Implicit Euler approximation and Optimization of one-sided Lipschitzion differntial inclusions

TL;DR

This paper develops a numerical approximation method for solving generalized Bolza problems governed by differential inclusions with relaxed one-sided Lipschitz conditions, establishing convergence and optimality conditions.

Contribution

It introduces a new implicit Euler scheme-based discretization for ROSL differential inclusions and proves convergence to local solutions with derived optimality conditions.

Findings

Strong $W^{1,2}$ norm convergence of discretized solutions.

Justification of well-posedness for implicit Euler approximations.

Derivation of necessary optimality conditions for discretized problems.

Abstract

This paper concerns the study of the generalized Bolza problem governed by differential inclusions satisfying the so-called "relaxed one-sided Lipschitzian" (ROSL) condition with respect to the state variables subject to various types of nonsmooth endpoint constraints. We construct discrete approximations of differential inclusions with ROSL right-hand sides by using the implicit Euler scheme for approximating time derivatives, and then we justify an appropriate well-posedness of such approximations. Our principal result establishes the strong approximation (in the sense of the $W^{1,2}$ norm convergence) of an "intermediate" (between strong and weak minimizers) local optimal solution of the continuous-time Bolza problem under the ROSL assumption by optimal solutions of the implicitly discretized finite-difference systems. Finally, we derive necessary optimality conditions for the discretized Bolza problems via suitable generalized differential constructions of variational analysis. The obtained results on the well-posedness of discrete approximations and necessary optimality conditions allow us to justify a numerical approach to solve the generalized Bolza problem for one-sided Lipschitzian differential inclusions by using discrete approximations constructed via the implicit Euler scheme.