Subdifferential properties of the minimal time function of linear control systems

Abstract

Minimal time function, Linear control system, Subdifferentials, Normal cones,

Fréchet Analysis and Sensitivity Relations for the Optimal Time Problem

“…In this paper, we first present and prove similar formulas for computing Fréchet subdifferentials and Fréchet singular subdifferentials of the minimum time function T associated with a general target K for a differential inclusion without assuming any calmness assumption. Our results extend and improve corresponding results in [20], [23]. A connection between the Fréchet normal cones to sub-level sets and to the epigraph of T is also derived.…”

“…Similar characterizations for the proximal singular subdifferentials were presented in [22]. In [20], the authors proved formulas for computing the Fréchet subdifferentials of the minimum time function for linear control systems under a calmness assumption. Characterizations of the proximal/Fréchet (singular) subdifferentials of the minimum time function associated…”

“…Recalling that a function f : R n → R ∪ {+∞} is said to be calm at a point x 0 ∈ dom(f ) if there exists a positive number such that |f (x) − f (x 0 )| ≤ ||x − x 0 || for all x in a neighborhood of x 0 . Since, under fairly general hypotheses, the minimum time function is merely lower semicontinuous, the requirement for the minimum time function being calm in [20] is strong. Our aim is to relax the calmness of the minimum time function and obtain the same formulas for computing the Fréchet subdifferentials.…”

Proximal Analysis and the Minimal Time Function of a Class of Semilinear Control Systems

“…In this paper, we extend the results in [7][8][9][10][11][12][13][14][15] to semilinear systems in infinite dimensional spaces. We also obtain a complete characterization of the proximal subdifferential of the minimal time function.…”

“…This result is elegant, but certain important cases are not covered in the analysis. There have been some recent papers addressing the missing cases of [6] such as [7][8][9][10][11][12][13][14][15], but they are either on simpler systems (constant or linear) or restricted to finite dimensional spaces.…”