi's from our control changed to be identically 1.5, and with δ(t) = 0.05 sin(t). In Fig. 4, we show the paths for the tracking error components x1(t) and x2(t) and v(t -1) in this case. With the perturbation, the tracking errors no longer converge to zero. However, they oscillate around 0, which is consistent with Theorem 1, so this too demonstrates our theory.

VII. CONCLUSIONS

Stabilization under arbitrarily long feedback delays leads to controllers derived from reduction model or prediction techniques, which can be difficult to implement because of the need to store past control values or to introduce possibly unstable dynamic extensions and evaluate the distributed terms in practice. We proposed an alternative stabilization technique for nonlinear systems with an arbitrarily long input delay. Potential advantages of our method include (a) that it produces a predictive state estimate that can be used in the nominal Components of Initial Functions for x and the z i 's set to 1.5 with δ(t) = 0.05 sin(t). Top to Bottom: x 1 (t), x 2 (t), and v(t -1) from (45).

control for the corresponding undelayed system, so our control can satisfy input constraints when the constraints are satisfied by the nominal control, (b) that our control does not involve distributed terms, and (c) that the closed-loop systems enjoy an ISS robustness property. We hope to generalize our work to systems with sampled data, as in [START_REF] Omran | Stability analysis of bilinear systems under aperiodic sampled-data control[END_REF], using state values at aperiodic discrete times.

Fig. 1 .

 1 Fig. 1. Simulating x(t) for (35) with h = 1 using Control (7) with m = 23, k = 2, λa = 0.0225, and x(r) = z i (r) = x for all i and r ∈ [-1, 0], and Different Perturbations δ. Top: δ(t) = 0 with x = 1. Middle: δ(t) = e -t sin(t) with x = 1. Bottom: δ(t) = 0.1 sin(t) with x = 7.

Fig. 2 .

 2 Fig. 2. Simulation of Nonlinear Pendulum Tracking System (44) with allComponents of Initial Functions for x and the z i 's set to 1 on [-1, 0] with Perturbation δ(t) = 0. Top to Bottom: x 1 (t), x 2 (t), and v(t -1) from (45).

Fig. 3 .

 3 Fig. 3. Simulation of (44) from Fig. 2 (red) Showing Convergence, with Corresponding Simulations of Uncontrolled System Obtained by Setting u = δ = 0 in (44) (blue) Showing Divergence. Top to Bottom: x 1 (t) and x 2 (t).

Fig. 4 .

 4 Fig. 4. Simulation of Nonlinear Pendulum Tracking System (44) with allComponents of Initial Functions for x and the z i 's set to 1.5 with δ(t) = 0.05 sin(t). Top to Bottom: x 1 (t), x 2 (t), and v(t -1) from (45).

Supported by NSF-ECCS Grants 1102348 and 1408295.


Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu

Reduction Model Approach for Linear Time-Varying Systems with Input Delays based on Extensions of Floquet Theory *

Keywords: delay, time-varying, Floquet theory, reduction model. Corresponding

Reduction model approach for linear time-varying systems with input delays based on extensions of Floquet theory

Frédéric Mazenc, Michael Malisoff

Introduction

This note continues our search (begun in [START_REF] Mazenc | Local stabilization of nonlinear systems through the reduction model approach[END_REF] and [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF]) for extensions of the classical reduction model method that cover time-varying systems with input delays. Input delays are common when controllers are remotely implemented; see [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF][START_REF] Bresch-Pietri | Prediction-based stabilization of linear systems subject to input-dependent input delay of integral type[END_REF][START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF][START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF][START_REF] Krstic | Input delay compensation for forward complete and feedforward nonlinear systems[END_REF][START_REF] Witrant | Remote stabilization via communication networks with a distributed control law[END_REF] for more motivation. The reduction method has its origins in the works [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], [START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF], and [START_REF] Mayne | Control of linear systems with time delay[END_REF] by Artstein and others, who focused on continuous time time-invariant linear systems.

Stabilization problems for linear time-varying systems with delays have been studied in fewer works. In most of them, time-varying Lyapunov functions are needed; see for instance [START_REF] Anh | Stability analysis for linear non-autonomous systems with continuously distributed multiple time-varying delays and applications[END_REF] and [START_REF] Phat | An application of Razumikhin theorem to exponential stability for linear nonautonomous systems with time-varying delay[END_REF] for the use of strict Lyapunov functions, and [START_REF] Tiwari | On stability properties of time-varying systems with delays[END_REF] for Razumikhin-Lyapunov functions. One useful Lyapunov-based approach to delay systems entails solving the stabilization problem with the input delay set equal to zero, and then using Lyapunov-Krasovskii functionals to look for upper bounds on the input delays that the closed loop system can tolerate without sacrificing the stability performance; see [START_REF] Malisoff | Robustness of adaptive control under time delays for three-dimensional curve tracking[END_REF][START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]. Linear time-varying systems arise in the context of the local stabilization of a trajectory of a nonlinear system, but are beyond the scope of the classical reduction model method. The main differences between reduction approaches and the Lyapunov-Krasovskii functional approaches such as those in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] are that (a) under certain delay bounds, methods such as [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] lead to relatively simple controllers that do not require the distributed terms that are used in reduction model methods and (b) reduction model methods usually make it possible to compensate for arbitrarily long input delays, by using the delay value in the dynamic feedback control design.

Our work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] extends the reduction model method to linear time-varying systems, using two approaches. One approach in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] leads to a control formula that involves the fundamental matrix for the corresponding uncontrolled system (i.e., the time-varying system obtained from the original system by setting the input equal to zero in the original system), and so may be difficult to apply in practice. The other control design in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] does not require a formula for the fundamental matrix, but requires that the input delay stay below a suitable constant bound. By contrast, [START_REF] Mazenc | Local stabilization of nonlinear systems through the reduction model approach[END_REF] covers time-varying nonlinear systems whose nonlinear parts satisfy certain conditions, and then builds a reduction model control for the linearization of the system, and finally shows local stability of the original nonlinear system in closed loop with the reduction model control.

One natural research direction for addressing the challenges of extending the reduction model method to time-varying linear systems, as well as analogous problems for discrete time systems, is to seek analogues of Floquet's theory; see [START_REF] Perko | Differential Equations and Dynamical Systems[END_REF]Section 3.5] for the basic results of Floquet theory. Floquet's theory covers systems without controls. One of its basic results is that if a time-varying linear system is periodic, then it can be transformed into a time-invariant system through a periodic change of coordinates. This suggests the possibility of using Floquet theory to transform a time-varying linear control system into a new time-varying linear control system with a time-invariant draft, and then stabilizing the new system using the reduction approach. One of the key observations in this work is that such a transformation can indeed be done under periodicity of the coefficient matrices in the system, and that this simplifies the stabilization problem to one that involves globally asymptotically stabilizing systems with no delay. Our assumptions are novel as well. We also provide analogues for nonperiodic or discrete time systems.

Discrete time systems with delay are important because they can be used to model some engineering devices; see [START_REF] Bittanti | Companion forms and cyclic matrices for discrete-time periodic systems[END_REF], [START_REF] Cloosterman | Stability of networked control systems with large delays[END_REF], [START_REF] Van Dooren | When is a periodic discrete-time system equivalent to a time-invariant one?[END_REF], and [START_REF] Hernandez-Vargas | Discrete-time control for switched positive systems with application to mitigating viral escape[END_REF]. However, not many contributions are concerned with time-varying discrete time systems with delay. Our discrete time delayed systems in this work have the form

x k+1 = A k x k + B k u k-r (1)
where x k ∈ R n is the state, u k ∈ R p is the control, and r ∈ N is the delay. Here and in the sequel, the dimensions and delays are arbitrary. For the case of time-invariant coefficients, the work [START_REF] Cloosterman | Stability of networked control systems with large delays[END_REF] uses dynamic extensions to transform systems of the form (1) into systems with no delay, in the special case of networked control systems. See also [START_REF] Garcia | Robustness of a discrete-time predictorbased controller for time-varying measurement delay[END_REF] for a prediction based approach for (1) in the time-invariant case. For timevarying continuous time systems with delay in the input, the reduction model approach can be applied under conditions pertaining to the speed of variation of the time-varying matrices; see [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF]. However, to the best of our knowledge, no discrete time version of [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] exists. Also, [START_REF] Karafyllis | Robust predictor feedback for discrete-time systems with input delays[END_REF] is concerned with time-invariant systems.

