A shooting algorithm for problems with singular arcs

TL;DR

This paper introduces a shooting algorithm for optimal control problems with linearly appearing controls, demonstrating local quadratic convergence under certain conditions and validating the approach through numerical tests.

Contribution

It develops a shooting algorithm tailored for problems with linear controls, proving convergence and stability conditions, and validating it with numerical experiments.

Findings

Quadratic convergence of the shooting algorithm under weak optimality conditions

Stability of the solution when the system is square and conditions are met

Numerical validation confirms effectiveness of the proposed method

Abstract

In this article we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss-Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system) we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated to the perturbed problem. We present numerical tests that validate our method.