On the continuity of the state constrained minimal time function

We obtain results on the propagation of the (Lipschitz) continuity of the minimal time function associated with a finite dimensional autonomous differential inclusion with state constraints and a closed target. To this end, we first obtain new regularity results of the solution map with respect to initial data.


Introduction
Let S be a nonempty subset of R p , F a multifunction mapping S to nonempty subsets of R p and consider the state constrained differential inclusion y (t) ∈ F(y(t)). (1.1) A solution of (1.1) on [0, T] is an absolutely continuous function y : [0, T] → S that satisfies y (t) ∈ F(y(t)) for a.e.t ∈ [0, T] .A solution of (1.1) on a semi-open interval [0, T) is defined similarly.The S-constrained minimal time problem associated to a nonempty subset Σ of S (called the target set) is the problem in which the goal is to steer an initial point x ∈ S to Σ along a solution of (1.1) in minimal time.The minimal time value is denoted by T(x), which is defined to be +∞ if no solution of (1.1) from x can reach Σ.The function T is called the S-constrained minimal time function.When S = R p , T coincides with the well known (unconstrained) minimal time function associated with the target Σ.In this paper we study continuity properties of the S-constrained minimal time function.
The regularity properties of the minimal time function, being strongly connected to controllability properties of the system, have been the object of an extensive literature.For more details on controllability see, e.g., [3,12].The Lipschitz continuity of the unconstrained minimal time function associated to a point target was first studied in [29].In that paper, Petrov introduced a necessary and sufficient condition, called Petrov condition, for the Lipschitz continuity of the minimal time function in a neighborhood of the origin.That result was extended later to more general target sets in [4,32].In [33], Veliov obtained a necessary and sufficient condition for the local Lipschitz continuity of the unconstrained minimal time function for closed target sets, when the multifunction F is nonautonomous and depends measurably on time.In [35], in the absence of constraints, Wolenski and Zhuang showed that the Lipschitz continuity of the minimal time function near the target Σ is equivalent to the boundedness of the proximal subgradient of the minimal time function on Σ.
For the state constrained case we mention the paper [28], where the authors generalize the results obtained for the unconstrained minimal time function in [35].They gave necessary and sufficient conditions for the proto-Lipschitzness of T (the definition is given in Section 3), imposing some geometric assumptions for the pair (Σ, S) (the admissibility of Σ for S and conditions involving points near Σ which are exterior to S).Moreover, under further geometric assumptions on S, in [28] there are given necessary and sufficient conditions for T to be Lipschitz on a neighborhood of Σ in S.
In [27], a Petrov type condition is provided for the state constrained minimal time function T to be proto-Lipschitz.More exactly, the following result is proved.
Theorem 1.1.Let F : S → R p be an upper semi-continuous multifunction with nonempty compact convex values, S a nonempty closed subset of R p and Σ a closed subset of S. Suppose that there exist ρ > 0 and γ > 0 such that for all x ∈ S ∩ (Σ + ρB).Then the S-constrained minimal time function T is proto-Lipschitz.
We denoted by π Σ (x) the set of projections of x on Σ and T S (x) is the Bouligand tangent cone to S at x.Moreover, in [27] there are given examples where the hypothesis of Theorem 1.1 holds, but the geometric conditions from [28] are not satisfied.
This paper is a continuation of [27] and its goal is to get the propagation of the continuity of the state constrained minimal time function T around the target to the whole reachable set, without imposing explicitly the geometric assumptions from [28].Instead, we use some regularity properties of the multifunction 3) The propagation of the continuity properties of the S-constrained minimal time function was previously discussed in [9] and [17].In [9] the authors considered the control system y ∈ f (t, y, U) with state constraints and proved the Lipschitz continuity of T under Lipschitz hypotheses on f and some regularity assumptions on the set of constraints.In that paper, the set of constraints is the closure of an open set Ω ⊂ R p and the target Σ is a subset of Ω.In our paper we require only that Σ ⊂ S with S a closed subset of R p and we do not assume Lipschitz continuity of F. In [17] we imposed that F(x) ⊂ T S (x) for any x ∈ S, which, in fact, implies the invariance of S with respect to the solutions of the differential inclusion y ∈ F(y).For other results on the propagation of continuity properties of the minimal time function, in the absence of constraints, see, [10,13,15,35].A key role in obtaining these results is played by the dependence of the solutions on the initial conditions.In this paper, in order to obtain the propagation results, we first prove a Filippov type result for our state constrained differential inclusion (1.1), which is a main result of the paper.
There are also many papers on Filippov type results, in the state constrained case.We recall the paper of Frankowska and Rampazzo [25], where there are given Filippov and Filippov-Wazewski theorems in the case when the state variable is constrained to the closure of an open subset of R n .Nour and Stern [28], while investigating the Lipschitz continuity of the minimal time function, established the Lipschitz dependence of the solutions of (1.1) on the initial data, under Lipschitz hypothesis on F and certain assumptions on S. In [8], Bressan and Facchi established a result of this type, assuming that S is compact and convex, F is Lipschitz and satisfies a strict inward pointing condition at every boundary point x ∈ ∂S, that is co F(x) ∩ int T S (x) = ∅. (1.4) Filippov type results were also obtained in [5,7].We want to remark that in all the papers above, the Filippov type results were obtained under the Lipschitz hypothesis on F.
In this paper, we prove a Filippov type result for our state constrained differential inclusion (1.1), avoiding explicit geometric assumptions on S or Σ and using regularity properties of the multifunction defined by (1.3).We give examples that do not satisfy the conditions imposed in [28] and/or [8], but satisfy our hypotheses.It is important to remark that the technique for obtaining our result, by viability, was used for the first time in [14], for a semilinear system, with F Lipschitz.This technique was also used in [16,17,31].It requires the convexity of the values of F, as it was remarked also in [31].From this point of view, the Filippov type results of this paper are new compared to the previous ones, because this technique of the proof allows us to weaken the Lipschitz conditions; moreover, they are new and important even in the absence of state constraints.However, by these results we relax the Lipschitz hypothesis, but we impose F to have convex values.