We propose a rather general solution to the problem of exponential stabilization of (1) through the reduction model approach, including cases where the coefficient matrices are not necessarily periodic, with an arbitrarily large delay r. It decomposes into two steps. First, under reasonable assumptions, we transform (1) into a system that is autonomous when the control is set equal to zero. Then, we adapt the reduction method to the resulting dynamics, using a novel discrete time analogue of an operator that is used in the predictor based analysis in [START_REF] Karafyllis | Robust predictor feedback for discrete-time systems with input delays[END_REF]. Our treatment of (1) also has implications for using reduction model controllers in continuous time systems, because in practice, implementing controllers in continuous time systems uses discretizations, leading to discrete time delay systems of the form [START_REF] Anh | Stability analysis for linear non-autonomous systems with continuously distributed multiple time-varying delays and applications[END_REF]. We illustrate our theory in two examples, including a discrete time linear system in which the coefficient matrices are not periodic.

Preliminary Results in Continuous Time

We use the following notation and definitions. We let | • | denote the usual Euclidean norm of matrices and vectors, and I n is the n × n identity matrix. Here and in the sequel, the dimensions of our Euclidean spaces are arbitrary unless other indicated. Given any function φ : S → R p that is defined on any subset S of a Euclidean space, denote its supremum over any set J ⊆ S by |φ| J . We set N = {1, 2, . . .}. We often leave out the arguments of functions, when they are clear from the context.

Fundamental General Result

Consider the system ẋ = A(t)x + F (t)u(t -τ ) (2)
where the state x and the input u are valued in R n and R p respectively, the functions A : R → R n×n and F : R → R n×p are continuous, and τ > 0 is any positive constant delay. We introduce the following assumption. See below for ways to build the required function P .

Assumption 1. There exist a constant matrix

A c ∈ R n×n , a C 1 function P : R → R n×n such that P -1 (t)
is defined for all t, and a constant p M > 0 such that

|P (t)| + |P -1 (t)| ≤ p M (3)
and

Ṗ (t) = A c P (t) -P (t)A(t). ( 4
)
hold for all t ≥ 0.

We use the following key observations:

Lemma 1. Assume that the system (2) satisfies Assumption 1. Then the time-varying change of coordinates z = P (t)x ( 5)

transforms (2) into ż(t) = A c z(t) + P (t)F (t)u(t -τ ). (6)
Also, the operator

Z(t) = z(t) + t t-τ e Ac(t-m-τ ) P (m + τ )F (m + τ )u(m)dm (7)
transforms (6) into the system

Ż(t) = A c Z(t) + e -Acτ P (t + τ )F (t + τ )u(t) (8)
with the state variable Z.

Proof. Our choice (5) of z gives

ż(t) = Ṗ (t)x(t) + P (t) ẋ(t) = Ṗ (t)x(t) + P (t)[A(t)x(t) + F (t)u(t -τ )]. (9)
Then ( 6) follows from (4) and our choice of z. Also, the time derivative of ( 7) along all solutions of ( 6) is

Ż(t) = A c z(t) + P (t)F (t)u(t -τ ) + A c t t-τ e Ac(t-m-τ ) P (m + τ )F (m + τ )u(m)dm + e -Acτ P (t + τ )F (t + τ )u(t) -P (t)F (t)u(t -τ ) = A c Z(t) + e -Acτ P (t + τ )F (t + τ )u(t) (10)
which gives the second conclusion.

The message of Lemma 1 is that if one knows a function P that leads to a time-invariant system when the input is set to zero, then in practice, we can use the reduction model approach to obtain a new system without delays. We study the problem of finding P in the next subsection.

Remark 1. Before discussing ways to find P , we remark that similar reasoning applies to systems

ẋ = A(t)x + t t-τ F ()u()d ( 11
)
with distributed delay in the input and continuous matrix valued functions A and F . To see how, notice that if Assumption 1 is satisfied, and if we define z by (5) as before and redefine Z to be

Z(t) = z(t) + t t-τ e Ac(t-m-τ ) P (m + τ ) t m F ()u()d dm, (12)
then similar reasoning to the proof of Lemma 1 gives

ż(t) = A c z(t) + P (t) t t-τ F ()u()d ( 13
)
and then

Ż(t) = A c Z(t) + t t-τ e Ac(t--τ ) P ( + τ )d F (t)u(t) ( 14
)
so we eliminated the distributed delay.

Particular Case

In order to get a function P for which Assumption 1 is satisfied, we impose checkable conditions on the corresponding zero input subsystem ẋ = A(t)x

which we later show how to verify using Floquet theory when A is periodic. We introduce the following assumption:

Assumption 2. All solutions of the system (15) are of the form

x(t) = K(t)E(t)x(0), (16)
where K : R → R n×n and E : R → R n×n are everywhere invertible functions of class C 1 , K -1 is of class C 1 , and E is such that there exists a constant matrix E c such that

Ė(t)E -1 (t) = E c ( 17
)
holds for all t ≥ 0. Moreover, there is a constant k c > 0 such that

|K(t)| + |K -1 (t)| ≤ k c ( 18
)
for all t ≥ 0.

We can often find functions E satisfying the preceding requirements. For example, they hold if

E(t) = e Mt G,
where M ∈ R n×n and G ∈ R n×n are constant matrices and G is invertible. Indeed, in this case

Ė(t) = Me Mt G, E -1 (t) = G -1
e -Mt and therefore also Ė(t)E -1 (t) = M are satisfied. See below for cases where Assumption 2 is satisfied. We are ready to state and prove the following result: Lemma 2. Let the system (2) be such that (15) satisfies Assumption 2. Then the change of coordinates

z = L(t)x (19)
with L = K -1 gives the new system

ż(t) = E c z(t) + L(t)F (t)u(t -τ ) (20)
with a time invariant drift term.

Proof. From ( 15) and ( 16), it follows that

ẋ(t) = A(t)K(t)E(t)x(0) (21)
and ẋ(t) = K(t)E(t)x(0) + K(t) Ė(t)x(0) [START_REF] Mazenc | Local stabilization of nonlinear systems through the reduction model approach[END_REF] hold for all x(0) ∈ R n . This gives

A(t)K(t)E(t) = K(t)E(t) + K(t) Ė(t). ( 23
)
Here and in the sequel, all equalities or inequalities are to be understood to hold for all t ≥ 0. Since E is invertible everywhere, [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] gives K(t) + K(t) Ė(t)E -1 (t) = A(t)K(t). From [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF], it follows that K(t) + K(t)E c = A(t)K(t). This gives

L(t) K(t)L(t) + E c L(t) = L(t)A(t). (24)
Since L(t)K(t) = I n for all t, it follows that L(t) K(t) + L(t)K(t) = 0. Therefore, [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] gives -

L(t)K(t)L(t) + E c L(t) = L(t)A(t). We deduce that L(t) = E c L(t)-L(t)A(t)
. This equality and (18) ensure that Assumption 1 is satisfied with L = P and E c = A c . Therefore Lemma 1 applies, which proves Lemma 2.

We next turn to our general ways to find functions K and E that satisfy Assumption 2.

3 Results Based on Floquet's Theory in Continuous Time

Theoretical result

In the particular case where the function A is periodic, Floquet's theory (e.g., [26, Section 3.5]) guarantees that there are two functions K and E such that Assumption 2 is satisfied. More precisely, thanks to Floquet's theory, we can establish the following result:

Theorem 1. Consider the system (2) and assume that A is periodic of any period T > 0. Then there exist an everywhere invertible C 1 function H : R → R n×n that is periodic of period T and a constant matrix R ∈ R n×n such that the change of coordinates z = H(t)x transforms (2) into

ż(t) = Rz(t) + H(t)F (t)u(t -τ ) ( 25
)
for all t ≥ 0.

