Solving Bang-Bang Problems Using The Immersed Interface Method and Integer Programming

TL;DR

This paper introduces a numerical method combining the immersed interface method and integer programming to solve bang-bang optimal control problems without smoothing the discontinuous control functions.

Contribution

It presents a formal Lagrangian framework and an immersed interface method to accurately solve true bang-bang control problems using an adjoint-based gradient approach.

Findings

Effective numerical solution for true bang-bang control problems

Avoids smoothing discontinuous controls with traditional methods

Demonstrates success with detailed numerical results

Abstract

In this paper we study numerically solving optimal control problems with bang-bang control functions. We present a formal Lagrangian approach for solving the optimal control problem, and address difficulties encountered when numerically solving the state and adjoint equations by using the immersed interface method. We note that our numerical approach does not approximate the discontinuous control function with smooth functions, instead we solve the true bang-bang optimal control problem. Our approach for solving the optimal control problem uses an adjoint-based gradient. We use the gradient in our first-order trust-region method to generate a local minimizing control. We present detailed numerical results to demonstrate the effectiveness of our method.