Infimal Convolution and Duality in Convex Optimal Control Problems with Second Order Evolution Differential Inclusions

TL;DR

This paper develops a duality framework for second order evolution differential inclusions in optimal control, utilizing infimal convolution and discrete approximations to establish dual problems and extend to higher order cases.

Contribution

It introduces a novel method using infimal convolution to construct dual problems for second order differential inclusions and higher order cases, bridging discrete and continuous formulations.

Findings

Constructed dual problems for second order evolution differential inclusions.

Proved duality relations using Euler-Lagrange type inclusions.

Developed a computational method using Pascal's triangle for higher order problems.

Abstract

The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex functions, step by step we construct the dual problems for discrete, discrete-approximate and differential inclusions and prove duality results. It seems that the Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems and that the dual problem for discrete-approximate problem make a bridge between them. Finally, relying to the method described within the framework of the idea of this paper a dual problem can be obtained for any higher order differential inclusions. In this way relying to the described method for computation of the conjugate and support functions of discrete-approximate problems a Pascal triangle with binomial coefficients, can be successfully used for any "higher order" calculations.