Proof. Consider the system ẋ = A(t)x from (15) associated with the system (2). Let X : R → R n×n be the fundamental solution of ( 15) such that X(0) = I n , so Ẋ(t) = A(t)X(t) and X(t) is invertible for all t ∈ R. Then Floquet's theory (e.g., from [26, Section 3.5]) ensures that there exist a constant matrix R ∈ R n×n , and an everywhere invertible C 1 function Q : R → R n×n that is periodic of period T , such that

X(t) = Q(t)e Rt (26)
holds for all t ∈ R. Then Q -1 is C 1 , because X is C 1 and invertible. By [START_REF] Perko | Differential Equations and Dynamical Systems[END_REF], we deduce that Assumption 2 is satisfied with K(t) = Q(t), E(t) = e Rt , and E c = R. It follows from Lemma 2 that the change of coordinates

z(t) = Q -1 (t)x(t) (27)
transforms the system (2) into ż(t) = Rz(t) + Q -1 (t)F (t)u(tτ ). Since Q -1 is periodic of period T , this allows us to conclude by setting

H = Q -1 .
A challenge in applying Theorem 1 is that determining the function Q is not easy. This fact is a fundamental limitation of Floquet's theory. This remark motivates the next subsection.

More Applied Results Based on Floquet's Theory

We again consider the system (15) in the case where A is periodic of some period T . Then the decomposition (26) of the fundamental matrix gives I n = X(0) = Q(0) = Q(T ), so Q(T ) = I n and X(T ) = e RT . Through numerical simulations, we can determine X(T ), which is invertible. Recalling from [START_REF] Culver | On the existence and uniqueness of the real logarithm of a matrix[END_REF] that an invertible real matrix has a real matrix logarithm (denoted by log) if and only if each Jordan block in its Jordan decomposition corresponding to a negative eigenvalue occurs an even number of times, it follows that if X(T ) has no eigenvalues that are contained in (-∞, 0), then we can set R = 1

T log(X(T )); see [15, Section 1.6] for the definition of and formulas for matrix logarithms of invertible matrices.

Next, we return to [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF]. Let us show how we can determine a function Q such that X(t) = Q(t)e Rt , which makes it possible to apply Theorem 1 in practice. Since X(t) = Q(t)e Rt is a fundamental solution of ẋ = A(t)x, we get Ẋ(t) = Q(t)e Rt + Q(t)Re Rt = A(t)Q(t)e Rt , so cancelling e Rt from both sides gives

Q -1 (t) Q(t)Q -1 (t) + RQ -1 (t) = Q -1 (t)A(t). This gives RH(t) -H(t)A(t) = -Q -1 Q(t)Q -1 (t) = Ḣ(t), (28)
where H = Q -1 , and where the last equality follows by computing the derivative of Q(t)Q -1 (t) using the product rule. Therefore, we consider the system (2) with the dynamic extension (t) = R(t) -(t)A(t).

Then, since Q(0) = I n , it follows that when (0) = I n , we have H = . Hence, through a simple dynamic extension with the identity as the initial condition, we have H.

Then Theorem 1 implies that the change of coordinates

z(t) = H(t)x(t) gives ż(t) = Rz(t) + (t)F (t)u(t -τ ). ( 29
)
Therefore, when we apply the reduction model approach, we consider the operator Z from (7), namely,

Z(t) = z(t) + t t-τ e R(t-m-τ ) (m + τ )F (m + τ )u(m)dm. ( 30
)
Although this involves future values of , the function is periodic of period T . Therefore, we can replace

(m + τ ) by (m + τ -kT ),
where k is a sufficiently large integer. After such a substitution, ( 30) is only valid for sufficiently large t. Then the reasoning that gave the delay free system [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF] gives

Ż(t) = RZ(t) + e -Rτ (t + τ -kT )F (t + τ )u(t). ( 31
)
This means that if we stabilize this system using a state feedback, it will depend on the past values. Alternatively, we can consider the system

   ẋ(t) = A(t)x(t) + F (t)u(t -τ ) (t) = R(t) -(t)A(t) (lT ) = I n , l ∈ {0, 1, 2, . . .} (32)
to obtain the closed loop system.

Remark 2. If we consider the system ẋ(t) = f (x(t)) with x valued in R 2 with a periodic solution r(t) of some period T and with ṙ(0) = 0 and evaluate the linear approximation around this solution, then for the corresponding linearized system ẋ = A(t)x (where A(t) is the Jacobian matrix A(t) = Df (r(t))), we know that 1 is one of the eigenvalues of the corresponding matrix e RT . This follows because ṙ(t) is a solution of ẋ = A(t)x, so ṙ(0) = ṙ(T ) = Q(T )e RT ṙ(0) = e RT ṙ(0). Then, using the relation

ρ 1 ρ 2 = exp T 0 trace(A())d ( 33
)
between the eigenvalues ρ 1 and ρ 2 of e RT that is provided by Floquet's theory (e.g., [3, Proposition 2.39]), we get ρ 1 = 1 and

ρ 2 = exp T 0 trace(A())d . ( 34
)
Therefore one does not need to use a computer to get the two eigenvalues of e RT to compute the matrix logarithm for the Floquet theory decomposition of the fundamental matrix. Before illustrating our theory in the continuous time case in Section 6, we provide the following discrete time analogues of the preceding results.

Preliminary Results in Discrete Time

We propose a more general version of Floquet's theory for discrete time systems than what was developed in [START_REF] Hayakawa | Floquet transformations for discrete-time systems: Equivalence between periodic systems and time-invariant ones[END_REF] and [START_REF] Sreedhar | Discrete-time periodic systems: a Floquet approach[END_REF]. We wish to design delay compensating stabilizing controllers for systems of the form

x k+1 = A k x k + B k u k-r (35)
where x k ∈ R n is the state, u k ∈ R p is the control, and r ∈ N is the delay. We first consider the system

x k+1 = A k x k (36)
with k ∈ N and x k valued in R n . Since our extension does not require A k to be periodic, we believe that it is new. Moreover, our changes of coordinates are explicit in many interesting cases. The work [START_REF] Sreedhar | Discrete-time periodic systems: a Floquet approach[END_REF] shows that in the periodic context, matrices similar to the R k 's defined below can be found, but in [START_REF] Hayakawa | Floquet transformations for discrete-time systems: Equivalence between periodic systems and time-invariant ones[END_REF], the change of coordinates is not explicit, and [START_REF] Hayakawa | Floquet transformations for discrete-time systems: Equivalence between periodic systems and time-invariant ones[END_REF][START_REF] Sreedhar | Discrete-time periodic systems: a Floquet approach[END_REF] do not cover reduction model controls. We introduce two assumptions.

Assumption 3. For all k ≥ 0, the matrix A k is invertible.

Assumption 3 allows us to introduce a sequence of matrices defined by

P k = k-1 i=0 A -1 i (37)
for all k ≥ 1 and P 0 = I n . Then P k is invertible for all k ∈ N. Our second assumption is:

Assumption 4.
There are an invertible matrix A * ∈ R n×n and a real number c > 0 such that the sequence

R k = A k * P k (38) satisfies |R -1 k | + |R k | ≤ c (39)
for all k ∈ N.

