H Él

Ène Frankowska

Ihab Haidar

A relaxation result for state constrained delay differential inclusions

Keywords: Delay differential inclusions, relaxation, state constraints, inward pointing conditions

In this paper, we consider a delay differential inclusion ẋ(t) ∈ F (t, xt), where xt denotes the history function of x(t) along an interval of time. We extend the celebrated Filippov's theorem to this case. Then, we further generalize this theorem to the case when the state variable x is constrained to the closure of an open subset K ⊂ R n . Under a new "inward pointing condition", we give a relaxation result stating that the set of trajectories lying in the interior of the state constraint is dense in the set of constrained trajectories of the convexified inclusion ẋ(t) ∈ co F (t, xt).

I. INTRODUCTION

Mathematical models arising in population dynamics or engineering sciences often involve control systems with delays (see, e.g., [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Kuang | Delay differential equations with application in population dynamics[END_REF]). Systems with delays, express that at each instant the velocity of the state depends upon the history of its evolution up to this instant [START_REF] Hale | Introduction to functional differential equations[END_REF]. Such control systems can be described as follows:    ẋ(t) = f (t, x t , u(t)), a.e. t ∈ [t 0 , T ], u(t) ∈ U ⊂ R q , a.e. t ∈ [t 0 , T ],

x t0 = ϕ, (1)
where x(t) ∈ R n , represents the state at time t, x t : [-τ, 0] → R n is the standard notation for the history function defined by x t (θ) = x(t + θ), for τ > 0 and -τ ≤ θ ≤ 0, u(•) is a Lebesgue measurable function, f is a mapping from [0, T ] × C([-τ, 0], R n ) × U into R n , 0 ≤ t 0 ≤ T , and ϕ is the initial condition taken in C([-τ, 0], R n ). In the above, C([-τ, 0], R n ) denotes the Banach space of continuous functions from [-τ, 0] into R n , with the usual norm.

This work was partially funded by the DeMagma project of Programme Convergence from Idex Super at Sorbonnes Universités.

When the trajectories of (1) are subject to the state constraint

x(t) ∈ K ∀ t ∈ [t 0 , T ], (2)
where K is a closed subset of R n , the viability theory [START_REF] Aubin | Viability theory[END_REF] provides adequate mathematical tools to study the existence of feasible (or viable) solutions of such systems. Thanks to this theory, a necessary and sufficient condition (linking the dynamics of system [START_REF] Aiello | A time-delay model of singlespecies growth with stage structure[END_REF] to the geometry of the constraint set K) for the existence of feasible solutions is known. Under some regularity assumptions on f , this condition was first given in [START_REF] Haddad | Monotone viable trajectories for functional differential inclusions[END_REF]:

∀ t ∈ [0, T ], ∀ ψ ∈ C([-τ, 0], R n ) such that ψ(0) ∈ K, f (t, ψ, U ) ∩ T K (ψ(0)) = ∅, (3)
where T K (ψ(0)) is the contingent cone to K at ψ(0). In the framework of this theory, convexity conditions are imposed on the set-valued map F (t, ψ) := f (t, ψ, U ), i.e. for every t ∈ [0, T ] and every ψ ∈ C([-τ, 0], R n ), F (t, ψ) is a convex subset of R n . This convexity hypothesis may fail in some mathematical models and may be even difficult to verify. In the case of delay-free control systems, a vast literature (see, e.g., [START_REF] Bettiol | On trajectories satisfying a state constraint: W 1,1 estimates and counter-examples[END_REF], [START_REF] Bettiol | L ∞ estimates on trajectories confined to a closed subset[END_REF], [START_REF] Bettiol | Improved sensitivity relations in state constrained optimal control[END_REF], [START_REF] Forcellini | On nonconvex differential inclusions whose state is constrained in the closure of an open set. applications to dynamic programming[END_REF], [START_REF] Frankowska | A relaxation result for state constrained inclusions in infinite dimension[END_REF], [START_REF] Frankowska | Filippov's and Filippov-Wazewski's theorems on closed domains[END_REF], [START_REF] Frankowska | Existence of neighboring feasible trajectories: Applications to dynamic programming for state-constrained optimal control problems[END_REF]) allows to relax this convexity hypothesis, by assuming, as a counterpart, stronger tangential conditions and stronger regularity of F . These conditions rely on the possibility of directing a velocity into the interior of the constraint K whenever approaching the boundary of K. Known as inward pointing conditions, they allow to approximate relaxed feasible trajectories by feasible trajectories and provide estimates on the distance of a given trajectory of unconstrained control system from the set of feasible trajectories, see for instance [START_REF] Bettiol | On trajectories satisfying a state constraint: W 1,1 estimates and counter-examples[END_REF], [START_REF] Bettiol | L ∞ estimates on trajectories confined to a closed subset[END_REF], [START_REF] Frankowska | A relaxation result for state constrained inclusions in infinite dimension[END_REF], [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF], [START_REF] Frankowska | Filippov's and Filippov-Wazewski's theorems on closed domains[END_REF]. In the literature, these estimates have been referred to as neighboring feasible trajectory (NFT) estimates. In the case when F is Lebesgue measurable with respect to the time and Lipschitz with respect to the state, NFT estimates result from the following inward pointing condition (see [START_REF] Frankowska | Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints[END_REF], [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF]):

               ∀ t ∈ [0, T ], ∀ x ∈ ∂K, ∀ v ∈ F (t, x) such that max n∈N 1 K (x) n, v ≥ 0, ∃ w ∈ Liminf (s,y)→(t,x) co F (s, y) satisfying max n∈N 1 K (x) n, w -v < 0, (4)
where co F (s, y) is the convex hull of F (s, y), Liminf denotes the Kuratowski lower set limit (see [START_REF] Aubin | Set-Valued Analysis[END_REF]),

N 1 K (x) := N K (x) ∩ S n-1 , S n-1
is the unit sphere and N K (x) denotes the Clarke normal cone to K at x (see [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]). The above condition takes sometimes a simpler form depending on the regularity assumptions on F and the smoothness of the boundary ∂K (see, e.g., [START_REF] Bettiol | On trajectories satisfying a state constraint: W 1,1 estimates and counter-examples[END_REF], [START_REF] Bettiol | L ∞ estimates on trajectories confined to a closed subset[END_REF], [START_REF] Bettiol | Improved sensitivity relations in state constrained optimal control[END_REF], [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF], [START_REF] Frankowska | Existence of neighboring feasible trajectories: Applications to dynamic programming for state-constrained optimal control problems[END_REF]). When the viability condition fails to be fulfilled on the boundary of K, the largest subset of initial conditions starting from which at least one viable solution exists (called viability kernel) is considered. In the case of delay-free control systems, viability algorithms providing constructive methods for the computation of the viability kernel, have been developed (see, e.g., [START_REF] Frankowska | Viability kernels of differential inclusions with constraints: algorithm and applications[END_REF], [START_REF] Saint-Pierre | Approximation of the viability kernel[END_REF]). Thanks to these algorithms, efficient numerical methods have been established (see, e.g., [START_REF] Rouquier | A kd-tree algorithm to discover the boundary of a black box hypervolume[END_REF]) and used in order to exhibit approximating viability kernels for numerous examples coming from different fields (see, e.g., [START_REF] Aubin | An introduction to viability theory and management of renewable resources[END_REF], [START_REF] Bernard | Building strategies to ensure language coexistence in presence of bilingualism[END_REF], [START_REF] Sicard | Congress of Mathematicians ICM 2010 in Hyderabad. Her research activities are focused on optimal control, both in the deterministic and stochastic framework, control of systems under state constraints, viability theory, differential inclusions, setvalued and variational analysis[END_REF], [START_REF] Lara | Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management[END_REF], [START_REF] Haidar | Mathematical modeling of an urban pigeon population subject to local management strategies[END_REF], [START_REF] Regan | Robust decision-making under severe uncertainty for conservation management[END_REF]). We would like to underline that these algorithms are developed for setvalued maps with convex values. Two steps are needed to extended these numerical methods to delay differential inclusions: adapt the viability algorithms to this case and obtain relaxation theorems under state constraints. This latter point is the purpose of this paper. To our knowledge, NFT estimates for delay differential inclusions are not yet obtained in the literature. Here, we propose to extend such results to this case. Inspired by the viability condition given by (3), we propose to adapt the inward pointing condition (4) to delay differential inclusions. Let λ > 0. Define the set

