On the Strong Invariance Property for Non-Lipschitz Dynamics

We provide a new sufficient condition for strong invariance for differential inclusions, under very general conditions on the dynamics, in terms of a Hamiltonian inequality. In lieu of the usual Lipschitzness assumption on the multifunction, we assume a feedback realization condition that can in particular be satisfied for measurable dynamics that are neither upper nor lower semicontinuous.

This motivates the search for sufficient conditions for strong invariance for non-Lipschitz differential inclusions, which is the focus of this note. Donchev, Rios and Wolenski [10,11] recently developed necessary and sufficient conditions for strong invariance for so-called one-sided Lipschitz differential inclusions. See also [19] for an autonomous normal type characterization of strong invariance for certain systems with a discontinuous component. These works apply under special conditions on the structure of the dynamics (cf. section 3 for further details).
In this note, we pursue a very different approach. Rather than restricting the structure of the dynamics, we provide a sufficient condition for strong invariance under an appropriate feedback realization hypothesis. This hypothesis is related to Sussmann's 'unique limiting' property that was introduced in [21] in the context of exit time optimal control problems with continuous dynamics, and to Malisoff's "Lipschitz upper envelope" condition from [17,18]. Roughly speaking, our realization property states that each trajectory φ of the dynamics F is also a unique trajectory of a nonautonomous singleton-valued dynamics f for which f (t, x) ∈ cone {F (x)} for all t and all x near φ(0) (cf. section 2 below for a precise statement of our hypothesis). This is a less restrictive assumption than those of the known strong invariance characterizations because it can be satisfied by important classes of differential inclusions with measurable, but possibly neither upper nor lower semicontinuous, right-hand sides (cf. section 2 for examples). While our main theorem can be shown using Zorn's Lemma, the proof we give below is constructive, and in particular leads to a new approach to building viable trajectories for Carathéodory dynamics; see Remark 4.1.
In section 2, we state our realization hypothesis precisely and provide the necessary background on differential inclusions and nonsmooth analysis. We also illustrate the applicability of our hypothesis to a broad class of discontinuous dynamics that are beyond the scope of the well known strong invariance results. In section 3, we announce our strong invariance result and discuss its relationship to the known theorems in strong invariant system theory. Section 4 contains the proof of our strong invariance criterion, and we close in section 5 by proving a new necessary and sufficient Hamiltonian condition for strong invariance for general lower semicontinuous feedback realizable dynamics.

Basic Hypothesis
Our main object of study in this note is an autonomous differential inclusionẋ ∈ F (x). In this subsection, we state our hypothesis on F and illustrate its relevance using several applications. Our novel feature is the requirement that each trajectory of F be realizable as the unique solution to a nonautonomous local feedback selection of F . On the other hand, we will not require the Lipschitz property or other structural assumptions on F that are generally invoked in strong invariant system theory (cf. [7,9,10,15,19]).
To make our realization hypothesis precise, we require the following definitions and notation. By a trajectory ofẋ ∈ F (x) on an interval [0, T ] starting at a point x o ∈ R n , we mean an absolutely continuous function φ : We let Traj T (F, x) denote the set of all trajectories φ : [0, T ] → R n for F starting at x on all possible intervals [0, T ], and we set Traj(F, x) := ∪ T ≥0 Traj T (F, x) and Traj(F ) := ∪ x∈R n Traj(F, x).
