Limiting subgradients of minimal time functions in Banach spaces

Abstract

Abstract. The paper mostly concerns the study of generalized differential properties of the so-called minimal time functions associated, in particular, with constant dynamics and arbitrary closed target sets in control theory. Functions of this type play a significant role in many aspects of optimization, control theory: and Hamilton-Jacobi partial differential equations. We pay the main attention to computing and estimating limiting subgradients of the minimal value functions and to deriving the corresponding relations for Frechet type c-subgradients in arbitrary Banach spaces.

Well-Posedness of Minimal Time Problems with Constant Dynamics in Banach Spaces

“…Furthermore, we get from [17,Theorem 3.6] the representation Given a target set C ⊂ X and a pointx / ∈ C, define the minimal time enlargement of C relative tox by…”

“…The most recent results in this direction derived in [17] provide tight upper estimates as well as exact formulas for computing the ε-subdifferentials of the Fréchet type and the limiting/Mordukhovich subdifferential of the minimal time function at both in-set (x ∈ C) and out-of-set (x / ∈ C) points in arbitrary Banach spaces X. The results of [17] extend those obtained in [3,13,15,16] for the latter subdifferentials of the distance function (1.5). They are used in what follows to establish some regularity properties of the minimal time function (1.2).…”

“…The imposed requirements on the dynamics and target are our standing assumptions in this paper. Various properties of optimal solutions to (1.1) were studied in [8,9,10,12,17] and the references therein in finite and infinite dimensions. The major optimal characteristics of problem (1.1) are given by the optimal value function (known also as the minimal value/time function) defined by T F C (x) = T (x) := inf t > 0 (x + tF ) ∩ C) = ∅ = inf y∈C ρ F (y − x), (1.2) where ρ F (·) is the classical Minkowski functional/gauge, ρ F (u) := inf t > 0 t −1 u ∈ F , u ∈ X, (1.3) and by the generalized/minimal time projection 4) which is generally a set-valued mapping Π : X → → C with possibly empty values.…”

Subdifferential formulas for a class of non-convex infimal convolutions

“…Subdifferential formulas for this class functions in both convex and nonconvex settings have been of great interest in the literature; see [5][6][7][10][11][12] and the references therein. It is well known that the subdifferential in the sense of convex analysis of the distance function (1.1) can be computed using the following infimal convolution representation: 4) where δ Ω is the indicator function associated with Ω given by δ(x; Ω) = 0 if x ∈ Ω, and δ(x; Ω) = ∞ otherwise.…”

Generalized Differentiation and Characterizations for Differentiability of Infimal Convolutions

“…The readers are referred to [4,5,8,9,12,14,15,17,19,21,22,25,26] and the references therein for the study of the minimal time function as well as its specification to the case of the distance function.…”