Extension of Switch Point Algorithm to Boundary-Value Problems

TL;DR

This paper extends the Switch Point Algorithm to boundary-value problems with initial and terminal constraints, enabling solutions for more complex optimal control problems with structured discontinuities.

Contribution

The paper introduces modifications to the existing Switch Point Algorithm to handle boundary-value problems with both initial and terminal constraints, broadening its applicability.

Findings

The derivative of the cost with respect to a switch point equals the jump in the Hamiltonian.

The extended algorithm effectively solves boundary-value optimal control problems.

Substantial theoretical modifications are developed for the generalized setting.

Abstract

In an earlier paper (https://doi.org/10.1137/21M1393315), the Switch Point Algorithm was developed 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. The class of control problems that were considered had a given initial condition, but no terminal constraint. The theory is now extended to include problems with both initial and terminal constraints, a structure that often arises in boundary-value problems. Substantial changes to the theory are needed to handle this more general setting. Nonetheless, the derivative of the cost with respect to a switch point is again the jump in the Hamiltonian at the switch point.