A multifunction G : R n ⇒ R n is said to have linear growth provided there exist positive constants c 1 and c 2 such that ||v|| ≤ c 1 + c 2 ||x|| for all v ∈ G(x) and x ∈ R n , where || · || denotes the Euclidean norm. For any interval I, a function f : I × R n → R n is said to have linear growth (on I) provided x → G(x) := {f (t, x) : t ∈ I} has linear growth. For any sets D, M ⊆ R n and η ∈ R, we set M + ηD := {m + ηd : m ∈ M, d ∈ D}, and cone {D} := ∪{ηD : η ≥ 0}. Also, B n (p) := {x ∈ R n : x − p ≤ 1} for all p ∈ R n and B n := B n (0). A function ω(·) : [0, ∞) → [0, ∞) is called a modulus provided it is nondecreasing and continuous with ω(0) = 0. For each T ≥ 0, we let C[0, T ] denote the set of all functions f : [0, T ] × R n → R n that satisfy (C 1 ) For each x ∈ R n , the map t → f (t, x) is measurable; (C 2 ) For each compact set K ⊆ R n , there exists a modulus ω f,K (·) such that, for all t ∈ [0, T ] and It is noteworthy that ω f,K (·) in the previous definition is independent of t ∈ [0, T ]. For eachx ∈ R n , denote by C F ([0, T ],x) those f ∈ C[0, T ] that are also selections of the cone of F for almost all t ∈ [0, T ] and all x ∈ R n sufficiently nearx; that is, Notice that while elements f ∈ C F ([0, T ],x) are defined on all of [0, T ] × R n , they are only required to satisfy and all x ∈ R n . We will assume the following: Notice that hypothesis (U ) is weaker than requiring a continuous selection from the dynamics F that realizes the trajectory. This is because f is allowed to depend on time as well as the state, and need only be a local selection. Moreover, f is allowed to depend on the choice of the trajectory φ, and need not be continuous. In practice, hypothesis (U ) can be checked using open or closed loop controls, and may be satisfied for non-Lipschitz dynamics. The following examples illustrate these points and also show how to use cones to check condition (U ).
Example 2.1. Assume F : R n ⇒ R n is Lipschitz and nonempty and compact-convex valued. We claim that F satisfies condition (U ). To see why, letx ∈ R n , T > 0, and φ ∈ Traj T (F,x) be given, and set (i.e., f (t, x) is the closest point toφ(t) in F (x), which is well defined by the convexity of F (x)). Then f ∈ C F [0, T ] satisfies the requirement. If on the other hand F : R ⇒ R is defined by F (x) = {1} for x < 0, F (0) = {0} ∪ [1,2], and F (x) = [0, 2] for x > 0, and if φ ∈ Traj(F ), then f (t, x) ≡φ(t) ∈ cone{F (x)} for almost all t and all x ∈ R n . Therefore, condition (U ) is again satisfied, even though F is neither upper nor lower semicontinuous nor convex valued.
where A ⊆ R m is compact, and g : R n × A → R n is continuous in x ∈ R n and measurable and satisfies (H) For each compact set K ⊆ R n , there exists L K > 0 such that for all x 1 , x 2 ∈ K and a ∈ A, Also, x → g(x, A) has linear growth.
Applying the Filippov lemma as in the previous example, we find measurable functions u and r such thatφ for all x and y, by our stated conditions. This implies that

Preliminaries in Nonsmooth Analysis
The principal nonsmooth objects used in this note are the proximal subgradient and normal cone, and here we review these concepts; see [7] for a complete treatment. Let S ⊆ R n be closed and s ∈ S. A vector ζ ∈ R n is called a proximal normal vector of S at s provided there exists σ = σ(ζ, s) > 0 so that The set of all proximal normals of S at s is denoted by N P S (s) and is a convex cone. One can show (cf. [7, p. 25]) that for each δ > 0 and s ∈ S, ζ ∈ N P S (s) if and only if there exists σ = σ(ζ, s) > 0 so that Recall that the distance function For the related functional concept, assume f : The set of all proximal subgradients for f at x is denoted by ∂ P f (x). This set could be empty at some points, even for C 1 functions (e.g., We next state the version of the Clarke-Ledyaev Mean Value Inequality needed for our strong invariance results. Let [x, Y ] denote the closed convex hull of x ∈ R n and Y ⊆ R n . Theorem 1. Assume x ∈ R n , Y ⊆ R n is compact and convex, and Ψ : R n → (∞, +∞] is lower semicontinuous. Then for any δ < min y∈Y Ψ(y) − Ψ(x) and λ > 0, there exist z ∈ [x, Y ] + λB n and ζ ∈ ∂ P Ψ(z) so that δ < ζ, y − x for all y ∈ Y .
For the proof, see [7, p. 117]; an infinite dimensional version also holds (see [5]), but is not needed here.