We use the operator

Γ k = A r * z k + r i=1 A i-1 * G k+r-i u k-i . (46)
Then,

Γ k+1 = A r * (A * z k + G k u k-r ) + r i=1 A i-1 * G k+1+r-i u k+1-i = A r+1 * z k + A r * G k u k-r + r i=2 A i-1 * G k+1+r-i u k+1-i + G k+r u k = A r+1 * z k + r+1 i=2 A i-1 * G k+1+r-i u k+1-i + G k+r u k . (47)
Using the definition of Γ k , we get

Γ k+1 = A * Γ k - r i=1 A i-1 * G k+r-i u k-i + r+1 i=2 A i-1 * G k+1+r-i u k+1-i + G k+r u k = A * Γ k - r i=1 A i * G k+r-i u k-i + r i=1 A i * G k+r-i u k-i + G k+r u k = A * Γ k + G k+r u k . (48)
Then we can choose

u k = K k R -1 k Γ k .
This feedback is equivalent to the feedback in (44). Then (42) gives

Γ k+1 = A * + G k+r K k R -1 k Γ k = R k+1 A k R -1 k + G k+r K k R -1 k Γ k = R k+1 A k + R -1 k+1 R k+r+1 B k+r K k R -1 k Γ k = R k+1 H k R -1 k Γ k . (49)
From Assumption 5, it follows that R -1 k Γ k is solution of a uniformly exponentially stable system. Hence, there are constants c 1 > 0 and c 2 > 0 such that for all k 0 ≥ 1 and k ≥ k 0 , the inequality

|Γ k | ≤ c 1 e c2(k0-k) |Γ k0 | (50)
is satisfied. Recalling that the sequence K k R -1 k is bounded, we conclude that there is a constant c 3 > 0 such that for all k 0 ≥ 1 and k ≥ k 0 , the inequality

|u k | ≤ c 3 e c2(k0-k) |Γ k0 | (51) is satisfied, namely, c 3 = c 1 sup k |K k R -1 k |.
Next, observe that (46) gives

x k = R -1 k (A r * ) -1 Γ k -(A r * ) -1 r i=1 A i-1 * G k+r-i u k-i . (52)
Also, the boundedness of the sequences B k and R k gives boundedness of the sequence G k . From (50), (51), (52) and the properties of the matrices R k , we can conclude.

Remark 3. If we know a strict quadratic Lyapunov function for (43), then, by adapting ideas of [START_REF] Mazenc | Lyapunov-Krasovskii functionals and application to input delay compensation for linear time-invariant systems[END_REF], we can build a function of Lyapunov-Krasovskii functional type for (35) in closed loop with (44).

Important Remark on Discretizations

Consider a continuous time time-varying system

ẋ(t) = M (t)x(t) + N (t)u(t -τ ) ( 53
)
where the delay τ > 0 is constant. Assume that M is periodic, and that M and N are continuous. Since we do not assume that a fundamental solution of ξ(t) = M (t)ξ(t) is known, no standard stabilization technique available in the literature applies.

It is natural to approximate the system (53) by

ẋ(t) = M (t k )x(t) + N (t k )u(t k-r ), (54)
where t k+1t k = ν, ν > 0 is a sufficiently small constant, and r > 0 is an integer. By integrating (54), we obtain

x(t k+1 ) = e νM(t k ) x(t k ) + t k+1 t k e M (t k )(t k +ν-) d N (t k )u(t k-r ). (55)
That way, we can handle cases which are not covered by the results of our work [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], e.g., because we do not need an upper bound on the allowable delays, nor do we need a formula for the fundamental matrix for ẋ = M (t)x to use in the reduction model control. Notice that, to simplify, we can approximate (55) by

x k+1 = [I n + νM (t k )]x k + νN (t k )u t k-r . ( 56
)
If ν is sufficiently small, then all of the matrices A k = I n + νM (t k ) are invertible. Moreover, we can select the sequence t k so that the sequence A k is periodic.

6 Examples

Continuous Time Example

Consider the system ẋ

(t) = sin(t) + 1 2 x(t) + u(t -τ ), (57)
where x and u are scalar valued. It is unstable when the input is set to zero, but Theorem 1 applies to (57).

In fact, the fundamental matrix of ẋ = sin(t) + 1 x is

X(t) = e 1 2 t+1
e -cos(t) .

Therefore, Theorem 1 leads us to consider the change of coordinates [START_REF] Phat | An application of Razumikhin theorem to exponential stability for linear nonautonomous systems with time-varying delay[END_REF], which in our case leads to our choice z = e cos(t)

x. It yields ż = 1 2 z + e cos(t) u(tτ ).

(59)

Applying the reduction model approach from Lemma 1, we obtain the undelayed system

Ż(t) = 1 2 Z(t) + e cos(t+τ )-τ 2 u(t), (60)
where

Z(t) = e cos(t) x(t) + t t-τ e 1 2 (t-m-τ ) e cos(m+τ ) u(m)dm. (61)
This system is asymptotically stabilized by

u(t) = -e -cos(t+τ )+ τ 2 Z(t) = -e -cos(t+τ )+ τ 2 e cos(t) x(t) + t t-τ e 1 2 (t-m-τ ) e cos(m+τ ) u(m)dm , (62)
which gives the closed loop system Ż(t) = -1 2 Z(t).

Discrete Time Example

Consider the system (35) with B k = 1 for all k and with the sequence of real numbers A k defined as follows:

A 2l = 2 and A 2l+1 = 2l 2 + 1 l 2 + 1 for all l ≥ 0 (63)
Observe that this sequence is not periodic. With the notation of Section 4, we have P 0 = 1, P 1 = 1 2 , and the following for all l ≥ 1:

P 2l+1 = 2l i=0 A -1 i = 1 2 l+1 l-1 i=0 i 2 + 1 2i 2 + 1 and P 2l = 2l-1 i=0 A -1 i = P 2l+1 A 2l = 1 2 l l-1 i=0 i 2 + 1 2i 2 + 1 (64)
Let us choose A * = 2. Then, for all l ≥ 1, the functions R k = A k * P k from Assumption 4 are

R 2l+1 = 2 2l+1 1 2 l+1 l-1 i=0 i 2 + 1 2i 2 + 1 = 2 l l-1 i=0 i 2 + 1 2i 2 + 1 = R 2l . (65)
Consequently, for all l ≥ 1, we have

0 < R 2l = R 2l+1 ≤ 2 l l-1 i=0 i 2 + 1 2i 2 + 1 = exp l-1 i=0 ln 2i 2 + 2 2i 2 + 1 = exp l-1 i=0 ln 1 + 1 2i 2 + 1 ≤ exp l-1 i=0 1 2i 2 + 1 ≤ exp 1 + 1 2 ∞ i=1 1 i 2 < ∞ (66)
and

0 < R -1 2l = R -1 2l+1 ≤ 1 2 l l-1 i=0 2i 2 + 1 i 2 + 1 ≤ 1. ( 67
)
Hence, we can satisfy the bound requirement (39) from Assumption 4 with the bound

c = exp 1 + 1 2 ∞ i=1 1 i 2 . ( 68
)
We now apply Theorem 2 to the corresponding discrete time system (35), with the preceding choices of A k , B k = 1 for all k, and r = 4. With the choices

K k = - A k R -1 k+1 R k+5 , (69)
the matrices H k from Assumption 5 become

H k = A k + R -1 k+1 R k+5 K k = 0, (70)
so Assumption 5 is satisfied. Then Theorem 2 provides us with the stabilizing control law

u k = -16 A k R k+1 R k+5 x k - 4 i=1 A k R k+1 R k+5-i R k R k+5 2 i-1 u k-i . (71)
Using our formulas (65) for the R k 's and (71), we obtain the formulas for all l, which ensure the desired exponential stability property from Theorem 2.

u 2l = -32 R 2l R 2l+4 x 2l -2 4 i=1 R 2l+5-i R 2l+4 2 i-1 u 2l-i = 2 2(l + 1) 2 + 1 (l + 1) 2 + 1 -2 2l 2 + 1 l 2 + 1 (2x 2l + u 2l-4 ) -u 2l-2 -2u 2l-3 -u 2l-1 and
u 2l+1 = - 2l 2 + 1 l 2 + 1 16 R 2l+2 R 2l+6 x 2l+1 + 4 i=1 R 2l+2 R 2l+6-i R 2l+1 R 2l+6 2 i-1 u 2l+1-i = - 2l 2 + 1 l 2 +

2

 2

Conclusions

We provided new reduction model method input delay compensating controllers for time-varying continuous and discrete time systems, using Floquet's theory and its extensions. Compared with other reduction model methods for time-varying continuous time linear systems (such as our recent work [24]), two potential advantages of our approach are that (a) we do not require closed form expressions for the fundamental matrix for the corresponding uncontrolled drift systems and (b) our new results allow arbitrarily long input delays. In future works, we will investigate stability problems for systems with several and time-varying delays.

