Optimal control problems with oscillations, concentrations and discontinuities

TL;DR

This paper develops a mathematical framework using anisotropic parametrized measures to accurately define costs in optimal control problems involving oscillations and concentrations, enabling effective numerical solutions.

Contribution

It introduces a novel approach combining functional analysis with the Lasserre hierarchy to handle complex control behaviors in optimal control problems.

Findings

Provides a rigorous definition of integral costs with oscillations and concentrations.

Demonstrates the integration of anisotropic measures with semidefinite programming.

Enables numerical approximation of solutions in complex optimal control scenarios.

Abstract

Optimal control problems with oscillations (chattering controls) and concentrations (impulsive controls) can have integral performance criteria such that concentration of the control signal occurs at a discontinuity of the state signal. Techniques from functional analysis (anisotropic parametrized measures) are applied to give a precise meaning of the integral cost and to allow for the sound application of numerical methods. We show how this can be combined with the Lasserre hierarchy of semidefinite programming relaxations.