Background in Differential Inclusions
In this subsection, we review invariant systems theory and a standard result on compactness of trajectories for discontinuous dynamics. The following definition of escape times was introduced in [23]: Then T is called an escape time of x(·) from G provided at least one of the following conditions hold: We next define strong and weak invariance. Assume G ⊆ R n is open and x 0 ∈ G. The set of all trajectories of F originating from x 0 that remain in G over a maximal interval is denoted by Υ For Hamiltonian characterizations of strong invariance for locally Lipschitz dynamics, see [7]. See also [10] for a characterization of strong invariance for systems satisfying appropriate one-sided Lipschitzness and dissipativity conditions, and [11] for general one-sided Lipschitz dynamics (with a modified Hamiltonian). Our main contribution will be a new sufficient condition for strong invariance for dynamics satisfying the realizability condition (U ), including cases where F is neither lower nor upper semicontinuous and not tractable by the known strong invariance results. Our condition is a Hamiltonian inequality involving a lower semicontinuous verification function. However, necessity is not true generally, and it is not clear what or if a modification of the inequality can be made to ensure a complete characterization.
The following is a variant of the well known "compactness of trajectories" lemma. This result says more than just that a bounded set of solutions is relatively compact. Rather, a stronger conclusion holds in that approximate trajectories have subsequences that converge to a trajectory. The proof is a special case of the compactness of trajectories proof in [7].
for all i, where {δ i (·)} is a sequence of nonnegative measurable functions that converges to 0 in L 2 as i → ∞, {r i (·)} is a sequence of measurable functions converging uniformly to 0 as i → ∞, and {τ i (·)} is a sequence of measurable functions converging uniformly to t on [0, T ] as i → ∞. Then there exists a trajectory y ofẏ =f (t, y), y(0) =x such that a subsequence of y i converges to y uniformly on [0, T ].
We will apply Lemma 2.7 to continuous mollifications of our feedback maps f ∈ C[0, T ]. More precisely, set where the constant C > 0 is chosen so that R η(s)ds = 1. For each ε > 0 and t ∈ R, set Notice for future use that Define the following convolutions of f ∈ C[0, T ] in the t-variable: with the convention that f (s, x) = 0 for s ∈ [0, T ]. Then f ε ∈ C[0, T ] and is continuous for all ε > 0. (In fact, f ε is a C ∞ function of t for each x ∈ R n , but we will not need this fact. See [12,13] for the well known theory of convolutions and mollifiers.) We will apply Lemma 2.7 to a sequencef := f ε(i) with ε(i) > 0 converging to zero. In this case, we will use ideas from the standard proof that to build trajectories of f ∈ C[0, T ] that respect the state constraint. We prove Theorem 2 in section 4 by constructing appropriate Euler polygonal arcs; see also Remark 4.1 for an alternative nonconstructive proof based on Zorn's Lemma. Theorem 2 differs from the usual strong invariance statements in the manner in which the set S is described, but it allows for some interplay between constraint and data assumptions. Note that we require the Hamiltonian inequality in a neighborhood U of S, for the result is not true in general if the Hamiltonian condition is placed only on S, even if Ψ and F are smooth. For example, take n = 1, Ψ(x) = x 2 , and F (x) ≡ {1}. In this case, S = {0} and H F (0, ∂ P Ψ(0)) = 0, but (F, S) is not strongly invariant. On the other hand, Example 2.2 is covered by Theorem 2, once we choose the verification function Ψ(x) = x 2 . In this case, the Hamiltonian condition reads H F (x, Ψ ′ (x)) = −2x sign(x) = −2|x| ≤ 0 for all x ∈ R, so our sufficient condition for strong invariance is satisfied.
Theorem 2 contains the usual sufficient condition for strong invariance for an arbitrary closed set S ⊆ R n by letting Ψ be the characteristic function I S of S; that is, I S (x) = 0 if x ∈ S and is 1 otherwise. Then ∂ P Ψ(x) = {0} for all x ∈ boundary (S), and ∂ P Ψ(x) = N P S (x) for all x ∈ boundary (S). This implies the following special case of Theorem 2: Corollary 3.1. Let F : R n ⇒ R n satisfy (U ) and S ⊆ R n be closed. If H F (x, N P S (x)) ≤ 0 for all x ∈ boundary (S), then (F, S) is strongly invariant in R n .