Preliminaries
For any subset K ⊆ R p we denote by int K the interior of K, K the closure of K, π K (x) the set of projections of x ∈ R p in K and by d K (x) the Euclidean distance from x to the set K. The open unit ball is denoted by We denote by T K (ξ) the set of all tangent vectors to K at ξ ∈ K.For each ξ ∈ K, the set T K (ξ) is a closed cone.A well-known characterization by sequences is the following: η ∈ T K (ξ) if and only if there exist two sequences We recall that a closed set K is called sleek if the multifunction is lower semicontinuous.For more details on tangent cones we refer for instance to [2].
O. Cârjȃ and A. I. Lazu Let F : K R p be a given multifunction and consider the differential inclusion w (t) ∈ F (w(t)). (2.1) The set K is viable with respect to F if for each ξ ∈ K there exists θ > 0 such that (2.1) has at least one solution w : [0, θ] → K with w(0) = ξ.
Theorem 2.1.Let K be a nonempty, locally closed subset in R p and let F : K R p be an upper semicontinuous multifunction with nonempty, compact and convex values.A necessary and sufficient condition in order that K be viable with respect to F is the following tangency condition: for each ξ ∈ K.
The following conditions on a multifunction, weaker than the Lipschitz continuity, introduced in [20,23], will be used in the next sections of the paper.

Definition 2.2. A multifunction
2) one-sided Perron if for any x, y ∈ K, any v ∈ F(x) there exists w ∈ F(y) such that By a Perron function we mean a continuous function ϑ : [0, ∞) → [0, ∞) with ϑ(0) = 0 such that the differential equation z = ϑ(z) has the null function as the unique solution with z(0) = 0.This function was introduced by Perron in [30].It is clear that the class of one-sided Perron multifunctions is larger than the class of one-sided Lipschitz ones.