Grant 1102348.

We prove the following: Lemma 3. Let Assumptions 3-4 hold. Then for each solution sequence x k of (36), the sequence

for all k.

Proof. We have

Here and in the sequel, all equalities or inequalities should be understood to hold for all integers k ≥ 0, unless otherwise indicated. Our definitions give

By right multiplying through (42) by z k , it follows that z k is solution of (41), which proves the lemma.

If A * is known, then R k is given by an explicit formula. Finding A * can be difficult. However, if A k is periodic with some period p, and if A p-1 A p-2 . . . A 0 has no eigenvalues that lie in the interval (-∞, 0], then [START_REF] Bini | Algorithms for the matrix pth root[END_REF] and [START_REF] Higham | Functions of Matrices: Theory and Computation[END_REF]Theorem 7.2] provide a matrix A * ∈ R n×n such that A p * = A p-1 A p-2 . . . A 0 , as well as algorithms for computing this pth root matrix. Since (42

k-1 for all k ≥ 1, which gives R p = I n and periodicity of R k . Hence, this A * satisfies Assumption 4. To check that this pth root matrix is invertible, notice that if v ∈ R n \ {0} were in the null space of A * , then 0 = A p * v = A p-1 (A p-2 . . . A 0 v), which contradicts the invertibility of the A k 's. See Section 6 for a nonperiodic example where we find A * .

Model Reduction in Discrete Time

Main Result

In this section, we use Lemma 3 to extend the reduction model approach to a general family of systems of the type (35). We introduce the following assumption, which is a natural analogue of the usual requirement of controllable pairs of coefficient matrices, generalized to time-varying systems with delays: Assumption 5. The system (35) is such that the sequence A k satisfies Assumptions 3-4, and the sequence B k is bounded. Also, there is a sequence K k such that the sequence K k R -1 k is bounded and such that the system

is uniformly exponentially stable, where R k is the sequence defined in (38).

We prove the following:

Theorem 2. Assume that the system (35) satisfies Assumption 5. Then we can find positive constants c1 and c2 such that all trajectories of (35), in closed loop with

) for all integers k 0 ≥ 1 and k ≥ k 0 .

Proof. We can use (42) to show that the change of coordinates z k = R k x k applied to (35) gives

where G k = R k+1 B k . Since A * is time-invariant, we can adapt ideas from [START_REF] Garcia | Robustness of a discrete-time predictorbased controller for time-varying measurement delay[END_REF] to the problem of stabilizing the system (45), as follows.


Frederic Mazenc

Michael Malisoff

Extension of Razumikhin's Theorem for Time-Varying Systems with Delay

Keywords: delay, Lyapunov, Razumikhin, stability

We provide an extension of Razumikhin's theorem for continuous time time-varying systems with time-varying delays. Our result uses a novel 'strictification' technique for converting a nonstrict Lyapunov function into a strict one. Our examples show how our method can sometimes allow broader classes of allowable delays than the results in the literature.

I. INTRODUCTION

Input delays are ubiquitous in engineering, where they are useful for modeling time consuming information gathering or latencies in engineering processes; see [START_REF] Bekiaris-Liberis | Nonlinear Control under Nonconstant Delays[END_REF], [START_REF] Dieulot | Tracking control of a nonlinear system with input-dependent delay[END_REF], [START_REF] Downey | Compensating for fatigue-induced time-varying delayed muscle response in neuromuscular electrical stimulation control[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF], [START_REF] Marquez | Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach[END_REF], [START_REF] Mazenc | Generating positive and stable solutions through delayed state feedback[END_REF], [START_REF] Mondié | Finite spectrum assignment of unstable time-delay system with a safe implementation[END_REF], and [START_REF] Petit | Motion planning for two classes of nonlinear systems with delays depending on the control[END_REF]. However, such systems are usually too complicated to be covered by standard methods for undelayed systems [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]. This note builds on our research (begun in [START_REF] Mazenc | Trajectory based approach for the stability analysis of nonlinear systems with time delays[END_REF], [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF], and [START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF]) on novel methods to prove important stability properties for time delayed systems.

Since the flow map for a nonlinear system usually cannot be written in explicit closed form, it is natural to use indirect methods such as Lyapunov approaches to prove stability for undelayed systems. Lyapunov functions provide a generalized notion of energy in dynamical systems, so the decay of the Lyapunov function in a suitable sense often implies asymptotic convergence of the solutions towards an equilibrium. Classical Lyapunov function approaches require that the time derivative of the Lyapunov function be nonpositive along all solutions of the system, which can sometimes be a demanding condition, especially for timevarying or time delay systems. While classical Lyapunov functions are suited for proving stability of systems without delays, one often replaces Lyapunov functions by Lyapunov-Krasovskii functionals [START_REF] Fridman | On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays[END_REF] or Ruzumikhin functions to help solve stability problems for delayed systems.

As explained in [START_REF] Mazenc | Reduction model approach for linear time-varying systems with delays[END_REF] and [START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF], time-varying systems with delay are very important, e.g., to model tracking problems; see also [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF]. The works [START_REF] Bresch-Pietri | Robust compensation of a chattering time-varying input delay[END_REF], [START_REF] Mazenc | Reduction model approach for systems with a time-varying delay[END_REF], [START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF], and [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] on time delay systems are significant, in part because they use Lyapunov functional ideas to prove stability but allow the time derivatives of the functionals to take positive and negative values along trajectories of the systems. For delayed systems, two important analogs of classical Lyapunov functions are often used. The first are Lyapunov-Krasovskii functionals, which can often be built by adding together a Lyapunov function for the corresponding undelayed system and a second integral term involving the delay; see [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]. A different approach involves Razumikhin's theorem; see [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Hale | Introduction to Functional Differential Equations[END_REF], and [START_REF] Zhou | Truncated Predictor Feedback for Time-Delay Systems[END_REF]Theorem B.2]. The Razumikhin approach is especially useful for systems with time-varying delay.

This note provides an extension of Razumikhin's theorem for time-varying systems. We extend the strictification technique, developed in particular in [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]. Our first result does not use periodicity. However, due to the importance of periodic systems, we present a slightly simpler result in the periodic case. We obtain less conservative stability conditions than those in [START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF] and, at least in some cases, those in [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF].

II. DEFINITIONS AND NOTATION

Throughout this work, all dimensions are arbitrary, unless indicated otherwise. The usual Euclidean norm, and its matrix norm, are denoted by | • |, and | • | I denotes the (essential) supremum over any interval I ⊆ R. Let C 1 be the set of all continuously differentiable functions, whose domains and ranges will be clear from the context. For each constant delay bound τ , let C([-τ, 0], R n ) be the set of all continuous R n -valued functions defined on [-τ, 0]. Let C in be the set of all absolutely continuous functions φ ∈ C([-τ, 0], R n ), which we call the set of all initial functions. For each continuous function ϕ : [-τ, ∞) → R n and t ≥ 0, set ϕ t (m) = ϕ(t + m) for all m ∈ [-τ, 0]. A locally bounded function defined on an interval I ⊆ R is called piecewise continuous provided it is continuous except at finitely many points on each bounded subset of I. For a function φ defined on [0, ∞) × R n as being differentiable, we view its partial derivative φ t (0, x) with respect to its first argument at 0 as being a right derivative at 0.

Let K denote the set of all strictly increasing continuous functions

α : [0, ∞) → [0, ∞) such that α(0) = 0; if, in addition, α is unbounded, then we say that α is of class K ∞ . A function F : [0, ∞) × C in → R n is called uniformly bounded with respect to its first argument provided there is a function α ∈ K and a constant c > 0 such that |F (t, φ))| ≤ c + α(|φ| [-τ,0]
) holds for all t ≥ 0 and φ ∈ C in ; it is called Lipschitz continuous with respect to its second argument provided for each constant K > 0, we have

|F (t, φ) -F (t, ψ)| ≤ α(K)|φ -ψ| [-τ,0]
for all t ≥ 0 and all φ and ψ in C in such that max{|φ| [-τ,0] , |ψ| [-τ,0] } ≤ K.