The converse of Corollary 3.1 does not hold, as illustrated by the simple example given in the introduction. This means that the converse of Theorem 2 does not hold.

Remark 3.2. Theorem 2 remains true (by the same proof) if its Hamiltonian inequality is replaced by
x ∈ U, and p ∈ ∂ P Ψ(x).

Relationship to Known Strong Invariance Results
Theorem 2 improves on the known strong invariance results because it does not require the usual Lipschitz or other structural assumptions on the dynamics. The papers [4,6,15,16] provide strong invariance results for locally Lipschitz dynamics (see also [7,Chapter 4]). In [4], Clarke showed that strong invariance of (F, S) in R n is equivalent to where T C S is the Clarke tangent cone (cf. [7]). Recall that v ∈ T C S (x) if and only if for all sequences x i ∈ S converging to x and all sequences t i > 0 decreasing to 0, there exists a sequence v i ∈ R n converging to v such that x i + t i v i ∈ S for all i. In particular, if S = {0}, then T C S (0) = {0}. Later, Krastanov [16] gave an infinitesimal characterization of normal-type, by showing strong invariance is equivalent to the following: H F (x, N P S (x)) ≤ 0 for all x ∈ S. See [3,6] for Hilbert space versions, and [15,22] for other strong invariance results for Lipschitz dynamics and nonautonomous versions.
Donchev [9] extended these characterizations beyond the autonomous Lipschitz case to "almost continuous, one-sided Lipschitz" multifunctions. Rios and Wolenski [19] proved an autonomous normal-type characterization that allows for a discontinuous component. Donchev, Rios, and Wolenski [10] proved a necessary and sufficient condition for strong invariance for a discontinuous nonautonomous differential inclusion F : R n × I ⇒ R n whose right-hand side is the sum of an almost upper semicontinuous dynamic D(t, x) with nonempty compact convex values that is dissipative in x, and an almost lower semicontinuous multifunction G(t, x) that is one-sided Lipschitz in x. In terms of the nonautonomous Hamiltonians defined for any dynamics R by the main result of [10] says: If S ⊆ R n is closed, then (D + G, S) is strongly invariant in R n if and only if there exists a subset I ⊆ I of full measure in I such that This result applies to cases where the Clarke tangency condition (7) is not satisfied, and covers the example in the introduction. They have gone further in [11] to provide a characterization of the general one-sided Lipschitz case, in which the Hamiltonian is replaced by a limiting condition. On the other hand, Theorem 2 does not make any structural assumptions on the dynamics. Moreover, our feedback realizability hypothesis (U ) can be satisfied for dynamics that are not tractable by the well known strong invariance results. For instance, see the examples in section 2.

Proof of Strong Invariance Theorem
This section is devoted to the proof of Theorem 2.
Fix T > 0 andx ∈ S. We first develop some properties that hold for all f ∈ C F ([0, T ],x). Fixing f ∈ C F ([0, T ],x) and ε > 0, and fixing γ > 0 such that f (t, x) ∈ cone{F (x)} for all x ∈ γB n (x) and almost all t ∈ [0, T ], set for each t ∈ [0, T ], x ∈ R n and k ∈ N, where f ε is the regularization of f defined by (6) and co denotes the closed convex hull. This is well defined because f ∈ C[0, T ]. By reducing γ > 0 as necessary, we can assume that γB n (x) ⊆ U. We also set for all t ∈ [0, T ], x ∈ R n , and k ∈ N. Note that where c 1 and c 2 are the constants from the linear growth requirement on f , so the sets G ε f [t, x, k] are compact. The following estimate is based on Theorem 1 from section 2: Proof. Suppose the contrary. Fix x ∈ γ 2 B n (x), t ≥ 0, k ∈ N, and h > 0 satisfying (10) but such that It follows that where . This is because Ψ is lower semicontinuous. Let λ ∈ (0, 1 2k ) be such that Next we apply Theorem 1 with the choices Y = x + hG ε f [t, x, k] and δ defined by (13). It follows that there exist z ∈ [x, Y ] + λB n and ζ ∈ ∂ P Ψ(z) for which Note that z ∈ γB n (x) ⊆ U, by (14). Since z ∈ [x, Y ] + λB n , (10) combined with the choice of λ gives x, k]. Since f (s, z) ∈ cone{F (z)} for a.a. s ∈ [0, T ] (by our choice of γ > 0), our Hamiltonian hypothesis gives ζ, f (s, z) ≤ 0 for almost all s ∈ [0, T ]. Therefore, (15) gives The contradiction (16) concludes the proof of Claim 4.1.