K λ := {ψ ∈ C([-τ, 0], R n ) : ψ is λ-Lipschitz and ψ(0) ∈ ∂K} , (5)
and consider the following relaxed inward pointing condition:

(IP λ rel )                ∀ t ∈ [0, T ], ∀ ψ ∈ K λ , ∀ v ∈ F (t, ψ) such that max n∈N 1 K (ψ(0)) n, v ≥ 0, ∃ w ∈ Liminf (s,φ)→(t,ψ) co F (s, φ) satisfying max n∈N 1 K (ψ(0)) n, w -v < 0.
Assuming (IP λ rel ), we give a relaxation result stating that the set of feasible trajectories is dense in the set of relaxed feasible ones. This is proved by using several preliminary results. The first one is an extension of the Filippov theorem, given by [START_REF] Filippov | Classical solutions of differential equations with multivalued right hand side[END_REF], to delay differential inclusions, which is an essential step to construct feasible trajectories. Then, we provide NFT estimates on the distance of a given trajectory from the set of feasible trajectories.

The paper is organized as follows. Section II presents the list of notations, definitions and assumptions in use. In Section III we state our main results. The proofs and useful technical tools are postponed to Section V. In Section IV we discuss an example, where the model corresponds to the problem considered in the present work.

II. PRELIMINARIES

In this section we list the notations and the main assumptions in use.

A. Notations and definitions

Consider the Euclidean space (R n , • ), where n is a positive integer. We denote by •, • the inner product, by B(x, r) the closed ball of center x ∈ R n and radius r > 0 and by B the closed unit ball in R n centered at 0. Let co A stands for the convex hull of a subset A ⊂ R n . For every pair (a, b) ∈ R 2 , set a ∨ b = max{a, b} and a ∧ b = min{a, b}.

Given I ⊂ R, (C(I, R n ), • C ) denotes the Banach space of continuous functions from I into R n , where

• C is the norm of uniform convergence. Given τ > 0, B C (ϕ, r) denotes the closed ball of center ϕ ∈ C([-τ, 0], R n ) and radius r > 0 and B C is the closed unit ball in C([-τ, 0], R n ) centered at 0. Given t ∈ R, we denote by B((t, ϕ), r) the closed ball B(t, r) × B C (ϕ, r).

We denote by µ the Lebesgue measure on the real line, and by L 1 (I, R n ) the space of Lebesgue integrable functions from I to R n .

Let K be a nonempty closed subset of R n , Int K be its interior and ∂K its boundary, dK is the oriented distance from x ∈ R n to K defined by

dK (x) = d K (x) if x / ∈ K -d R n \K (x)
otherwise, where d K (x) = inf y∈K x -y . We will use the following notion of solution:

Definition 1: Let 0 ≤ t 0 ≤ T , τ > 0 and ϕ ∈ C([-τ, 0], R n ). A function x ∈ C([t 0 -τ, T ], R n ) is called an F -trajectory, if x(•) is absolutely continuous on [t 0 , T ] and ẋ(t) ∈ F (t, x t ) a.e. t ∈ [t 0 , T ], (6)
x t0 = ϕ. (7)
An F -trajectory which verifies the state constraint ( 2) is called feasible F -trajectory. A trajectory associated to the relaxed differential inclusion

ẋ(t) ∈ co F (t, x t ), a.e. t ∈ [t 0 , T ], x t0 = ϕ (8)
is called relaxed F -trajectory, and relaxed feasible Ftrajectory if in addition (2) holds true.

B. Assumptions

Let 0 ≤ t 0 ≤ T , τ > 0 and F :

[t 0 , T ] × C([-τ, 0], R n )
R n be a set-valued map with nonempty closed images. In our main theorems, we will assume the following regularity conditions on F : (H1) for every

ψ ∈ C([-τ, 0], R n ) the set-valued map F (•, ψ) is measurable; (H2) the set-valued map F (t, •) is locally Lipschitz in the following sense: ∀ R > 0, ∃ ζ R (•) ∈ L 1 ([t 0 , T ], R + ) such that, for a.e. t ∈ [t 0 , T ] and any ϕ, ψ ∈ RB C F (t, ϕ) ⊂ F (t, ψ) + ζ R (t) ϕ -ψ C B;
(H3) the set-valued map F has a sublinear growth, i.e.

there exists σ > 0 such that, for a.e. t ∈ [t 0 , T ] and any

ψ ∈ C([-τ, 0], R n ) F (t, ψ) ⊂ σ (1 + ψ C ) B;
(H4) for a given λ > 0, the set-valued map F is upper semicontinuous on [t 0 , T ] × K λ , i.e. for all t ∈ [t 0 , T ] and all ϕ ∈ K λ , we have F (t, ϕ) = ∅ and for every ε > 0 there exists δ > 0 such that

F (s, ψ) ⊂ F (t, ϕ) + εB ∀ (s, ψ) ∈ B((t, ϕ), δ).

III. MAIN RESULTS

A. Filippov's Theorem

The following theorem extends the celebrated Filippov's theorem, given by [START_REF] Filippov | Classical solutions of differential equations with multivalued right hand side[END_REF], to differential inclusions of type [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF].

Theorem 1: Let β > 0 and δ 0 ≥ 0 and assume (H1), (H2).

Let y ∈ C([t 0 -τ, T ], R n ) be such that y(•) is ab- solutely continuous on [t 0 , T ]. Set R = max t∈[t0-τ,T ] y(t) , γ 1 (t) = d F (t,yt) ( ẏ(t)), γ 2 (t) = exp t t0 ζ R+β (s)ds , γ 3 (t) = γ 2 (t) δ 0 + t t0 γ 1 (s)ds . (9) If γ 3 (T ) < β, then for all ϕ ∈ C([-τ, 0], R n ) with ϕ -y t0 C ≤ δ 0 , there exists x ∈ C([t 0 -τ, T ], R n ) such that x(•) is an F -trajectory and for all t ∈ [t 0 , T ] x t -y t C ≤ γ 3 (t) and for almost every t ∈ [t 0 , T ], ẋ(t) -ẏ(t) ≤ ζ R+β (t)γ 3 (t) + γ 1 (t).
The following theorem establishes the possibility of approximating any relaxed F -trajectory by an F -trajectory starting from the same initial condition.