III. GENERAL RESULT A. Statement of Result

We consider a nonlinear time-varying system

ẋ = F (t, x t ) (1)
whose delay is bounded by some constant τ , having initial conditions in C([-τ, 0], R n ). We assume: Assumption 1: The function F is uniformly bounded with respect to its first argument and Lipschitz continuous with respect to its second argument. There is a function

V : [0, ∞) × R n → [0, ∞) that is of class C 1 on ([0, ∞) × R n ) \ {0} and functions α 1 and α 2 of class K ∞ such that α 1 (|x|) ≤ V (t, x) ≤ α 2 (|x|) (2)
hold for all (t, x) ∈ [0, ∞) × R n and such that there are bounded piecewise continuous functions a

: [0, ∞) → R and b : [0, ∞) → [0, ∞) such that V (t) ≤ a(t)V (t, x(t)) + b(t) sup ∈[t-τ,t] V ( , x( )) (3)
holds along all trajectories of (1). Assumption 2: There are positive constants β, , and and a piecewise continuous function

: [0, ∞) → [0, ∞) such that (t) ∈ [ , ]
for all t ≥ 0 and such that the function

µ(t) = a(t) + b(t) (4)
satisfies t 0 ( ( ) + µ( ))d ≤ β (5)
for all t ≥ 0. Moreover

κ(t) = sup ∈[t-τ,t] t (-(s) -µ(s))ds (6)
is a real number for each t ≥ 0, and

sup t≥0 e κ(t) -1 b(t) -(t) ≤ - (7)
holds.

See Section V for interesting applications where we can easily check the preceding assumptions. Our main result is:

Theorem 1: If Assumptions 1-2 hold, then the origin of (1) is a globally uniformly asymptotically stable equilibrium.

B. Discussion of Theorem 1

Consider the special case where there is a constant µ > 0 such that the function (4) satisfies

µ(t) ≤ -µ (8)
for all t ≥ 0, and b be any constant bound for b. Then, if

q ∈ 1, 1 + 1 2b µ , (9)
then

a(t) + qb(t) ≤ µ(t) + (q -1)b(t) ≤ -1 2 µ ( 10
)
is satisfied for all t ≥ 0. Therefore, for all t ≥ 0 such that

qV (t, x(t)) ≥ sup ∈[t-τ,t] V ( , x( )), (11)
the inequalities

V (t) ≤ a(t)V (t, x(t)) + b(t)qV (t, x(t)) ≤ - µ 2 V (t, x(t)) (12)
are satisfied. Then Razumikhin's theorem ensures the global uniform asymptotic stability of the origin of (1). However, our objective is precisely to establish stability results in cases where ( 8) is not satisfied. Notice that our assumptions include cases where the function a takes positive and negative values. Our discussion of Corollary 1 in the next section will explain the motivation for Assumption 2 and how it compares with the stability conditions obtained in [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF].

C. Proof of Theorem 1

Throughout the proof, all inequalities and equalities should be understood to hold for all t ≥ 0 and along all solutions of (1) unless otherwise indicated. Assumption 1 ensures the existence and the uniqueness of the solutions of (1). Let ā ≥ 0 and b ≥ 0 be any constant bounds for |a(t)| and b, respectively. We define the following functions

θ(t) = exp - t 0 ( (s) + µ(s))ds (13)
and

U (t, x) = θ(t)V (t, x) . ( 14
)
Then for all ∈ [t -τ, t], we have

θ(t) θ( ) ≤ e κ(t) . (15)
Also, ( 3)-( 4) give

d dt (U (t, x(t))) = θ(t)V (t, x(t)) + θ(t) d dt (V (t, x(t))) ≤ -( (t) + µ(t))U (t, x(t)) + a(t)U (t, x(t)) + θ(t)b(t) sup ∈[t-τ,t] V ( , x( )) = (-(t) -b(t))U (t, x(t)) + b(t) sup ∈[t-τ,t] θ(t)V ( , x( )).
It follows from [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] that

d dt (U (t, x(t))) = -( (t) + b(t))U (t, x(t)) + b(t) sup ∈[t-τ,t] θ(t) θ( ) U ( , x( )) ≤ -( (t) + b(t))U (t, x(t)) + b(t) sup ∈[t-τ,t] θ(t) θ( ) × sup ∈[t-τ,t] U ( , x( )) ≤ -( (t) + b(t))U (t, x(t)) + e κ(t) b(t) sup ∈[t-τ,t] U ( , x( )). (16)
Also, we can use the nonnegativity of b to get

e κ(t) b(t) ≤ exp t t-τ | (s) + µ(s)|ds b(t) ≤ e τ ( +a+b) b. (17)
Consequently, for all r > 0 such that

r ≤ 2e τ ( +a+b) b (18)
the inequality

re τ ( +a+b) b ≤ 2 (19)
is satisfied. This inequality combined with [START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF] ensures that

re κ(t) b(t) ≤ 2 . ( 20
)
Also, Assumption 2 ensures that for all t ≥ 0, we have

e κ(t) -1 b(t) -(t) ≤ -. (21)
From ( 20) and ( 21), it follows that for all t ≥ 0, we have

e κ(t) -1 b(t) + re κ(t) b(t) -(t) ≤ -2 . ( 22
)
Set q = 1 + r. Then grouping terms gives

qe κ(t) -1 b(t) -(t) ≤ -2 . ( 23
)
Next note that when qU (t, x(t)) ≥ sup ∈[t-τ,t] U ( , x( )), the second inequality in ( 16) and [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] give

U (t) ≤ -( (t) + b(t))U (t, x(t)) + e κ(t) b(t)qU (t, x(t) ≤ -2 U (t, x(t)).
(24) Also, (5) combined with Assumption 1 gives

e -β α 1 (|x|) ≤ U (t, x) ≤ e β α 2 (|x|) . (25)
Hence, the theorem follows from the classical Razumikhin theorem, by combining the decay estimate (24) with (25).

IV. PERIODIC CASE

A. Main Result for Periodic Case

We now consider key cases where the following holds: Assumption 3: Assumption 1 holds, and there is a constant T > 0 such that a and b in Assumption 1 are periodic of period T . Also, there exist a piecewise continuous function : R → R that is periodic of period T and constants > 0 and > 0 such that (t) ∈ [ , ] for all t ≥ 0 and

T 0 [ (m) + a(m) + b(m)]dm = 0 . (26)
Moreover, the function κ defined in ( 6) is such that

sup t∈[0,T ] e κ(t) -1 b(t) -(t) < 0 (27)
is satisfied. We are ready to state and prove the following result: Corollary 1: Let (1) satisfy Assumption 3. Then the origin of (1) is globally uniformly asymptotically stable.

B. Discussion of Corollary 1

A crucial question is whether Assumption 2 or Assumption 3 is restrictive with respect to the conditions obtained in [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF]. When the delays are small, our conditions are less restrictive than those imposed in [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] in the general nonlinear case.

