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knowledge of **the** restitution rule is assumed at **the** collision time instants.
To facilitate **the** exposition capabilities of **the** developed synthesis and its robustness features are first illustrated **with** a simple linear 1-DOF mass-spring-damper system, impacting against a barrier. In addition to **the** numerical study, made for **the** **state** feedback regulation of this Mass- Spring-Damper-Barrier (MSDB) testbed, two different scenarios are invented and tested side by side for **the** MSDB position feedback tracking of an impact reference model, generating a stable limit cycle. In **the** first scenario, an impact reference output to follow is constructed off-line based on **the** impact Van der Pol oscillator (?). In **the** second scenario, **the** same reference output is updated on-line to synchronize its impacts to **the** time instants when **the** plant hits **the** **unilateral** constraint. As theoretically predicted, **the** disturbance attenuation is actually enforced and good performance of **the** closed-loop system is concluded from **the** numerical study being conducted for **the** former scenario. However, **the** disturbance-free closed-loop system proves to be asymptotically unstable because of **the** potential impact desynchronization (?). In **the** latter scenario, **the** reference trajectory tracking is tested under on-line synchronization of **the** reference velocity jumps to **the** collision time instants of **the** plant. Simulation runs are additionally conducted for this scenario to support **the** theory in that **the** closed-loop system is capable of retaining attractive robustness features while also presenting **the** asymptotic stability in **the** disturbance-free environment.

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Proving **the** Lipschitz continuity of **the** solution of an **optimal** **control** problem is of interest for obtaining error bounds for **the** discretization, see Dontchev and Hager [10].
There has been a renewed interest on these questions in **the** recent years. Shvartsman and Vinter [19] considered **the** case of first-order **state** **constraints** combined **with** **control** **constraints**, **the** latter possibly in an abstract form. Do Rosario de Pinho and Shvartsman [9] extended **some** of these **results** to **the** case when mixed **state** and **control** **constraints** are also present. A standard hypothesis in **the** field is **the** one of linear independence of gradients w.r.t. **the** **control** of active **constraints** (more precisely, active mixed **constraints** and total derivatives of active **state** **constraints**). In these two references, this standard hypothesis is weakened by introducing a sign condition on **the** regularity hy- pothesis related to **the** combinations of derivatives of **the** **state** **constraints** (see (34)). Independently, Hermant [13] showed how to extend Hager’s result when all **state** **constraints** are of second-order.

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0 ( ¯ x(1)) ¯ y(1), ¯ y(1)i ≥ 0 , (5)
for any suitable second-order tangent z to **the** **constraints**, see **the** exact definition of tangents in Section II.
We work in a quite general infinite dimensional frame- work, hence our **results** apply to **optimal** **control** problems involving **some** classes of PDEs, see [17], [18], [21] where reduction of **some** PDEs to **the** form (1) is discussed. In many phenomena, such as heat conduction, reaction-diffusion, population dynamics, economics, one seeks to optimize measures of best performances. **The** **optimal** **control** theory involving PDEs represents **the** natural framework to deal **with** such models. In this setting, second order analysis has been largely studied, **with** particular emphasis on sufficient second order conditions, due to their application in numerical analysis. It is impossible to provide here an exhaustive list of papers. **Some** significant contributions can be found in [3], [4], [5], [6], [7], [16], [19], [20], where evolution equations are analyzed **with** a particular interest to **the** parabolic case, see also **the** bibliographies therein. Second order optimality conditions are usually obtained by rewriting **the** **control** problem as an abstract mathematical programming one. However this approach requires Robinson like constraint qualification conditions, implying severe restrictions on **the** data. **Control** **constraints**, mixed (**control**-**state**) **constraints**, and **some** particular cases of **state** **constraints** were already investigated in **the** literature. Nevertheless, to our knowledge a general theory involving pure **state** **constraints** and end- point **constraints** is still lacking.