Theorem 2: Let y(•) be a relaxed F -trajectory. Assume (H1), (H2) and (H3). Then for every δ > 0 there exists an F -trajectory x(•) satisfying x t0 = y t0 and sup t∈[t0,T ] x(t) -y(t) ≤ δ.

B. Neighboring feasible trajectories theorems

Let λ > 0. Consider the following inward pointing condition:

(IP C λ )                ∀ t ∈ [0, T ], ∀ ψ ∈ K λ , ∀ v ∈ F (t, ψ) such that max n∈N 1 K (ψ(0)) n, v ≥ 0, ∃ w ∈ Liminf (s,φ)→(t,ψ) F (s, φ) satisfying max n∈N 1 K (ψ(0)) n, w -v < 0,
where K λ is defined by [START_REF] Aumann | Integrals of set-valued functions[END_REF]. Before stating our first NFT theorem, a crucial result is given by the following lemma which shows that (IP C λ ) implies an uniform inward pointing condition on a neighborhood of K λ . Lemma 1: Let λ > 0 and assume (H1)-(H4) and

(IP C λ ). Then ∀ R > 0, ∃ ρ > 0 and η > 0 such that for every t ∈ [0, T ], ψ ∈ (K λ + ηB C ) ∩ RB C and for every v ∈ F (t, ψ) with max n∈N 1 K (x),x∈∂K∩B(ψ(0),η) n, v ≥ 0 we can find w ∈ F (t, ψ) satisfying n, w ≤ -ρ and n, w -v ≤ -ρ ∀ n ∈ N 1 K (x), ∀ x ∈ ∂K ∩ B(ψ(0), η). (10)
The following theorem shows the existence of a feasible F -trajectory and provides an estimate of the distance (in the norm of uniform convergence) of this trajectory from a specified F -trajectory. Theorem 3: Assume (H1)-(H3). Let τ > 0, r 0 > 0 and λ 0 > 0 and suppose that, for

λ = max{λ 0 , (1 + (1 + λ 0 τ + r 0 )e σT )σ}, (11)
assumptions (H4) and (IP C λ ) hold true. Then there exists a constant C > 0 such that for any t 0 ∈ [0, T ] and every F -trajectory x(•) on [t 0 -τ, T ] with λ 0 -Lipschitz xt0 and x(t 0 ) ∈ K ∩ r 0 B, and for any ε 0 > 0, we can find a feasible F -trajectory on [t 0 -τ, T ] satisfying

x t0 = xt0 , x((t 0 , T ]) ⊂ Int K and x t -xt C ≤ C max t∈[t0,T ] d K (x(t)) + ε 0 . (12)
Theorem 3 together with Theorem 2 imply that under the inward pointing condition (IP C λ ), the set of Ftrajectories lying in the interior of the constraint set K, for t ∈ (t 0 , T ] and starting at xt0 , is dense in the set of feasible relaxed F -trajectories. This results from the following corollary: Corollary 1: Under all the assumptions of Theorem 3, for any feasible relaxed F -trajectory x(•) with λ 0 -Lipschitz xt0 and x(t 0 ) ∈ K ∩ r 0 B, and any δ > 0, there exists a feasible F -trajectory x(•) such that x t0 = xt0 , x((t 0 , T ]) ∈ Int K and x t -xt C < δ for all t ∈ [t 0 , T ]. Now, assume the relaxed inward pointing condition given by (IP λ rel ). As before, we have the following lemma which is similar to Lemma 1 but in the framework of the relaxed set-valued map.

Lemma 2: Let λ > 0 and assume (H1)-(H4) and (IP λ rel ).

Then ∀ R > 0, ∃ ρ > 0 and η > 0 such that for every t ∈ [0, T ], ψ ∈ (K λ + ηB C ) ∩ RB C and for every v ∈ co F (t, ψ) with max n∈N 1 K (x),x∈∂K∩B(ψ(0),η) n, v ≥ 0, we can find w ∈ co F (t, ψ) satisfying n, w ≤ -ρ and n, w -v ≤ -ρ ∀ n ∈ N 1 K (x), ∀ x ∈ ∂K ∩ B(ψ(0), η).
The following theorem is related to Theorem 3, however neither one is contained in another.

Theorem 4: Assume (H1)-(H3). Let τ > 0, r 0 > 0 and λ 0 > 0 and suppose that, for λ given by [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], assumptions (H4) and (IP λ rel ) hold true. Then there exists a constant C > 0 such that for any t 0 ∈ [0, T ] and every relaxed F -trajectory x(•) on [t 0 -τ, T ] with λ 0 -Lipschitz xt0 and x(t 0 ) ∈ K ∩ r 0 B, and for any ε 0 > 0, we can find a relaxed feasible F -trajectory on [t 0 -τ, T ] satisfying x t0 = xt0 , x((t 0 , T ]) ⊂ Int K and

x t -xt C ≤ C max t∈[t0,T ] d K (x(t)) + ε 0 . ( 13
)
The proof of Theorem 4 is a straightforward consequence of Lemma 2, together with Theorem 3 applied with co F instead of F . Theorem 4 and the constructive argument of [9, Proof of Lemma 5.2] imply the following Corollary:

Corollary 2: Under all the assumptions of Theorem 4, for any relaxed feasible F -trajectory x(•) with λ 0 -Lipschitz xt0 and x(t 0 ) ∈ K ∩ r 0 B, and any δ > 0, there exists a feasible F -trajectory x(•) such that

x t0 = xt0 , x((t 0 , T ]) ∈ Int K and x t -xt C < δ for all t ∈ [t 0 , T ].

C. Neighboring feasible trajectories theorem: constant delay case

Consider the constant-delay differential inclusion

ẋ(t) ∈ F(t, x(t), x(t -τ )), a.e. t ∈ [t 0 , T ], x t0 = ϕ, (14) where
F : [0, T ] × R n × R n
R n is a set-valued map having closed nonempty images and ϕ ∈ C([-τ, 0], R n ). Let λ > 0. Consider the following inward pointing condition:

(IP λ eq )                ∀ t ∈ [0, T ], ∀ x ∈ ∂K, ∀ y ∈ x + τ λB, ∀ v ∈ F(t, x, y) such that max n∈N 1 K (x) n, v ≥ 0, ∃ w ∈ Liminf (s,z,ξ)→(t,x,y) co F(s, z, ξ) satisfying max n∈N 1 K (x) n, w -v < 0.
Assume the following regularity conditions on F: (A1) for every

X = (x, y) ∈ R n × R n the set-valued map F(•, X) is measurable; (A2) the set-valued map F(t, •) is locally Lipschitz, i.e. ∀ R > 0, ∃ ζ R (•) ∈ L 1 ([t 0 , T ], R + ) such that, for a.e. t ∈ [t 0 , T ] and any X = (x 1 , y 1 ), Y = (x 2 , y 2 ) ∈ RB × RB F(t, X) ⊂ F(t, Y ) + ζ R (t) X -Y B;
(A3) there exists σ > 0 such that, for a.e. t ∈ [t 0 , T ] and any