To see why, consider the case where a and b have some period T and choose the constant function [(e τ sµ -1) b(t)] < * (30) is satisfied. This is the case if τ ∈ (0, τ * ), where

τ * = 1 sµ ln 1 + * b . (31)
Another sufficient condition can be obtained when µ is Lipschitz. Let us consider this case, and let λ µ > 0 denote a Lipschitz constant of µ. Then, since * = -1

T s s-T µ( )d , (32)
for all s ≥ 0, it follows that

κ(t) = sup ∈[t-τ,t] t 1 T s s-T µ(m)dm -µ(s) ds . ( 33
)
Since the function

ϕ(t) = 1 T t t-T t s µ(m)dmds (34)
satisfies

φ(t) = µ(t) -1 T t t-T µ(m)dm, (35)
we can integrate φ over [ , t] to get

κ(t) = sup ∈[t-τ,t] (-ϕ(t) + ϕ( )) . (36)
Since we can change variables to get

ϕ( ) = 1 T t t-T t s µ(m -t + )dmds, (37)
our choice λ µ > 0 of the Lipschitz constant and (36) give

κ(t) = sup ∈[t-τ,t] 1 T t t-T t s [-µ(m) + µ(m -t + )]dmds ≤ sup ∈[t-τ,t] λ µ T t t-T t s (t -)d ds ≤ T λ µ τ 2 , (38)
by upper bounding the last integrand in (38) by τ . Hence, Assumption 3 is satisfied if * > 0 and the inequality e

T λµτ 2 -1 b < * ( 39
)
are satisfied. This leads to the delay dependent condition

λ µ τ T < 2 ln 1 + * b . ( 40
)
Similar conditions cannot be derived from [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF], because the conditions in [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] do not depend on a Lipschitz constant and are delay independent, except in the particular case of the linear systems. In this particular case, our conditions can be less restrictive than those of [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] too. To understand why, consider the simple one-dimensional time-invariant system

ẋ(t) = -qx(t) + (q -1)x(t -τ ) ( 41
)
where q > 1 and τ ≥ 0 and constants. We consider (41) because it is easier to compare techniques in the timeinvariant case than in the time-varying case. It is well known that for any τ ≥ 0, the origin of ( 41) is globally exponentially stable. We show that our method makes it possible to prove this result, while the results in [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] lead to an extra constraint. Taking V (x) = |x|, Assumption 3 is satisfied with a = -q, b = q -1, κ = 0, and = q -(q -1) = 1. Hence, Corollary 1 ensures that the origin of ( 41) is exponentially stable. This cannot be deduced from the Razumikhin result for linear systems in [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF], since this earlier result only proves stability of (41) for values of τ smaller than a function of q.

C. Proof of Corollary 1

We show that Assumption 3 implies that Assumption 2 is satisfied, as follows. Let t > 0 be given, and let k be the integer such that t ∈ [kT , (k + 1)T ). Then

t 0 ( ( ) + µ( ))d ≤ kT 0 ( ( ) + µ( ))d + t kT ( ( ) + µ( ))d . (42) Since (26) implies that kT 0 ( ( ) + µ( ))d = 0, (43)
it follows that

t 0 ( ( ) + µ( ))d ≤ t kT ( + a + b)d ≤ T ( + a + b) . (44)
Next, we prove that κ is periodic of period T . We have

κ(t + T ) = sup ∈[t+T -τ,t+T ] t+T (-(m) -µ(m))dm = sup ∈[t-τ,t] t+T +T (-(m) -µ(m))dm = sup ∈[t-τ,t] t (-(m + T ) -µ(m + T ))dm.
Since both µ and are periodic of period T , it follows that

κ(t + T ) = sup ∈[t-τ,t] t (-(m) -µ(m))dm = κ(t) . (45)
From the fact that κ, and µ are all periodic of period T , we easily deduce that for all t ≥ 0, we have

sup t≥0 e κ(t) -1 b(t) -(t) ≤ - (46)
where

= -sup t∈[0,T ] e κ(t) -1 b(t) -(t) . (47)
The corollary now follows from Theorem 1.

V. EXAMPLES A. First Example

First consider the one dimensional system

ẋ(t) = -x(t) + b(t)x(t -τ (t)) (48)
where b : R → R is a periodic function of period 1 such that there are constants c ∈ (0, 1) and d > 0 satisfying (i) b(t) = 0 for all t ∈ [0, c) and (ii) b(t) = d for all t ∈ [c, 1). Assume that τ : [0, +∞) → [0, +∞) is continuous and bounded by a constant τ > 0. This system is slightly more general than the one used in [START_REF] Mazenc | Stability analysis for systems with time-varying delay: trajectory based approach[END_REF] and [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF], since it includes a time-varying delay. See below for higher dimensional examples. Since (48) is periodic, we apply Corollary 1. With V (x) = |x| and a(t) = -1, Assumption 1 is satisfied. Let * = -

1 0 [-1 + b(m)]dm = 1 -d(1 -c) (49)
We consider the case where τ < 1 4 and give conditions ensuring that Assumption 3 is satisfied.

To get less restrictive conditions than those obtained with (t) = * , pick any constant ν ∈ (0, min {1, * /c}), and let be the periodic 1 function such that (a) (t) = ν for all t ∈ [0, c) and (b)

(t) = * -cν 1-c for all t ∈ [c, 1]. Then (t) = 1 -cν 1 -c -d for all t ∈ [c, 1]. (50)
Also, for all t ≥ 1, we have (t) ∈ [ , ], where

= min ν, 1-cν 1-c -d > 0 and = max ν, 1-cν 1-c -d . (51)
We have

κ(t) = sup ∈[t-τ ,t] t [-(m) + 1 -b(m)]dm (52)
From the definition of b, it follows that (54) is equivalent to

exp sup t∈[c,1] κ(t) < 1 -cν d(1 -c) . (55)
Now observe that

-(m) + 1 -b(m) = -1-cν 1-c -d + 1 -d = c ν-1 1-c (56) for all m ∈ [c, 1], while -(m) + 1 -b(m) = -ν + 1 for all m ∈ [0, c]. We deduce that sup t∈[c,1] κ(t) ≤ τ (1 -ν). (57)
Then (55) is satisfied if

τ < 1 1-ν ln 1-cν d(1-c) . ( 58
) Recall that ν < * c = 1-d(1-c) c ( 59
)
and ν < 1. Taking the limit when ν goes to zero, we get the inequality τ < τ * , where τ * = -ln(d(1 -c)).

The work [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] uses a Razumikhin approach as well, but it does not provide a delay dependent result. However, [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF] uses a Krasovskii aproach to provide a delay dependent result, which only applies to (48) when τ is constant.

B. Second Example

Consider the system ẋ = -m(t) m(t)u(t -τ (t)) ( 60
)
where x is valued in R n for any n, the function m : R → R n is a continuous and has some period ω > 0, u ∈ R n is the input, and the delay τ is a time-varying piecewise continuous function that bounded by a constant τ > 0.

We studied (60) in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], but only in the case where τ is constant. The approach in [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF] is based in the construction of a Lyapunov-Krasovskii functional, which is written as the sum of a strict Lyapunov function for the corresponding undelayed system plus a double integral term. In this section, we use Corollary 1 to provide stabilizability conditions in the case where the delay is time-varying. Notice also that the result of [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF] does not apply to systems with a time-varying delay. We first introduce this assumption:

Assumption 4: The matrix M ∈ R n×n defined by

M = 1 ω ω 0 m(s) m(s)ds (61)
(x(t)) = x(t) (65)
is globally uniformly exponentially stabilized to the origin. Proof: The feedback (65) produces the system

ẋ = -m(t) m(t)x(t -τ (t)). ( 66
)
Our objective is now to prove that the origin of (66) is globally uniformly exponentially stable. This system is a linear time-varying system and therefore we can restrict our stability analysis to the time interval [2τ , +∞). For all t ≥ 2τ , we can rewrite the system as ẋ = -m(t) m(t)x(t)m(t) m(t) (67)

Let us consider ν 1 (x) = where M is the matrix defined in (61).

Let us introduce the function

V (t, x) = 2ν 1 (x) + ν 2 (t, x). (70)
Then for all t ≥ 0 and x ∈ R n , the inequalities

  (t) = * , where * = -µ = a + b as before. Then the function (6) satisfies κ(t) = sup ∈[t-τ,t] t (- * -µ(s))ds ≤ τ s µ (29) for all t ≥ 0, where s µ = sup s∈[0,T ] (- * -µ(s)). Then Assumption 3 holds if (a) * > 0 and (b) the inequality sup t∈[0,T ]

and t 0 (

 0 (t) -1 + b(t)) dt = 0. (53) Assumption 3 is satisfied if sup t∈[0,1] e κ(t) -1 b(t) -(t) < 0 .

  )| 2 ds and s m = sup t∈[0,ω] |m(t)| (62) and let λ M be the smallest eigenvalue of M . Then k m > 0, s m > 0 and λ M > 0. We now add these three assumptions: Assumption 5: The inequality 2λ M 1 + 2ωs

  (t) m(s) m(s)x(s -τ (s))ds.


