Positivity certificates in optimal control

TL;DR

This paper introduces positivity certificates as an accessible framework for approximating solutions in optimal control, demonstrating their application in region of attraction, direct, and inverse optimal control with numerical examples.

Contribution

It provides a unified, simplified presentation of positivity-based relaxations in optimal control, connecting them with existing methods like occupation measures and SOS hierarchies.

Findings

Positivity certificates effectively approximate control problems.

The approach applies to multiple control applications with numerical validation.

Connections to broader control theory concepts are established.

Abstract

We propose a tutorial on relaxations and weak formulations of optimal control with their semidefinite approximations. We present this approach solely through the prism of positivity certificates which we consider to be the most accessible for a broad audience, in particular in the engineering and robotics communities. This simple concept allows to express very concisely powerful approximation certificates in control. The relevance of this technique is illustrated on three applications: region of attraction approximation, direct optimal control and inverse optimal control, for which it constitutes a common denominator. In a first step, we highlight the core mechanisms underpinning the application of positivity in control and how they appear in the different control applications. This relies on simple mathematical concepts and gives a unified treatment of the applications considered. This presentation is based on the combination and simplification of published materials. In a second step, we describe briefly relations with broader literature, in particular, occupation measures and Hamilton-Jacobi-Bellman equation which are important elements of the global picture. We describe the Sum-Of-Squares (SOS) semidefinite hierarchy in the semialgebraic case and briefly mention its convergence properties. Numerical experiments on a classical example in robotics, namely the nonholonomic vehicle, illustrate the concepts presented in the text for the three applications considered.