X = (x, y) ∈ R n × R n F(t, X) ⊂ σ(1 + |X|)B;
(A4) for a given λ > 0, the set-valued map F is upper semicontinuous on [t 0 , T ] × (∂K + τ λB). Theorem 5: Assume (A1)-(A3). Let τ > 0, r 0 > 0 and λ 0 > 0 and suppose that, for λ given by [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], assumptions (A4) and (IP λ eq ) hold true. Then there exists a constant C > 0 such that for any t 0 ∈ [0, T ] and every F-trajectory x(•) on [t 0 -τ, T ] with λ 0 -Lipschitz xt0 and x(t 0 ) ∈ K ∩ r 0 B, and for any ε 0 > 0, we can find a feasible F-trajectory on [t 0 -τ, T ] satisfying x t0 = xt0 , x((t 0 , T ]) ⊂ Int K and

x t -xt C ≤ C max t∈[t0,T ] d K (x(t)) + ε 0 . (15)
IV. EXAMPLE: POPULATION DYNAMICS Here, we present an example that motivated us for this study. It concerns the management of urban pigeon population. In fact, urban pigeon population can reach high densities in cities and disturb the cohabitation with urban citizens. In view of some ecological considerations, this increased population may lead to a citizen dissatisfaction. A model describing the evolution of such population which is subject to some management strategies is proposed in [START_REF] Haidar | Mathematical modeling of an urban pigeon population subject to local management strategies[END_REF], and the urban citizen tolerance formulated as a state constraint. The population dynamics is given by 1

ẋ1 = n(x 2 , u)x 2 -m 1 (x 1 , u)x 1 -p 1 (x 1 , u)x 1 ẋ2 = -m 2 (x 2 , u) x 2 + p 1 (x 1 , u)x 1 , (16)
where x 1 and x 2 denote the size of juvenile and adult pigeon populations and u is the control parameter relative to a management strategy (resource limitation, egg removal, sterilization, capturing). The function n(•) describes the reproduction of adult pigeon; m 1 (•) and m 2 (•) describe the mortality of juvenile and adult pigeons. The function p 1 (•) represents the transfer rate from juvenile to adult class. The urban citizen tolerance is described through the following state constraints set

K = M ≤ x 1 + x 2 ≤ M , ∀t ≥ 0, x 1 ≥ 0, x 2 ≥ 0, ∀t ≥ 0,
where M and M represent the lower (the presence of some pigeons) and upper (not too many pigeons) limits. Thanks to the viability theory, viability kernels describing the existence of efficient management strategies are calculated (see [START_REF] Haidar | Mathematical modeling of an urban pigeon population subject to local management strategies[END_REF] for more details). The model ( 16) is not sufficiently precise, because it 1 By abuse of notation, we omit writing explicitly the dependence of ẋi , x i , and u on t, for i = 1, 2.

does not take into account the fact that juvenile pigeons start to reproduce only after becoming adults. This leads naturally to a model with time delay that we describe next. Actually, the transfer from juvenile to adult class is modeled as a function with a delay involving the adult pigeons, taking into account the following observation: the juveniles which born a time t -τ and survive to time t exit to adult pigeon class, where τ is the time from birth to maturity. This can be formulated by the following equations (see [START_REF] Aiello | A time-delay model of singlespecies growth with stage structure[END_REF] for more details)

ẋ1 = n(x 2 , u)x 2 -m 1 (x 1 , u)x 1 -p 2 x 2 (t -τ ) ẋ2 = -m 2 (x 2 , u)x 2 + p 2 x 2 (t -τ ), (17)
where p 2 is the corresponding survival rate of juvenile pigeons born at τ -time ago, which is taken constant in order to simplify the presentation (in fact, it depends on x 2 (t -τ ) and u). Let x = (x 1 , x 2 ) and y = (y 1 , y 2 ) ∈ R 2 . System (17) can be written in the form of [START_REF] Frankowska | A priori estimates for operational differential inclusions[END_REF], where the set-valued map F : R 2 × R 2 R 2 is given by

F(x, y) = u∈U n(x 2 , u)x 2 -m 1 (x 1 , u)x 1 -p 2 y 2 -m 2 (x 2 , u) x 2 + p 2 y 2 .
To extend the study realized in [START_REF] Haidar | Mathematical modeling of an urban pigeon population subject to local management strategies[END_REF] to the time delayed case, Corollary 2 is crucial. Indeed, viability algorithms are conceived for convex maps. In the case of ( 17), in general, F is not convex. Knowing that n(•), m 1 (•) and m 2 (•) are sufficiently regular then F fulfills the assumption of Theorem 5. Thus, if condition (IP λ eq ) holds true on the boundary of K, Corollary 2 guaranties the existence of feasible trajectories for [START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF], approximating a feasible relaxed trajectory of the convexified problem.

V. PROOFS A. Proof of Theorem 1

We need the following lemma from [START_REF] Frankowska | A priori estimates for operational differential inclusions[END_REF]: Lemma 3: Let X be a separable Banach space, G be a set-valued map from [t 0 , T ] × X into closed nonempty subsets of X and z : [t 0 , T ] → X be a continuous function such that 1)

∀ x ∈ X the set-valued map G(•, x) is measurable. 2) ∃ β > 0, ζ(•) ∈ L 1 ([t 0 , T ], R + ) such that for almost all t ∈ [t 0 , T ] the map G(t, •) is ζ(t)-Lipschitzian on z(t) + βB X
, where B X is the closed unit ball in X centered at 0. Let x ∈ C([t 0 , T ], X) be such that x -z C ≤ β. Then the set-valued map t G(t, x(t)) is measurable. In addition to Lemma 3, the proof of Theorem 1 requires the following two lemmas. The first one states that, for every x ∈ C([t 0 -τ, T ], R n ) taken in a neighborhood of the reference trajectory y, the map t F (t, x t ) is measurable.

Lemma 4: Let β > 0. Assume (H1), (H2) and let y be as in Theorem 1. Let x ∈ C([t 0 -τ, T ], R n ) be such that x(t) -y(t) ≤ β, for every t ∈ [t 0 -τ, T ]. Then the set-valued map t F (t, x t ) is measurable. Proof. Since x and y are continuous on [t 0 -τ, T ], we can easily prove (see [START_REF] Hale | Introduction to functional differential equations[END_REF]Lemma 2.1] for more details) that x t and y t are also continuous functions of t on [t 0 , T ]. By (H2),

F (t, •) is ζ(t)-Lipschitzian on y t + βB C with the Lipschitz constant ζ(•) = ζ R+β (•). Then, by Lemma 3 (with X = C([-τ, 0], R n ), G = F and z(•) = y(• + θ), θ ∈ [-τ, 0]), we obtain that the set-valued function [t 0 , T ] t F (t, x t ) is measurable, which concludes the proof.
The following lemma proves that, starting from a reference trajectory y, we can construct a sequence (x n ) n≥0 in C([t 0 -τ, T ], R n ) approximating a solution of ( 6)- [START_REF] Bernard | Building strategies to ensure language coexistence in presence of bilingualism[END_REF].

