The Switch Point Algorithm

TL;DR

The paper introduces the Switch Point Algorithm, a novel method for efficiently solving optimal control problems with singular, bang-bang, or mixed solutions by reducing the problem to optimizing over switch points and computing derivatives simultaneously.

Contribution

It presents a new algorithm that simplifies solving complex optimal control problems by deriving formulas for derivatives and enabling gradient-based optimization over switch points.

Findings

Effective in problems with known solutions

Outperforms existing algorithms in test cases

Allows simultaneous derivative computation

Abstract

The Switch Point Algorithm is a new approach for solving optimal control problems whose solutions are either singular or bang-bang or both singular and bang-bang, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. Problems in this class can often be reduced to an optimization over the switching points. Formulas are derived for the derivative of the objective with respect to the switch points, the initial costate, and the terminal time. All these derivatives can be computed simultaneously in just one integration of the state and costate dynamics. Hence, gradient-based unconstrained optimization techniques, including the conjugate gradient method or quasi-Newton methods, can be used to compute an optimal control. The performance of the algorithm is illustrated using test problems with known solutions and comparisons with other algorithms from the literature.