Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

TL;DR

This paper introduces and compares three projection-based numerical methods for solving the minimum-energy control problem of a double integrator with control constraints, a problem previously not tackled with such approaches.

Contribution

The paper applies three projection algorithms—Dykstra's, Douglas--Rachford, and Aragón Artacho--Campoy—to continuous-time optimal control, pioneering their use in infinite-dimensional Hilbert space problems.

Findings

Douglas--Rachford and Aragón Artacho--Campoy algorithms show promising convergence behavior.

Numerical error analysis demonstrates the effectiveness of the proposed methods.

Parameter tuning significantly impacts the algorithms' performance.

Abstract

We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i)~Dykstra's algorithm, (ii)~the Douglas--Rachford (DR) method and (iii)~the Arag\'on Artacho--Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.