Lemma 5: Let β > 0 and δ 0 ≥ 0. Assume (H1), (H2) and let y, γ 1 , γ 2 , γ 3 be as in Theorem 1. If

γ 3 (T ) < β, then for any ϕ ∈ C([-τ, 0], R n ) with ϕ -y t0 C ≤ δ 0 there exist sequences x n ∈ C([t 0 -τ, T ], R n ) and f n ∈ L 1 ([t 0 , T ], R n ), f or n ≥ 0, such that x 0,t = y t , f 0 = ẏ, t ∈ [t 0 , T ], (18)
f 1 (t) -f 0 (t) = γ 1 (t), a.e. t ∈ [t 0 , T ]; (19)
and for n ≥ 1

   x n (t) = ϕ(0) + t t0 f n (s)ds, t ∈ [t 0 , T ],
x n,t0 = ϕ,

f n (t) ∈ F (t, x n-1,t ), t ∈ [t 0 , T ], (20)
with

f n+1 (t) -f n (t) ≤ ζ R+β (t) x n,t -x n-1,t C , (22)
for almost every t ∈ [t 0 , T ]. Proof. By Lemma 4, the set-valued map t F (t, y t ) is measurable. Since the function t → γ 1 (t) is measurable (see, [14, Lemma 1.5]), the set-valued map U 1 defined by

U 1 (t) := {v ∈ F (t, y t ) : v -f 0 (t) = γ 1 (t)}
is measurable (see, e.g., [START_REF] Aubin | Set-Valued Analysis[END_REF]). Hence, by the measurable selection theorem the set-valued map U 1 admits a measurable selection f 1 : [t 0 , T ] → R n . From the definition of U 1 , we have f 1 (t) ∈ F (t, y t ) for t ∈ [t 0 , T ] and

f 1 (t) -f 0 (t) = γ 1 (t) a.e. t ∈ [t 0 , T ]. ( 23
) Let ϕ ∈ C([-τ, 0], R n ) be such that ϕ -y t0 C ≤ δ 0 and define x 1 ∈ C([t 0 -τ, T ], R n ) by    x 1 (t) = ϕ(0) + t t0 f 1 (s)ds, t ∈ [t 0 , T ],
x

1 (t 0 + θ) = ϕ(θ), θ ∈ [-τ, 0].
Observe that

x 1,t -y t ≤ δ 0 + t t0 γ 1 (s)ds, t ∈ [t 0 , T ]. ( 24
)
Indeed, for θ ∈ [-τ, 0] and t ∈ [t 0 , T ] such that t + θ ≥ t 0 , we have

x 1 (t + θ) -y(t + θ) ≤ ϕ(0) -y(t 0 ) + t+θ t0 f 1 (s) -ẏ(s)ds ≤ ϕ -y t0 C + t+θ t0 γ 1 (s)ds ≤ δ 0 + t t0 γ 1 (s)ds.
In the case when t + θ < t 0 , we have

x 1 (t + θ) -y(t + θ) = ϕ -y t0 C ≤ δ 0 . With x 1 ,
: [t 0 , T ] → R n of U 2 such that f 2 (t) -f 1 (t) ≤ ζ R+β (t) x 1,t -x 0,t C , for almost every t ∈ [t 0 , T ].
Then, we conclude that ( 20)-( 22) hold true for n = 1. Assume that we already have constructed [START_REF] Frankowska | Existence of neighboring feasible trajectories: Applications to dynamic programming for state-constrained optimal control problems[END_REF], ( 21) and [START_REF] Haidar | Mathematical modeling of an urban pigeon population subject to local management strategies[END_REF]. Before extending to n = N + 1, we prove that the constructed sequence x n verifies the following:

x n ∈ C([t 0 -τ, T ], R n ) and f n ∈ L 1 ([t 0 , T ], R n ), for n = 1, • • • , N , verifying
Claim 1: x n,t -y t C ≤ β, ∀ t ∈ [t 0 , T ], ∀ n = 1, • • • , N.
In fact, for n = 1, the claim follows directly from [START_REF] Kuang | Delay differential equations with application in population dynamics[END_REF]. For n ≥ 2, (22) implies the following inequalities:

x n,t -x n-1,t C ≤ t t0 f n (s 1 ) -f n-1 (s 1 ) ds 1 ≤ t t0 ζ R+β (s 1 ) x n-1,s1 -x n-2,s1 C ds 1 ,
which can be repeated recursively (see the proof of [START_REF] Frankowska | A priori estimates for operational differential inclusions[END_REF]Theorem 1.2] for more details) to obtain the following property: 24) and the last inequality, we get (see the proof of [14, Theorem 1.2] for more details)

x n,t -x n-1,t C ≤ (δ 0 + t t0 γ 1 (s)ds) [ln(γ 2 (t))] n n! , ( 25
) for every n ∈ {2, • • • , N }. From (
x N,t -y t C ≤ N i=1 x i,t -x i-1,t C ≤ γ 3 (t) ≤ β. ( 26
)
Again, define the set-valued map

[t 0 , T ] t U N +1 (t) by U N +1 (t) := {v ∈ F (t, x N,t ) : v -ẋN (t) = d F (t,x N,t ) ( ẋN (t))}.
Knowing that x N,t -y t C ≤ β and using the same reasoning as before, we deduce the existence of

f N +1 : [t 0 , T ] → R n , a measurable selection of U N +1 , such that f N +1 (t) -f N (t) ≤ ζ R+β (t) x N,t -x N -1,t C , for almost every t ∈ [t 0 , T ]. The function x N +1 , associated to f N +1 , is defined by (20), for n = N + 1.
Proof of Theorem 1. By [START_REF] Lara | Multi-criteria dynamic decision under uncertainty: A stochastic viability analysis and an application to sustainable fishery management[END_REF], for each t ∈ [t 0 , T ], the sequence {x n,t } is Cauchy in the Banach space C([-τ, 0], R n ). Thus, for each t ∈ [t 0 , T ] we may define x t ∈ C([-τ, 0], R n ) as the limit of x n,t . In addition, by [START_REF] Haidar | Mathematical modeling of an urban pigeon population subject to local management strategies[END_REF], for almost every t ∈ [t 0 , T ] the sequence {f n (t)} is Cauchy in R n . Furthermore, from ( 22) and [START_REF] Hale | Introduction to functional differential equations[END_REF], it follows that for n ≥ 1

f n (t) -ẏ(t) ≤ n-1 i=0 f i+1 (t) -f i (t) ≤ γ 1 (t) + ζ R+β (t) n-1 i=1 x i,t -x i-1,t C ≤ γ 1 (t) + ζ R+β (t)γ 3 (t) a.e. t ∈ [t 0 , T ], (27)
from which we conclude that the sequence {f n } is integrably bounded. Thus we may define

f ∈ L 1 ([t 0 , T ], R n ) by f (t) = lim n→+∞ f n (t).
By arguments similar to [14, Theorem 1.2], we obtain that x(•) is an F -Trajectory satisfying ẋ(t) = f (t) a.e. in [t 0 , T ]. Passing to the limits in ( 26) and ( 27) yields the desired estimations on x and ẋ.

