A Proximal Characterization of the Reachable Set

We show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a Hamilton-Jacobi equation involving proximal normals. Résumé On démontre que le graphe de l’ensemble des points accessibles d’un système de contrôle décrit par une inclusion différentielle est caractérisé par une équation Hamilton-Jacobi impliquant les normales proximales. We study a control system defined via a differential inclusion ẋ(t) ∈ F ( t, x(t) ) a. e. (1) As usual, a trajectory of (1) refers to an absolutely continuous function x(·) satisfying (1) on a given interval [a, b]. The equivalence of (1) to a classical control system ẋ = f(t, x, u), u ∈ U is well-understood; we shall not dwell upon it. For a given choice of initial time t0 and nonempty compact subset A of R, we consider the set R defined as follows: R = {( t, §(t) ) : t ≥ t′, §(·) is a trajectory on [t′,t], §(t′) ∈ A } . The assumptions on the multifunction F are as follows: (H1) For each (t, x) ∈ [t0,∞)× R, the set F (t, x) is a nonempty, convex, compact subset of R. (H2) For some constants γ and c, and for all (t, x) in [t0,∞)× R, one has v ∈ F (t, x)⇒ |v| ≤ γ|x|+ c. (H3) F is locally Lipschitz on [t0,∞)×R; i.e., for any bounded subset S of [t0,∞)×R there is a constant K such that, for all (ti, xi) ∈ S (i = 1, 2), we have F (t2, x2) ⊆ F (t1, x1) +K ∣∣(t2 − t1, x2 − x1)∣∣B, where B denotes the closed unit ball in R. It is a well-known fact that under these hypotheses the set R is closed, and that its “slice” at time T , the reachable set RT := { § : (T , §) ∈ R } is compact and nonempty for each T ≥ t0. A proximal normal [3] ζ to a closed set S at a point x ∈ S is a vector ζ such that, for some σ ≥ 0, one has 〈ζ, x′ − x〉 ≤ σ|x′ − x| ∀x′ ∈ S. The set of proximal normals to S at x is a cone; we denote it ∂PS(x). Note that 0 ∈ ∂PS(x) ∀x ∈ S, and that ∂PS(x) is undefined when x / ∈ S. It is known that ∂PS(x) reduces to the set of normals in the usual sense when S is a smooth manifold (with or without boundary), or when S is a convex set. The set of points x for which ∂PS(x) is nontrivial (i.e., 6= {0}) can be “small”, but is always dense in the boundary of S. The (upper) Hamiltonian H : R× R × R → R is the function defined by H(t, x, p) := max { 〈p, v〉 : v ∈ F (t, x) } . Theorem 1. R is the unique closed subset S of [t0,∞)× R satisfying: (i) θ +H(t, x, ζ) = 0 ∀(θ, ζ) ∈ ∂PS(t, x), ∀(t, x) ∈ (t0,∞)× R, (ii) limT↓0 ST = A. Remark 1. (a) Since ∂PS(t, x) is only defined when (t, x) lies in S, the “proximal Hamilton-Jacobi equation” in (i) is in issue only at such points. Since H(t, x, 0) = 0, it holds automatically at any point (t, x) ∈ S for which ∂PS(t, x) is trivial. (b) The initial condition (ii) is to be understood in the Hausdorff metric ρ; that is, for any ε > 0 there exists δ > 0 such that T ∈ [t0, t0 + δ)⇒ ρ(ST , A) < ε. It follows in particular that St0 = A. It is equivalent to (ii) to require this last equality together with the uniform boundedness of ST for T near t0. Proof of the Theorem. Let us verify first that R satisfies (i). Given any point (τ, α) in R, let x be any trajectory on [τ,∞) with x(τ) = α. We claim that ( t, x(t) ) ∈ R for all t > τ . If τ = t0, then α ∈ A necessarily, and so ( t, x(t) ) ∈ R by the very definition of R. If τ > t0, there is a trajectory y on [t0, τ ] with y(t0) ∈ A, y(τ) = α. But then “y followed by x” is a trajectory on [t0,∞) beginning in A, whence ( t, x(t) ) ∈ R ∀t > τ as claimed.