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1.1. Structure of **the** paper. In Section 2 we introduce **the** problem and **the** assumptions adopted in **the** paper, and we **state** our main result (Theorem 2.6), i.e., a O(¯ h) error estimate for **the** **control**, **state**, costate and multiplier. A key role in our construction is played by an homotopy path introduced in Section 3. **The** path links **the** continuous problem to **the** discrete one, involving **the** **control**, **state**, costate and multipliers, and creating a class of auxiliary problems. Through **the** study of **the** regularity of each auxiliary problem (Section 4) and checking that, under appropri- ate hypotheses, **the** application obtained (homotopy path) has bounded directional derivatives (in a sense clarified in **the** devoted section 5), we can establish **the** an- nounced convergence estimates for **the** discrete problem. More precisely, due to **some** coercivity properties of **the** Hessian of **the** Lagrangian, we first obtain a bound in **the** L 2 norm from which respective estimates in **the** L ∞ norm easily follow. In this

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In section 2, **the** Cauchy problem is studied in details. Section 3 deals **with** **the** existence of **optimal** controls, **the** material of these sections is rather standard but is included for **the** sake of completeness. **The** main novelty consists of **the** derivation of optimality conditions obtained in section 4. In **the** case of an unconstrained **control**, we first establish **the** Euler-Lagrange equations of **the** problem, then apply **the** optimality conditions to obtain **some** regularity **results**. We then consider **the** case of (convex) **constraints** on **the** **control** and prove that in this case, **the** optimality conditions can be written in **the** form of a Pontryagin principle **with** a nonlocal Hamiltonian. Finally, section 5 extends **the** previous result to **the** more general **state** equation (2).

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then **the** graph of **the** multivalued map Γ opt : Θ Ñ Γ, px, vq ÞÑ Γ opt rx, vs is closed in **the**
sense given in Proposition 2.1.
4 One dimensional problems: more accurate **results**
In dimension one and for a running cost quadratic in α, it is possible to obtain more accurate **results** under a slightly stronger assumption on **the** running cost, namely that it does not favor **the** trajectories which exit **the** domain. In particular, **the** closed graph property can be proved to hold on **the** whole set Ξ ad , and concerning mean field games, no assumptions are needed on **the** support of m 0 by contrast **with** Theorem 3.2.

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is that **the** dynamics F is considered to be uppersemi-continuous on **the** set K whereas in [20] it is assumed to be Lipschitz continuous.
**The** paper is organized as follows. In section 2, Mayer problem is introduced along **with** **the** assumptions. Then a characterisation of **the** value function is stated in Theorem 2.2. This result is discussed on a more specific example in subsection 2.4. Section 3 is devoted to **the** analysis of **some** increasing principles for stratified systems and gives a proof to **the** main result Theorem 2.2. Section 5 focused on **the** characterization for a **state**-constrained minimum time problem, and section 6 gives **some** comments on **the** main features of **the** paper. Finally, an appendix gathers **some** technical **results** that are needed in section 3.

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Lipschitz stability **results** for **state** **constraints** of first-order have been obtained by Malanowski [18] and Dontchev and Hager [9]. **The** difficulty of pure **state** **constraints** is **the** low regularity of multipliers, which are bounded Borel measures. For first-order **state** **constraints**, **the** multipliers are actually more regular (they are Lipschitz continuous func- tions, see Hager [14]). This additional regularity of solutions and multipliers is strongly used in **the** analysis in [18] and [9]. In those two papers, strong second-order sufficient conditions were used (that do not take into account **the** active **constraints**). **The** sufficient condition was recently weakened by Malanowski [21, 20].

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In **the** general case where **the** set of **state** **constraints** K is a non-empty closed subset of R d (K ( R d ), **the** value
function is merely l.s.c and its characterization as unique solution of a HJB equation requires **some** assumptions that involve an interplay between **the** dynamics f and **the** set of **constraints** K. A most popular assumption, called inward pointing condition, has been introduced in [27] and requires, at each point of **the** boundary of K, **the** existence of a **control** variable that lets **the** dynamics points in **the** interior of **the** set K. This assumption, when it is satisfied, provides a nice framework for analysing **the** value function and also **the** **optimal** trajectories. However, in many applications **the** inward pointing condition is not satisfied and then **the** characterization of **the** value function as solution of a HJB equation becomes much more delicate, see for instance [1, 19].