B. Proof of Theorem 2

Fix δ > 0 and let R := max t∈[t0-τ,T ] y(t) . It is not restrictive to assume that t 0 = 0 and

T 0 ζ R+δ (t)dt > 0.
Choose any positive α such that

α < min δ 2 , δ 2γ 2 (T ) T 0 ζ R+δ (t)dt , (28)
where γ 2 (•) is as in Theorem 1, with β = δ. Let n ≥ 1 be so large such that σ(1 + R)/n < α/2, where σ is given by (H3). Let I j be the interval [(j -1) T n , j T n ], for j = 1, • • • , n. By Lemma 4, the set-valued map t F (t, y t ) is measurable. In addition, by (H3), t F (t, y t ) is integrably bounded because for almost every t ∈ [0, T ], F (t, y t ) ⊂ σ(1 + R)B. Then, by Aumann's Theorem [START_REF] Aumann | Integrals of set-valued functions[END_REF], there exists a measurable selection

f j (t) ∈ F (t, y t ) such that Ij f j (t)dt = Ij ẏ(t)dt, j = 1, • • • , n.
Let f be the function which is equal to f j on I j , and define the continuous function z :

[-τ, T ] → R n by    z(t) = y(0) + t 0 f (s)ds, t ∈ [0, T ], z 0 = y 0 .
Observe that

z t -y t C < α, ∀ t ∈ [0, T ].
Indeed, for every t ∈ [0, T ] and every θ ∈ [-τ, 0] such that t + θ ≥ 0, there exists j ∈ {1, • • • , n} for which t + θ ∈ I j and

z(t + θ) -y(t + θ) = t+θ 0 (f (s) -ẏ(s)) ds ≤ Ij f (s) -ẏ(s) ds < α. If t + θ < 0, then z(t + θ) -y(t + θ) = 0. Since for almost every t ∈ [0, T ], F (t, •) is ζ R+δ (t)- Lipschitz on y t + δB C , we obtain d F (t,zt) ( ż(t)) ≤ sup{d F (t,zt) (ξ) : ξ ∈ F (t, y t )} +d F (t,yt) ( ż(t)) (29) ≤ ζ R+δ (t) z t -y t C ≤ αζ R+δ (t).
Inequality [START_REF] Sicard | Congress of Mathematicians ICM 2010 in Hyderabad. Her research activities are focused on optimal control, both in the deterministic and stochastic framework, control of systems under state constraints, viability theory, differential inclusions, setvalued and variational analysis[END_REF] together with [START_REF] Saint-Pierre | Approximation of the viability kernel[END_REF] imply that γ 2 (T ) T 0 αζ R+δ (t)dt < δ/2. Then, by Theorem 1 applied with β = δ and δ 0 = 0, there exists a trajectory x of (6) satisfying x 0 = z 0 = y 0 and

x t -z t C ≤ γ 2 (T ) T 0 αζ R+δ (s)ds < δ 2 .
Finally, we obtain

x t -y t C ≤ x t -z t C + z t -y t C < δ 2 + δ 2 = δ,
which concludes the proof.

C. Proof of Lemma 1.

We proceed in three steps.

Step 1. Let R > 0, t ∈ [0, T ] and ψ ∈ K λ ∩ RB C be fixed. Knowing that F is locally bounded and using exactly the same argument as [17, Lemma 3.5], we prove the existence of ρ t, ψ > 0 such that for all v ∈ F ( t, ψ) with max

n∈N 1 K ( ψ(0))
n, v ≥ 0, there exists

w ∈ Liminf (s,φ)→( t, ψ) F (s, φ) satisfying max n, w , n, w -v | n ∈ N 1 K ( ψ(0)) ≤ -2ρ t, ψ . (30)
Step 2. We show the existence of η t, ψ > 0 such that for every t ∈ B( t, η t, ψ ), for every ψ ∈ K λ ∩B C ( ψ, η t, ψ )+ η t, ψ B C and for every v ∈ F (t, ψ) with

max n∈N 1 K (x),x∈∂K∩B(ψ(0),ηt , ψ ) n, v ≥ 0, there exists w ∈ F (t, ψ) satisfying n, w ≤ -ρ t, ψ and n, w -v ≤ -ρ t, ψ ∀ n ∈ N 1 K (x), ∀ x ∈ ∂K ∩ B(ψ(0), η t, ψ ). (31)
Suppose by contradiction that there exist

t i → t, ψ i → ψ in C([-τ, 0], R n ), v i ∈ F (t i , ψ i ), x i → ∂K ψ(0) and n i ∈ N 1 K (x i ) such that n i , v i ≥ 0 and for every w i ∈ F (t, ψ i ) we can find x i → ∂K ψ(0), n i ∈ N 1 K (x i ) satisfying n i , w i ∨ n i , w i -v i > -ρ t, ψ . ( 32
)
Since F is upper semicontinuous at every point of [0, T ] × K λ , taking subsequences and keeping the same notations we may assume that v i converge to some v ∈ F ( t, ψ), n i → n and n i → n . Since the map x N 1 K (x) is upper semicontinuous, we have n, n ∈ N 1 K ( ψ(0)) and n, v ≥ 0. Then max

n∈N 1 K ( ψ(0))
n, v ≥ 0. Consider w as in (30) corresponding to this v and let w i ∈ F (t, ψ i ) be such that w i → w. From (32) we deduce that n , w ∨ n , w -v ≥ -ρ t, ψ , contradicting the choice of w.

Step 3. Consider a covering of [0, T ] × (K λ ∩ 2RB C ) by the open balls B(( t, ψ), η t, ψ ) satisfying the following requirement:

[0, T ] × (K λ ∩ 2RB C ) ⊂ ( t, ψ)∈[0,T ]×K λ ∩2RB C B(( t, ψ), η t, ψ
) such that for every t ∈ B( t, η t, ψ ), for every ψ ∈ K λ ∩ B C ( ψ, η t, ψ ) + η t, ψ B C and for every v ∈ F (t, ψ) with

max n∈N 1 K (x),x∈∂K∩B(ψ(0),ηt , ψ )
n, v ≥ 0, there exists w ∈ F (t, ψ) satisfying (31). We claim that the set K λ ∩ 2RB C is compact. Indeed, Thanks to Ascoli's Theorem, we know that a subset of C([-τ, 0], R n ) is compact if and only if it is closed, bounded, and equicontinuous. The set K λ is closed (the uniform limit of λ-Lipschitz functions is λ-Lipschitz) and equicontinuous (by assumption). The boundedness follows from the fact that for all θ ∈ [-τ, 0]

ψ(θ) ≤ ψ(θ) -ψ(0) + ψ(0) ≤ λτ + R.
Now, consider a finite subcovering

[0, T ] × (K λ ∩ 2RB C ) ⊂ i=1••• ,N B((t i , ψ i ), η ti,ψi ).
Then, for ρ = min{ρ ti,ψi , i = 1

• • • , N }, for some 0 < η < min{R, η ti,ψi , i = 1 • • • , N } and for all (t, ψ) ∈ [0, T ] × (K λ + ηB C ) ∩ RB C , there exists 1 ≤ i ≤ N such that (t, ψ) ∈ B(t i , η ti,ψi ) × (K λ ∩ B C (ψ i , η ti,ψi ) + η ti,ψi B C
). This complete the proof.

D. Proof of Theorem 3.

The proof is inspired by the construction proposed in [START_REF] Frankowska | Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints[END_REF].

Lemma 6: Assume (H1)-(H3). Let τ > 0, r > 0 and λ 0 > 0 and suppose that, for λ given by [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], assumptions (H4) and (IP C λ ) hold true. Then there exist positive constants δ and c such that for every t ∈ [0, T ] and every F -trajectory x(•) on [ t -τ, T ] with λ 0 -Lipschitz xt and x( t) ∈ K ∩ rB, and for any ε > 0, we can find an F -trajectory on [ t -τ, T ] satisfying

     xt = xt, x(t) ∈ Int K, ∀ t ∈ ( t, ( t + δ) ∧ T ] x t -xt C ≤ c max t∈[ t,T ] d K (x(t)) + ε.
Proof. Let R := (1 + λ 0 τ + r)e σT , R := (1 + R)σ and R := 2 RT + R, where σ is as in (H3). Fix t ∈ [0, T ] and an F -trajectory x(•) on [ t -τ, T ] such that xt is λ 0 -Lipschitz and x( t) ∈ K ∩ rB. Let δ > 0 and 0 < β < ρ be such that δ < We proceed in four steps.