Now set
Let ω f,K be a modulus of continuity for x → f (t, x) on K := D + B n for all t ∈ [0, T ]. Such a modulus exists by condition (C 2 ). Then ω f,K is also a modulus of continuity of and ε > 0. The following estimate follows from Carathéodory's Lemma (cf. [20, p. 55]): In particular, x j ∈ K for all j. This gives as desired.
Since (24) has the form (4) from our compactness of trajectories lemma and f ε is continuous, we can find a subsequence of {x π(k) (·)} that converges uniformly to a trajectory y ε oḟ y = f ε (t, y), y(0) =x.
By possibly passing to a subsequence without relabelling, we can assume that . . , c(k) − 1 and k ≥ N , conditions (20) and (21) along with Claim 4.1 give Summing these inequalities and recalling that h k ≤ γ gives for all t ∈ [0,T ]. It follows from (27) and (28) that Since Ψ is lower semicontinuous, it follows from (29)-(30) that Ψ(y ε (t)) ≤ Ψ(x) for all t ∈ [0,T ]. Now we consider a sequence {ε(i)} of positive numbers converging to zero. Let y i := y ε(i) : [0,T ] → R n be the trajectories obtained by the preceding argument for ε = ε(i) for all i ∈ N. Note that y i (t) ∈ D for all i and t, because each of the polygonal arcs x π(k) constructed above joins points in D and D is closed and convex. Moreover,ẏ for all i and almost all t ∈ [0,T ]. Since ||ẋ π(k) (t)|| ≤ δ(D) for all k and a.a. t ∈ [0,T ] for all the polygonal arcs x π(k) defined above, we get Since the y i are uniformly bounded and equicontinuous, we can assume (possibly by passing to a subsequence without relabelling) that there is a continuous function y : [0,T ] → D such that We next show that y is a trajectory of f . To this end, we prove the following claim: Proof. Since y i → y uniformly on [0,T ] and f is locally bounded and locally uniformly continuous in the x variable, it suffices to show that as i → ∞. We do this by adapting a standard mollification argument (see for example [13,Chapter 8]) as follows. We first extend y to all of R by defining y(t) ≡ y(0) for all t ≤ 0 and y(t) ≡ y(T ) for all t ≥T . Recall that we also defined f (s, x) ≡ 0 for all s ∈ [0, T ]. It follows from (32) and (33) that as i → ∞. Moreover, for 0 ≤ t ≤T , we can change variables to get Notice that g ∈ L 1 [0,T ], and that ||g ε(i)z − g|| 1 ≤ 2||g|| 1 ∀i.