Motivation

The construction of strict Lyapunov functions is important for proving stability and robustness properties for nonlinear control systems. In some cases, stabilization problems can be solved with the help of nonstrict Lyapunov functions, which are proper and positive definite functions whose time derivatives are nonpositive along all solutions of the closed loop system. By properness and positive definiteness of a function V , we mean that V is zero at the equilibrium, positive at all other states, and satisfies V (x) → ∞ as |x| → ∞ or as x approaches the boundary of the state space. However, nonstrict Lyapunov functions by themselves are insufficient to solve asymptotic stabilization problems, since they do not ensure convergence to the equilibrium. Instead, one often uses nonstrict Lyapunov functions, combined with LaSalle invariance or a Matrosov approach [START_REF] Mazenc | Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem[END_REF].

However, even if one uses LaSalle invariance or standard Matrosov approaches, there is usually no guarantee of robustness, e.g., under control or model uncertainty. This helped motivate the lecturers' 'strictification' approach for converting nonstrict Lyapunov functions into strict ones [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]. A strict Lyapunov function is a proper positive definite function whose time derivative is negative along all solutions of the system outside the equilibrium. Strict Lyapunov functions also allow us to robustify controls, e.g., to prove robustness in the key sense of input-to-state stability (or ISS).

To see how this 'robustification' approach can be done in the special case of time invariant nonlinear control affine systems of the form ẋ = f (x) + g(x)u(x), assume that we found a control u(x) such that the closed loop system is globally asymptotically stable to 0, and that we have a strict Lyapunov function V for the closed loop system such that -V (x) is proper and positive definite, or equivalently, there is a class K ∞ function α such that V (x) ≤ -α(|x|) holds along all trajectories of the closed loop system [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]. Then the closed loop system ẋ = f (x) + g(x)(u (x) + δ) has the ISS property with respect to the set of all measurable essentially bounded functions δ when we use u (x) = u(x) -(∇V (x)g(x)) , i.e., we get ISS with respect to actuator errors δ [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF].

However, to use u , one needs formulas for the gradient ∇V of the strict Lyapunov function, and this was another motivation for the strictification approach, but there are other motivations. For instance, having closed form strict Lyapunov functions leads to explicit formulas for comparison functions in ISS estimates, and can make backstepping possible. Analogous results can be shown for systems with delays, where strict Lyapunov functions for undelayed systems are replaced by strict Lyapunov-Krasovskii functionals, whose domains are infinite dimensional sets of functions, and this makes it possible to quantify the effects of input delays on the control performance.

Objectives

The preceding discussion motivates this mini-course, which presents several developments involving constructions of strict Lyapunov functions for systems with uncertainties, state constraints, or input delays. It will be understandable and beneficial to those who are familiar with the material in a standard graduate course on nonlinear control systems, and will consist of three 50-minute lectures.

Lecture 1: Matrosov's Approach (Lecturer: Frederic Mazenc)

The Matrosov approach involves combining known nonstrict Lyapunov functions for nonlinear systems with one or more so-called auxiliary functions, to build strict Lyapunov function for the systems [START_REF] Mazenc | Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem[END_REF]. While very general, it may not always be easy to find the auxiliary functions. In this lecture, we review a method for building the required auxiliary functions, based on iterated Lie derivatives, and we illustrate the method using a Lotka-Voltera dynamics [START_REF] Mazenc | Strict Lyapunov function constructions under LaSalle conditions with an application to Lotka-Volterra systems[END_REF]. We also provide an alternative Matrosov type construction for chains of exponential integrators where the right sides of the systems saturate, so they are not completely controllable [START_REF] Malisoff | Stabilization and robustness analysis for a chain of exponential integrators using strict Lyapunov functions[END_REF]. This leads to stabilization and tracking results for a broad class of predator-prey and other time-varying biologically inspired control systems that have input delays or positivity constraints on the states of the systems.

Lecture 2: Lyapunov-Krasovskii Methods (Lecturer: Michael Malisoff )

Starting from a strict Lyapunov function V for a controlled system without delays, it is often possible to add integral terms to V to obtain a Lyapunov-Krasovskii functional for the corresponding input delayed system, and to find a range of admissible values for the delays that ensure robust asymptotic stability properties under control or model uncertainties. This contrasts with prediction or reduction methods that take the delay value into account in the control design. Although prediction and reduction often compensate for arbitrarily long delays, a potential advantage of the Lyapunov-Krasovskii approach is that it allows us to use simpler controllers that were originally designed for systems without delays. This lecture will discuss the Lyapunov-Krasovskii approach, including illustrations involving a key curve tracking dynamics that arises in marine robotics [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF][START_REF] Mukhopadhyay | Collaborative autonomous surveys in marine environments affected by oil spills[END_REF].

Lecture 3: Robust Forward Invariance (Lecturer: Michael Malisoff ) Many applications such as collision avoidance problems require that tracking objectives be met, while also keeping the state in certain safe regions of the state space. Robust forward invariance can be used with Lyapunov-Krasovskii functionals to provide predictable tolerance and safety bounds that ensure that systems respect state constraints. It is a novel variant of the strong invariance property for differential inclusions, with the additional useful feature that it provides maximum allowable perturbation sets that the system can tolerate without violating the required tolerance and safety bounds. This lecture will discuss applications of robust forward invariance to humanpointer interactions under pointer acceleration [START_REF] Varnell | Stability and robustness analysis for human pointing motions with acceleration under feedback delays[END_REF], and to 3D curve tracking under delays and state constraints where an adaptive controller is used to identify unknown control gains [START_REF] Malisoff | Robustness of adaptive control under time delays for threedimensional curve tracking[END_REF].

Speaker Biographies

Frederic

  Mazenc was born in Cannes, France in 1969. He received his Ph.D. in Automatic Control and Mathematics from the CAS at Ecole des Mines de Paris in 1996. He was a Postdoctoral Fellow at CESAME at the University of Louvain in 1997. From 1998 to 1999, he was a Postdoctoral Fellow

  Constructions of Strict Lyapunov Functions: Stability, Robustness, Delays, and State Constraints

	Proposal for MTNS 2016 Mini-Course
	Lecturers: Frederic Mazenc

* and Michael Malisoff †

at the Centre for Process Systems Engineering at Imperial College. He was a CR at INRIA Lorraine from October 1999 to January 2004. In January 2004, he became CR1 at INRIA Sophia-Antipolis, and he has been CR1 at INRIA Saclay since January 2010. He received a best paper award from IEEE Transactions on Control Systems Technology at the 2006 IEEE Conference on Decision and Control, and a Best Presentation Award in an American Control Conference session. His current research interests include nonlinear control theory, differential equations with delay, robust control, and microbial ecology. He has more than 200 peer reviewed publications. Together with Michael Malisoff, he authored a research monograph entitled Constructions of Strict Lyapunov Functions in the Springer Communications and Control Engineering Series. He serves as Associate Editor for Asian Journal of Control, European Journal of Control, IEEE Transactions on Automatic Control, Journal of Control and Decision, and Mathematical Control and Related Fields. Michael Malisoff was born in the City of New York and received his Ph.D. in 2000 from the Department of Mathematics at Rutgers University in New Brunswick, NJ. In 2000, he became a DARPA Research Associate at Washington University in Saint Louis, as part of the Joint Force Air Component Commander Project. In 2001, he joined the tenure track faculty of the Department of Mathematics at Louisiana State University in Baton Rouge, where he is currently the Roy Paul Daniels Professor #3 in the College of Science. His awards include the First Place Student Best Paper Award at the 1999 IEEE Conference on Decision and Control, two three-year US National Science Foundation Mathematical Sciences Priority Area grants, and 9 Best Presentation awards in American Control Conference sessions. He has more than 100 publications on optimal control, feedback design, and engineering applications. He has served as Associate Editor for Automatica, IEEE Transactions on Automatic Control, and SIAM Journal on Control and Optimization.