Subgradients of Distance Functions at Out-of-Set Points

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

“…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).…”

“…Note that both inclusions in (6.7) are generally strict even in finite dimensions and that we can equivalently put ε k ≡ 0 in (6.6) if X is Asplund and ϕ is lower semicontinuous aroundx; see [15,16] and [14,Subsection 1.3.3] for these and other properties of the right-sided limiting subdifferential (6.6). ∂T (x) = N (x; C r ) ∩ x * ∈ X * ρ F • (x * ) ≤ 1 (6.8) in terms of the enlargement (6.5); for F = B it was shown in [3,13].…”

Limiting subgradients of minimal time functions in Banach spaces

“…Tb proceed with calculating and estimating the limiting subdifferential, we establish first the corresponding results for c -subgradients of the Frechet type for the minimal time function in general Banach spaces; some of the latter results are fully new while the others are extensions and clarifications of those obtained in [10] for the case of Frechet subgradients (c = 0). In particular, we are able to fill the gap in the proof of [10, The results derived in this paper can be viewed as extensions of our previous developments [14,15] for the distance function (1.3); see also [12,Subsection 1.3.3]. Similarly to the distance function, we pay the main attention here to evaluating subgradients of (1.1) at out-of set points x rfc 0, which is essentially more involved in comparison with the inset case "' E 0.…”

“…Subdifferential properties of the distance function (1.3) have been well investigated and applied in many publications; see, e.g., [2,3,5,12,13,14,15,18] and the references therein. Much less has been done for the minimal time function (1.1).…”

On variational analysis for general distance functions

“…This function is a natural generalization of the classical distance function. In fact, if the dynamics F is the closed unit ball B of X, then the minimal time function is reduced to the classical distance function to Ω which is formed by x ∈ X, (1.2) whose subdifferential properties have been extensively studied (see, e.g., [1,5,15,16,17] and references therein). Let G : Z ⇒ X be a set-valued mapping between normed spaces.…”