It therefore follows from Remark 2.8 that a subsequence of {y i } converges to a trajectory of f uniformly on [0,T ]. This must be the aforementioned function y, as desired. Again using the lower semicontinuity of Ψ, we therefore get The strong invariance asserted in the theorem is now immediate. Indeed, let x o ∈ S, T ≥ 0, and φ ∈ Traj T (F, x o ) be given. We next show that which would imply that φ remains in S on [0, T ]. To this end, note that if this supremum were some timet ∈ [0, T ), then the lower semicontinuity of Ψ would give In particular,x := φ(t) ∈ S. Let f ∈ C F ([0, T −t],x) satisfy the requirement (U ) for F and the trajectory [0, T −t] ∋ t → y(t) := φ(t +t), and let γ > 0 be such that f (t, x) ∈ cone{F (x)} for almost all t ∈ [0, T −t] and all x ∈ γB n (x). By reducing γ > 0 as necessary, we can assume that γB n (x) ⊆ U. By uniqueness of solutions of the initial value probleṁ , the first part of the proof applied to f and the initial valuex = φ(t) ∈ S would givẽ t ∈ (0, T −t ) such that Here we use the fact that the trajectory on [0,T ] constructed above for f starting atx can be extended to [0, T −t], by the linear growth assumption (C 3 ), and therefore coincides with y by our uniqueness assumption in (U). Since φ remains in S on [0,t], summing (37)-(38) would then contradict the definition of the supremumt. This establishes (36) and proves the theorem.
Remark 4.1. The preceding proof provides a constructive approach to finding viable trajectories for our Carathéodory feedback realizations f that remain in S. As suggested by [14], an alternative but highly nonconstructive proof of Theorem 2 would proceed as follows. Let x(t) be any trajectory of F starting in S. By Condition (U ), x(t) admits a feedback realization f ∈ C[0, T ], and our Hamiltonian assumption gives (f (t, x), 0), q ≤ 0 for all q ∈ N P epi(Ψ) (x, Ψ(x)) = {(ξ, −1) : ξ ∈ ∂ P Ψ(x)}, almost all t ≥ 0, and all x ∈ U, where epi (Ψ) := {(x, r) : r ≥ Ψ(x)} is the epigraph of Ψ. It is not hard to deduce from this (cf. [7]) that (f (t, x), 0) ∈ T B epi (Ψ) (x, r) for almost all t ≥ 0, all x ∈ U, and all (x, r) ∈ epi(Ψ), where T B denotes the Bouligand tangent cone. Applying the measurable viability theorem (see [15,Section 4]) to the dynamics F (t, x) = {(f (t, x), 0)} provides a trajectory t → (φ(t), Ψ(x(0))) forẋ = f (t, x),ẏ = 0 starting at (x(0), Ψ(x(0))) that stays in epi(Ψ). This requires f to be modified outside U in the usual way. Hence, Ψ(φ(t)) ≤ Ψ(x(0)) ≤ 0 for all t. By the uniqueness part of Condition (U ), φ(t) ≡ x(t) so x(t) stays in S. Unfortunately, the preceding alternative argument is highly nonconstructive, since the proof of the measurable viability theorem relies on Zorn's Lemma to construct the trajectory φ(t). One natural question which should be considered is how our Euler constructions from our proof could be used to build numerical schemes for approximating viable trajectories for Carathéodory dynamics. This type of result could be useful in physical applications. This question will be addressed by the authors in future research.

Strong Invariance Characterization
As we saw in Example 2.2, the Hamiltonian condition that H F (x, N P S (x)) ≤ 0 for all x ∈ boundary(S) is not necessary for strong invariance for (F, S); there, (F, {0}) is strongly invariant in R n , but the Hamiltonian condition is not satisfied, and F is upper semicontinuous but not lower semicontinuous. On the other hand, if we strengthen our assumption on F to (U ♯ ) Condition (U ) holds; and F is lower semicontinuous, and closed, convex, and nonempty valued. then we get the following strong invariance characterization: Theorem 3. Let F : R n ⇒ R n satisfy (U ♯ ) and S ⊆ R n be closed. Then (F, S) is strongly invariant in R n if and only if H F (x, N P S (x)) ≤ 0 for all x ∈ boundary (S). Proof. We showed the sufficiency of the Hamiltonian condition for strong invariance in Theorem 2, so it remains to show the necessity. We do this by extending an argument from the appendix of [1] to non-Lipschitz F . Assume (F, S) is strongly invariant. Fix x ∈ boundary(S), v ∈ F (x), and ζ ∈ N P S (x).
Taking the supremum over all v ∈ F (x) and noting that x ∈ boundary(S) was arbitrary gives the desired result.
Theorem 3 is no longer true if the requirement that F be lower semicontinuous is dropped, as shown by Example 2.2.