Best Approximation Optimal Control for Infeasible Double Integrator and Douglas--Rachford Algorithm

TL;DR

This paper develops an analytical approach for optimal control of infeasible double integrators by minimizing a gap function, and explores the Douglas--Rachford algorithm for solving the problem numerically.

Contribution

It provides an explicit analytical solution for the best approximation control in infeasible double integrator problems and discusses numerical methods including the Douglas--Rachford algorithm.

Findings

Analytical solution for bang--bang control with one switch.

Reduction of the problem to solving two algebraic equations.

Numerical experiments demonstrating algorithm performance.

Abstract

We consider the problem of finding (in some sense) the best approximation control for an infeasible double integrator. The control function is constrained by upper and lower bounds that are too tight and thus cause infeasibility. The infeasibility is characterized by a gap function (representing the separation between two constraint sets) whose squared ${\cal L}^2$-norm is to be minimized to find the best approximation control solution. First, we review the existing results for problems involving a general linear control system. Then, for the infeasible double integrator problem, we present an analytical solution for the bang--bang control with at most one switching. The infinite-dimensional optimization problem is reduced to the problem of solving two algebraic equations in two variables, to compute the switching time and gap function. We discuss numerical approaches to solving the system of equations. Finally, we describe the (relaxed) Douglas--Rachford algorithm for the double integrator problem and carry out numerical experiments to illustrate the implementation of the algorithm and test performance.