Lipschitz continuity of the state constrained minimal time function
Let S ⊂ R p be a closed nonempty set and let Σ ⊂ S be a closed subset.The S-constrained minimal time function T : S → [0, +∞] is defined by T(x) = inf {τ ≥ 0; there exists a solution y of (1.1) with y(0) = x, y(τ) ∈ Σ} .
If no solution from x can reach Σ then T(x) = +∞.We denote by R the set of all points x ∈ S such that T(x) < +∞.Following [28], the minimal time function T is said to be proto-Lipschitz if there exist ρ > 0 and M > 0 such that for all x ∈ (Σ + ρB) ∩ S.
In the same spirit, we say that T is proto-continuous if there exist ρ > 0 and ω : [0, ρ] → [0, +∞) such that lim s→0 + ω(s) = 0 and for all x ∈ (Σ + ρB) ∩ S. As it is proved in [3, p. 229], when Σ is closed with compact boundary, T is proto-continuous iff it is continuous in each point of Σ.
We define the multifunction G : S R p by and we impose some regularity properties for G in order to obtain the Lipschitz/continuous dependence of the solutions of (1.1) on the initial data, that is the key for the propagation Theorems 3.4 and 4.5.First, we give an extension of the Filippov theorem, on the Lipschitz dependence of the solutions of (1.1) on the initial data, in the state constraints case.The proof is based on the viability Theorem 2.1 with F and K appropriately chosen as in [17, Theorem 2.1].Theorem 3.1.Let F : S R p be an upper semicontinuous multifunction, with convex and compact values.Assume that G, defined by (3.1), has nonempty convex values, is lower semicontinuous and one-sided Lipschitz of constant L.Then, for any x 1 , x 2 ∈ S, any solution y 1 : [0, σ] → S of (1.1) with y 1 (0) = x 1 , there exists a solution y for all t ∈ [0, σ] .
Remark 3.2.The lower semicontinuity and convexity hypotheses on G are satisfied, for instance, if S is sleek, F is lower semicontinuous and for any x ∈ S. Indeed, if the set S is sleek it is known that T S (x) is a convex cone (see, e.g., [2]), hence the multifunction G has convex values.If, in addition, F is lower semicontinuous and (3.5) is satisfied, then the multifunction G is lower semicontinuous (see [6,Lemma 3.1]).However, condition (3.5) is not necessary for the lower semicontinuity of G (see the Example below).It should be interesting to find general conditions on S and F to ensure that the multifunction G is one-sided Lipschitz.An interesting case when this happens is when F(x) ⊂ T S (x) for any x ∈ S (which, in fact, assures invariance) and F is one-sided Lipschitz.
Example 3.3.Consider the set S = {(x 1 , x 2 ); x 2 ≥ 0} and the multifunction F(x 1 , x 2 ) = B ∩ {(y 1 , y 2 ); y 2 ≤ 0} for all (x 1 , x 2 ) ∈ S. We have that T S (x 1 , x 2 ) = S for (x 1 , x 2 ) ∈ ∂S, so condition (3.5) is not satisfied.However, it is easy to see that the multifunction G, given by is lower semicontinuous.For this system, Lemma 1 from [8] can not be applied because S does not satisfy the following assumption required there, that there exist a non-zero vector a ∈ F = F(x 1 , x 2 ) and ρ > 0 such that where Γ a,ρ := {λy; λ ≥ 0, y − a ≤ ρ}.Indeed, for any a = (a 1 , a 2 ) ∈ F and ρ > 0 take It is easy to see that s + y / ∈ S, so (3.7) does not hold.Neither [28,Lemma 14] can be used because one of the conditions required is not fulfilled, that is where N C S (x) denotes the Clarke normal cone to S at x. Take, for instance, x = (0, 0) and η = (0, −1), then min v∈F(x) η, v = 0.