Step 1. We have xt C ≤ R for every t ∈ [ t, T ]. Indeed, for t ∈ [ t, T ] and θ ∈ [-τ, 0], we have

x(t + θ) =    x( t
S := {s ∈ [ t, δ] : ∃ x ∈ ∂K ∩ B(x(s), η), n ∈ N 1 K (x), n, ẋ(s) ≥ 0}. Fix any ε > 0 and ε > 0 such that 0 < ε < ε 2 R 1 + exp T 0 ζ R+β (s)ds T 0 ζ R(s)ds
, and let κ ∈ [ t, δ] be defined as follows:

• If µ(S) ≤ Γ max s∈[ t,T ] d K (x(s)) + ε then set κ = δ. • If µ(S) > Γ max s∈[ t,T ] d K (x(s)) + ε then take κ be the smallest number in [ t, δ] such that µ(S ∩ [ t, κ]) = Γ max s∈[ t,T ] d K (x(s)) + ε .
For each s ∈ S, we have xs

∈ (K λ + ηB C ) ∩ RB C . Indeed, let z ∈ ∂K be such that z -x(s) = d K (x(s)). Let us define the function ψ ∈ C([-τ, 0], R n ) by ψ(θ) = x(s + θ) -x(s) + z, θ ∈ [-τ, 0].
The function ψ belongs to K λ . In addition, we have

xs (θ) -ψ(θ) = x(s) -z = d K (x(s)) ≤ d K (x( t)) + |d K (x(s)) -d K (x( t))| < η.
Then xs ∈ (K λ +ηB C )∩RB C . Thanks to Lemma 1, for each s ∈ S, we can find w ∈ F (s, xs ) satisfying [START_REF] Bettiol | Improved sensitivity relations in state constrained optimal control[END_REF]. By the measurable selection theorem [START_REF] Aubin | Set-Valued Analysis[END_REF], there exists a measurable function w : S → R n such that w(s) ∈ F (s, xs ), and for a.e. s ∈ S n, w(s) ≤ -ρ, and n, w(s) -ẋ(s) ≤ -ρ

∀ x ∈ ∂K ∩ B(x(s), η), ∀ n ∈ N 1 K (x). ( 34
ẏ(t) := w(t) if t ∈ S ∩ [ t, κ] ẋ(t) if t ∈ ([ t, T ] \ S) ∩ [ t, κ]. ( 35
) For t ∈ [ t, T ] and θ ∈ [-τ, 0], we have y t (θ) -xt (θ) ≤ max{ t,t+θ} t ẏ(s) -ẋ(s) ds ≤ t t ẏ(s) -ẋ(s) ds = S∩[ t,t∧κ] w(s) -ẋ(s) ds ≤ 2 Rµ(S ∩ [ t, t ∧ κ]) implying that, y t -xt C ≤ 2 Rµ(S ∩ [ t, t ∧ κ]). (36)
Moreover, for a.e. t ∈ [ t, T ],

d F (t,yt) ( ẏ(t)) ≤ sup{d F (t,xt) (ξ) : ξ ∈ F (t, y t )} + d F (t,xt) ( ẏ(t)) ≤ ζ R(t) y t -xt ≤ 2 Rµ(S ∩ [ t, t ∧ κ])ζ R(t).
Hence, thanks to Theorem 1 applied with δ 0 = 0, there exists an F -trajectory x(•) on [ t-τ, T ] such that xt = yt and for every t ∈ [ t, T ]

x t -y t C ≤ 2 R exp T 0 ζ R+β (s)ds ω ζ R (|t -t|)µ(S ∩ [ t, t ∧ κ]).
By the definition of κ, the inequality (36) and the triangle inequality, we have

x t -xt C ≤ ε ε µ(S ∩ [ t, κ]) ≤ c max t∈[ t,T ] d K (x(t)) + ε,
for a constant c independent from t and x(•).

Step 4. We show next that {x(t) : t ∈ ( t, δ]} ⊂ Int K. We distinguish two different cases: (37) As in the proof of [START_REF] Frankowska | Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints[END_REF]Theorem 5], it follows that ξ(t) is a convex combination of m vectors ξ α ∈ N 1 K (y α ), with y α ∈ π ∂K (z(t)), such that for all s ∈ [ t, κ] and α ∈ {1, • • • , m} we have y α -x(s) ≤ η, where 1 ≤ m ≤ n + 1. Then, from (37) together with (34), we obtain that dK (x(t)) ≤ This completes the proof.

Proof of Theorem 3. Let r := (1 + λ 0 τ + r 0 )e σT and δ = δ(r), c = c(r) > 0 be as in Lemma 6. Fix t 0 ∈ [0, T ] and an F -trajectory x(•) such that xt0 is λ 0 -Lipschitz and x(t 0 ) ∈ K ∩ r 0 B. Let N be the smallest integer satisfying (t 0 +N δ)∧T = T . Set t i = (t 0 +iδ)∧T for all i = 1, • • • N . Fix any ε 0 > 0. Lemma 6 assures that, for any sequence of positive numbers ε 1 , • • • , ε N -1 , there exists a sequence of F -trajectories {x 0 (•) = x(•), x i (•) :

i = 1, • • • , N } such that for all i = 1, • • • , N      x i (t) = x i-1 (t), ∀ t ∈ [t 0 -τ, t i-1 ] x(t) ∈ Int K, ∀ t ∈ (t 0 , t i ] x i,t -x i-1,t C ≤ c max t∈[t0,T ] d K (x i-1 (t)) + ε i-1 .
Lemma 6 is applied recursively on the interval [(t i-1 -τ ) ∧ T, T ] with reference trajectory x i-1 (•) restricted to this interval, for i = 1, • • • , N . Note that, at each stage of this recursive construction, the same constant δ and c are used; this is justified by the fact that x i-1,ti-1 ∈ rB C , for all i = 1, • • • , N . Call x(•) = x N (•), then x t0 = xt0 and x(t) ∈ Int K for every t ∈ (t 0 , T ]. Using the same arguments as in [START_REF] Frankowska | Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints[END_REF]Theorem 5], we prove the existence of some C > 0, independent from t 0 , x(•) and ε 0 such that

x t -xt C ≤ C max t∈[t0,T ] d K (x(t)) + ε 0 .
This completes the proof.

E. Proof of Corollary 1.

Fix a relaxed feasible F -trajectory x(•) such that xt0 is λ 0 -Lipschitz and x(t 0 ) ∈ K ∩ r 0 B, and δ > 0. Let C be as in Theorem 3. By Theorem 2, there exists an F -trajectory x(•) on [t 0 -τ, T ] satisfying xt0 = xt0 and xt -xt C ≤ δ/3C for all t ∈ [t 0 , T ]. By Theorem 3, for every ε 0 > 0, there exists a feasible Ftrajectory x(•) such that x t0 = xt0 , x((t 0 , T ]) ∈ Int K

η

  4σ(1+R) (where ρ and η are given by Lemma 1) andν(δ) := 2 R exp T t ζ R+β (s)ds ω ζ R (δ) < β,where ζ R(•) and ζ R+β (•) are as in (H2), and ω ζ R (•) is the modulus of continuity of the map t → t t ζ R(s)ds.Consider Γ > 0 such that Γ(ρ -ν(δ)) > 1 and call δ := ( t + δ) ∧ T .

