Subdifferential analysis of differential inclusions via discretization

TL;DR

This paper develops a subdifferential analysis framework for differential inclusions using discretization, providing new estimates and conditions that connect discretized problems to continuous ones in optimal control.

Contribution

It introduces a novel approach to analyze differential inclusions through discretization, estimating subdifferentials and coderivatives to derive optimality conditions.

Findings

Estimated subdifferential dependence of optimal value on endpoints

Derived discretized Euler-Lagrange and transversality conditions

Extended results to continuous differential inclusions via limit processes

Abstract

The framework of differential inclusions encompasses modern optimal control and the calculus of variations. Necessary optimality conditions in the literature identify potentially optimal paths, but do not show how to perturb paths to optimality. We first look at the corresponding discretized inclusions, estimating the subdifferential dependence of the optimal value in terms of the endpoints of the feasible paths. Our approach is to first estimate the coderivative of the reachable map. The discretized (nonsmooth) Euler-Lagrange and transversality conditions follow as a corollary. We obtain corresponding results for differential inclusions by passing discretized inclusions to the limit.