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Chapter 4 contains **the** main **results** of this thesis, namely **the** study of reset **control** systems **with** magnitude limitation on **the** output. From an analysis point of view, we aim at estimating **the** stability region and performance level of such a saturated hybrid system. In both cases, constructive conditions are developed. This chapter addresses also ﬁrst steps for **the** design of additional **control** laws to improve **the** stability region of **the** system. **The** ﬁrst approach rests on **the** deﬁnition of new reset rules. By increasing **the** admissible set of controller initial conditions, we enlarge **the** stability domain of **the** closed- loop system. **The** second approach is certainly more usual, but applies for **the** ﬁrst time on reset **control** systems. Given a reset law which improves closed-loop system behavior, we propose constructive conditions to design an anti-windup compensator which enlarges **the** set of admissible initial **state** values of **the** system from which **the** asymptotic stability is ensured. Finally, we expose a result which combines **the** two previous methods: **the** resulting quasi-LMIs allow us to synthesize adapted reset rules and anti-windup controller. To use **results** of Chapters 3 and 4 in an eﬃcient way, corresponding convex opti- mization problems are needed. This is **the** purpose of Chapter 5. For each theorem or proposition, we associate **the** minimization (or **the** maximization) of **some** decision vari- ables in a speciﬁc way: best performance estimation, attraction domain as large as possible ... These objectives ask sometimes to deal **with** **the** minimum (or **the** maximum) of fur- ther variables. **The** classical paradigm is retrieved: to solve a multi-objective problem, a trade-oﬀ have often to be done.

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holds **with** **some** positive definite functions = : , > = 0, … , ? for all @ > 0 and ? ∈ ℤ such that C ≤ @. This
definition is consistent **with** **the** notion of dissipativity introduced by Willems (1972) and Hill & Moyan (1980), that has become standard in **the** literature, and represents a natural extension for hybrid systems (see, e.g., **the** works by Neši , Zaccarian & Teel (2008), Yuliar, James & Helton (1998), Lin & Byrnes (1996) and Baras & James (1993)).

Starting from this functional equation and under suitable regularity assumptions, it can be proved that **the** value function satisfies a special kind of nonlinear partial differ- ential equation called **the** Hamilton-Jacobi-Bellman (HJB) equation. A first order equa- tion is typically obtained for **optimal** **control** problems set in a deterministic framework, whereas second order equations arise in **the** stochastic case. **The** high nonlinearity of **the** problem requires to consider weak notions of solutions. A breakthrough in this direction was **the** introduction in **the** early 80’s of **the** notion of viscosity solution by Michael G. Crandall and Pierre-Louis Lions [ 86 ]. In [ 132 ] and [ 133 ] Lions characterized **the** value function associated **with** **optimal** **control** problems for controlled diffusion processes as **the** unique continuous viscosity solution of a second order HJB equation. It turned out that viscosity solutions theory is **the** suitable framework for providing existence and uniqueness for a wider class of nonlinear equations: **the** Hamilton-Jacobi (HJ) equations. This is also a very convenient context for analyzing **the** numerical methods related to such nonlinear equations. In **the** later years a wide literature was produces on **the** subject providing different existence and uniqueness **results** (see [ 110 , 83 , 113 , 109 , 111 , 116 ]). When **state** **constraints** are taken into account **the** characterization of **the** value function as **the** viscosity solution of an HJB equation becomes much more complicated and, in absence of further assumption, uniqueness cannot in general be proved. We mention **the** works of Soner [ 160 , 161 ], Frankowska et al. [ 101 , 100 ], Katsolulakis [ 117 ], Barles and Rouy [ 35 ], Bouchard and Nutz [ 62 ] and many others for **the** discussion of **the** suitable sets of assumptions to be considered.