  ) Define the absolutely continuous function y : [ t-τ, T ] → R n by yt = xt and

  ξ α , ẋ(s) ds + ν(δ)µ(S ∩ [ t, t ∧ κ]) ≤ (ν(δ) -ρ) µ(S ∩ [ t, t]) < 0.Case 2. t ∈ (κ, δ]: By the mean-value theorem, for some z(t) ∈ [x(t), y(t)] and ξ(t) ∈ ∂ dK (z(t)), dK (y(t)) = dK (x(t)) + ξ(t), y(t) -x(t) .Then,dK (x(t)) ≤ dK (y(t)) + ν(δ)µ(S ∩ [ t, t ∧ κ]) = dK (x(t)) + ξ(t), y(t) -x(t) +ν(δ)µ(S ∩ [ t, t ∧ κ]) = dK (x(t)) + t t ξ(t), ẏ(s) -ẋ(s) ds +ν(δ)µ(S ∩ [ t, t ∧ κ]) = dK (x(t)) + S∩[ t,t] ξ(t), w(s) -ẋ(s) ds +ν(δ)µ(S ∩ [ t, t ∧ κ]) (38)As in the first case, ξ(t) is a convex combination of m vectors ξ α ∈ N 1 K (y α ), with y α ∈ π ∂K (z(t)), such that for all s ∈ [ t, κ] and α ∈ {1, • • • , m} we havey α -x(s) ≤ η,where 1 ≤ m ≤ n+1. Then, from (38), we obtain that dK (x(t)) ≤ dK (x(t)) + ν(δ)µ(S ∩ [ t, t ∧ κ]) ξ α , w(s) -ẋ(s) ds ≤ dK (x(t)) -ρµ(S ∩ [ t, κ]) + ν(δ)µ(S ∩ [ t, t ∧ κ]) ≤ dK (x(t)) -(ρ -ν(δ)) Γ max s∈[ t,T ] d K (x(s)) + ε < 0.

  we associate the set-valued map [t 0 , T ] t U By (24), x 1,t -y t C ≤ β and, since F (t, •) is ζ R+β -Lipschitz on y t + βB C , we deduce (using the same arguments as before) the existence of a measurable selection f 2

2 (t) defined by U 2 (t) := {v ∈ F (t, x 1,t ) : v -ẋ1 (t) = d F (t,x1,t) ( ẋ1 (t))}.

  ) +

	t+θ
	ẋ(s)ds,			if t + θ ≥	t
	t xt(t + θ -t),					if t + θ < t.
	Then,
	max{ t,t+θ}
	x(t + θ) ≤ xt C +			σ(1 + xs C )ds.
	t
	Hence
	max{ t,t+θ}
	xt C ≤ xt C +		σ(1 + xs C )ds.
	t
	Thanks to Gronwall's Lemma, we can easily verify that
	for any t ∈ [ t, T ]
	xt < -η 4 + x(t) -x( t) ≤ -< -η 4 + (1 + R)σδ < 0,	η 4	+	t	t	ẋ(s) ds
	and x(•) = x(•) is as required. Step 3. If -η ≤ dK (x( t)) ≤ 0, we define the measurable 4 set

C ≤ (1 + xt C )e σ(t-t) , (33) from which conclude that xt C ≤ R for every t ∈ [ t, T ]. Step 2. If dK (x( t)) < -η 4 , then x(•) = x(•) satisfies our lemma. Indeed, if dK (x( t)) < -η

4

, then for all t ∈ [ t, δ], we have dK (x(t)) ≤ dK (x( t)) + | dK (x(t)) -dK (x( t))|

  Case 1. t ∈ ( t, κ]: The mean-value theorem (see, e.g.,[START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] Theorem 2.3.7]), affirms the existence of some z(t) ∈ [y( t), y(t)] and ξ(t) ∈ ∂ dK (z(t)), such that dK (y(t)) = dK (y( t)) + ξ(t), y(t) -y( t) . (t), ẋ(s) ds + ν(δ)µ(S ∩ [ t, t ∧ κ]).

	Then,
	dK (x(t)) ≤ dK (y(t)) + x(t) -y(t)
	≤ dK (y(t)) + ν(δ)µ(S ∩ [ t, t ∧ κ])
	≤ dK (y( t)) + ξ(t), y(t) -y( t) + ν(δ)µ(S ∩ [ t, t ∧ κ])
	≤	ξ(t), w(s) ds
	S∩[ t,t]
	+
	[ t,t]\S

ξ

F. Proof of Lemma 2.

Lemma 7: Let λ > 0 and assume (H1)-(H4) and (IP λ rel ). Then for every R > 0, ψ ∈ K λ ∩ RB C and every t ∈ [0, T ] there exists ρ t, ψ > 0 such that ∀ v ∈ co F ( t, ψ) with max Proof of Lemma 2. Let R > 0, t ∈ [0, T ] and ψ ∈ K λ ∩ RB C be fixed and let ρ t, ψ be as in Lemma 7. We claim the existence of η t, ψ > 0 such that for every

Suppose by contradiction that there exist

Since F is upper semicontinuous at every point of [0, T ] × K λ , so is co F . taking subsequences and keeping the same notations we may assume that v i converge to some v ∈ co F ( t, ψ), n i → n and n i → n . Since the map x N 1 K (x) is upper semicontinuous, we have n, n ∈ N 1 K ( ψ(0)) and n, v ≥ 0. Then max

n, v ≥ 0. Consider w as in (39) corresponding to this v and let w i ∈ co F (t, ψ i ) be such that w i → w. From (40) we deduce that n , w ∨ n , w -v ≥ -ρ t, ψ , contradicting the choice of w. The rest of the proof is similar to Step 3 of Lemma 1.

G. Proof of Theorem 5.

The proof of Theorem 5 is a straightforward consequence of Theorem 4. Indeed, fix t 0 ∈ [0, T ] and let us introduce for every (t, ψ) ∈ [t 0 , T ] × C([-τ, 0], R n ) the set-valued map F defined by F (t, ψ) := F(t, ψ(0), ψ(-τ )). It is easy to see that under the assumptions (A1)-(A4), the set-valued map F verifies (H1)-(H4). In addition, we can show that, under (IP λ eq ), condition (IP λ rel ) holds true. In fact, let ψ ∈ K λ . By definition of K λ , we have ψ(0) ∈ ∂K and ψ(-τ ) ∈ ψ(0) + λτ B. Then, from (IP λ eq ), for every v ∈ F (t, ψ) ≡ F(t, ψ(0), ψ(-τ )) such that max

n, w -v < 0.

From the inclusion

Liminf (s,z,ξ)→(t,ψ(0),ψ(-τ )) co F(s, z, ξ) ⊂ Liminf (s,φ)→(t,ψ) co F (s, φ), we deduce that (IP λ rel ) holds true. Hence, Theorem 4 concludes the proof.