Discrete approximations and optimality conditions for integro-differential inclusions

TL;DR

This paper develops a framework for approximating and deriving optimality conditions for complex integro-differential inclusion problems, enabling better analysis and solution of dynamic systems with nonconvex, time-dependent integral dynamics.

Contribution

It introduces a novel approximation scheme for integro-differential inclusions and derives new necessary optimality conditions of Volterra type for such systems.

Findings

Strong convergence of discrete solutions to continuous problems.

New necessary optimality conditions for integro-differential inclusions.

Effective approximation methods for nonsmooth dynamic systems.

Abstract

This paper addresses a new class of generalized Bolza problems governed by nonconvex integro-differential inclusions with endpoint constraints on trajectories, where the integral terms are given in the general (with time-dependent integrands in the dynamics) Volterra form. We pursue here a threefold goal. First we construct well-posed approximations of continuous-time integro-differential systems by their discrete-time counterparts with showing that any feasible solution to the original system can be strongly approximated in the $W^{1,2}$-norm topology by piecewise-linear extensions of feasible discrete trajectories. This allows us to verify in turn the strong convergence of discrete optimal solutions to a prescribed local minimizer for the original problem. Facing intrinsic nonsmoothness of original integro-differential problem and its discrete approximations, we employ appropriate tools of generalized differentiation in variational analysis to derive necessary optimality conditions for discrete-time problems (which is our second goal) and finally accomplish our third goal to obtain necessary conditions for the original continuous-time problems by passing to the limit from discrete approximations. In this way we establish, in particular, a novel necessary optimality condition of the Volterra type, which is the crucial result for dynamic optimization of integro-differential inclusions.