However, it is easy to see that our hypotheses from Theorem 3.1 hold.We shall only prove that G is one-sided Lipschitz.Take (x 1 , The other cases can be solved similarly.In conclusion, by Theorem 3.1, we get the Lipschitz dependence of solutions on initial states.Now we are ready to prove the propagation of the Lipschitz continuity of the state constrained minimal time function associated to (1.1).Theorem 3.4.Assume the hypotheses of Theorem 3.1.Suppose that T is proto-Lipschitz.Then R is open in S and T is locally Lipschitz on R, i.e., for every x ∈ R there exists a neighborhood U of x and a constant k Proof.Let ρ > 0 and M > 0 be from the definition of the proto-Lipschitzness of T and L be from the one-sided Lipschitzness of G. Let x ∈ R. We prove that if z ∈ S with z − x < ρe −L(T(x)+1) then z ∈ R and To this end, fix ε ∈ (0, 1) and consider τ < T(x) + ε and a solution y : [0, τ] → S of (1.1) with y(0) = x such that y(τ) ∈ Σ.Let z ∈ S be such that z − x < ρe −L(T(x)+1) .By Theorem 3.1 there exists y z : [0, τ] → S a solution of (1.1) with y z (0) = z such that Since T is the proto-Lipschitz, we get This implies that T(z) ≤ τ + Me L(T(x)+1) z − x .Further, T(z) ≤ T(x) + ε + Me L(T(x)+1) z − x .Finally, since ε ∈ (0, 1) is arbitrary, we get (3.8).Now, let x 0 ∈ R and let z 1 , z 2 ∈ S be such that To this end, we observe, by the first part of the proof, that z i ∈ R and T(z i ) ≤ T(x 0 ) + Me L(T(x 0 )+1) z i − x 0 ≤ T(x 0 ) + Mρ, for i = 1, 2.Moreover, for i = 1, 2. Therefore, by the first part of the proof, By symmetry, we get (3.9), as claimed.
In [9,Theorem 3.8] the Lipschitz continuity of the minimal time function is proved under some regularity assumptions on the set of constraints.We remind that in [9] S = Ω with Ω open and Σ ⊂ Ω.Moreover, the following condition on the boundary of Ω is imposed: there exist α > 0 and I a multifunction with some properties (called there uniformly hypertangent conical field) such that for any x ∈ ∂Ω (3.10) In the following example we present a system with Σ ⊂ Ω that does not satisfy (3.10) because F(x) ∩ {v ∈ R p ; v ≥ α} = ∅ for some x ∈ ∂Ω and any α > 0, but satisfies our hypotheses.

Small time controllability and continuity of the state constrained minimal time function
In the previous section we assumed that the multifunction G is one-sided Lipschitz and we obtained the Lipschitz continuity of the S-constrained minimal time function.In this section we study the propagation of the regularity of the S-constrained minimal time function when the proto-Lipschitz condition is replaced by a weaker one (proto-continuous, proto-Hölder continuous), related to small time controllability on Σ, studied in [3, Chapter IV].First, we give a Petrov-type condition that assures that the S-constrained minimal time function is proto-continuous and then we present a propagation result of this continuity property.In order to get the propagation result we consider a weaker condition on G than onesided Lipschitz, used in the previous section, that assures the continuity of the solution map of (1.1) in the sense of Hausdorff metric.
Step 1.We first prove that there exists an y : and z : for all t ∈ [0, τ).
To this aim, we consider the set The state constrained minimal time function for all (y, z) ∈ K and we apply Theorem 2.1.To this end, we use (4.1) to prove the tangency condition (2.2).For details, see the proof of Theorem 1.1 developed in [27], where µ(z) = −γ.

Theorem 4 . 1 .
Let F : S R p be an upper semicontinuous multifunction, with convex and compact values.Suppose that G, defined by (3.1), is nonempty valued and there exist ρ > 0 and µ :