Homotopic approach for turnpike and singularly perturbed optimal control problems

Olivier Cots,Joseph Gergaud,Boris Wembe,Didier Auroux,Roland Ruelle,Abderrahmane Habbal,Jean-Baptiste Caillau,Ludovic Rifford,Régis Duvigneau,Luc Pronzato,Chiara Soresi,Olivier Pantz,Christine Malot

doi:10.1051/proc/202171105

Abstract

The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.

Highlights

New investigations have been carried out on optimal control problems with turnpike properties [13]: the optimal solution remains for a long time close to the optimal steady-state solution associated to the static optimal control problem and at the beginning and at the end we have a transcient-short time arc
The first result we present here establishes that an optimal control problem with the turnpike property is a singularly perturbed optimal control problem with only fast variables
We propose here to improve this procedure with two homotopies: the first one is an homotopy on the two-point condition which connects the static optimal solution, i.e. the zero-order outer solution of the problem, to the original problem for a final time not too long and the second one is on the final time

Summary

Introduction

New investigations have been carried out on optimal control problems with turnpike properties [13]: the optimal solution remains for a long time close to the optimal steady-state solution associated to the static optimal control problem and at the beginning and at the end we have a transcient-short time arc. The first result we present here establishes that an optimal control problem with the turnpike property is a singularly perturbed optimal control problem with only fast variables. We propose here to improve this procedure with two homotopies: the first one is an homotopy on the two-point condition which connects the static optimal solution, i.e. the zero-order outer solution of the problem (seen as a singularly perturbed control problem), to the original problem for a final time not too long and the second one is on the final time. It is possible to extend this method to solve singularly perturbed optimal control problems. For this we have to adapt the first homotopy on the two-point condition to take into account the state equation on the fast variables. Several exemples from the literature show the efficiency of our approach

Objectives

Methods

Conclusion