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In this paper, **the** **control** problem of an aircraft in a climbing phase was modeled as a Mayer **optimal** **control** problem **with** two **state** **constraints** of order 1 and **with** affine and quadratic dependance **with** respect to **the** **control**. We have presented an approach which combines geometric analysis and numerical methods to compute candidates as minimizers which are selected among a set of extremals, solutions of a Hamiltonian system given by **the** maximum principle (section 3). **The** **optimal** trajectory is a concatenation of boundary and interior arcs. In section 3.6, we compute for each type of arcs we encounter in **the** numerical experiments, **the** **control** law, **the** multiplier associated to **the** constraint (scalar of order 1) and **the** jumps on **the** adjoint vector at junction times. Then we combined **the** theoretical **results** **with** indirect and direct methods to compute solutions which satisfy necessary conditions of optimality given by **the** maximum principle. One can find two technical **results** in section 4.1. First, proposition 4.1 justifies **the** use of **the** direct method to initialize **the** indirect method in **the** case of **state** constrained **optimal** **control** problems. It justifies also **the** direct adjoining approach, see 3.1. **The** second result from proposition 4.2 is a key tool for **the** definition of **the** multiple shooting functions which are solved by **the** indirect method, see section 4.1.3. At **the** end, we illustrate **the** approach **with** two examples: in section 4.2.1, we give a result about **the** minimum time problem while in section 4.2.2, one can find a more complex trajectory of **the** form σ + σ γ σ + σ v σ − in **the** case of a mixed criterion between time of flight and fuel consumption.

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This is in accordance **with** **the** convergence result given in § 5.3 of **the** minimal time for **the** discretized problem, to **the** minimal time of **the** continuous one, as N t and N x tend to +∞
(preserving **the** CFL condition).
It is interesting to note on Figure 2 that **the** two controls coincide and are identically equal to 0 over long time subintervals. We have shown that **the** controls can always be chosen to coincide; but, in general, they are not unique. And yet, **the** controls we obtain in our numerical experiments are always symmetric. This raises **the** question of knowing whether, at **the** minimal time, **the** controls are unique (if so, they must be equal). **The** numerical simulations also raise **the** interesting question of whether **the** controls in **the** minimal **control** time necessarily present a “sparse” structure **with** long lags where they are identically zero.

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of **the** set of C ∞ functions over Q, (78) holds. **The** conclusion follows.
3.4 Arc based alternative formulation
We next present a variant of **the** alternative formulation that is useful when for- mulating shooting algorithms; see Bonnans and Hermant [15] for related **results** in a finite dimensional setting. Let us start **with** **some** definitions. A boundary arc (resp. interior arc) is a maximal interval of positive measure I such that g(y)(t) = 0 (resp. g(y)(t) < 0), for all t ∈ I. Left and right endpoints of a boundary arc [τ en , τ ex ] are called entry and exit point, respectively. A touch

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6.2. Homotopy Algorithm and Numerical Simulations
6.2.1 Solving (OCP) τ by Shooting Methods and Homotopy on Delays
As pointed out previously, **the** critical behavior coming out from applying indirect methods to (OCP) τ consists of **the** integration of mixed-type equations that arise from system (6.4). **The** convergence **results** of Theorem 6.1 suggests **the** idea that, when considering indirect methods coupled to homotopies on **the** delay, one may solve **the** adjoint system (6.4) via usual iterative methods for ODEs, for example, by using **the** global **state** solution at **the** previous iteration. Moreover, **the** global adjoint vector of (OCP) could be used to initialize, from **the** beginning, **the** whole procedure. Under assumptions of nice enough extremals, these considerations lead us to Algo- rithm 3. For sake of brevity, we design this algorithm for problems (OCP) τ **with** pure **state** delays, i.e. τ = (τ 1 , 0) and fixed final time t f . Nevertheless, it is evident how to adapt Algorithm 3 to problems (OCP) τ **with** general constant delays and free final time t f . For sake of clarity, we do not consider any acceleration step in Algorithm 3, although, adapting this speed-up procedure from Algorithm 1 is straightforward. We prove **the** convergence of Algorithm 3, under appropriate assumptions, by apply- ing Theorem 6.1. Without loss of generality, we focus on **the** case where (OCP) τ has pure **state** delays. We assume that we are able to express **optimal** controls, via (6.5), as continuous functions of x and p (note that we do not remove nonregular extremals). Suppose that assumptions (A) and (B ) hold and that **the** delay τ = (τ 1 , 0) ∈ [0,∆] × {0} considered is such that τ 1 ∈ (0, τ 0 ), where τ 0 is provided by Theorem 6.1. Therefore,

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