Relaxation methods for optimal control problems

TL;DR

This paper introduces two relaxation methods for solving nonlinear optimal control problems with differential inclusions involving maximal monotone maps, accommodating systems with unilateral constraints, and proves their equivalence.

Contribution

It presents two novel relaxation techniques for optimal control problems with differential inclusions, extending existing methods to systems with unilateral constraints.

Findings

The two relaxation methods are shown to be equivalent.

The methods are admissible for the class of problems considered.

The framework includes systems with unilateral constraints.

Abstract

We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map $A:\mathbb{R}^N\rightarrow2^{\mathbb{R}^N}$. We do not assume that $D(A)=\mathbb{R}^N$, incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.