# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.06872