Solving Singular Control Problems in Mathematical Biology, Using PASA

TL;DR

This paper demonstrates the application of the Polyhedral Active Set Algorithm (PASA) to solve singular control problems in mathematical biology, introducing regularization techniques to handle control oscillations and showcasing its effectiveness through multiple biological case studies.

Contribution

It introduces a novel approach combining PASA with regularization for singular control problems, tailored for applications in mathematical biology.

Findings

PASA effectively solves complex singular control problems.

Regularization reduces control oscillations and artifacts.

The method is demonstrated on three biological applications.

Abstract

In this paper, we will demonstrate how to use a nonlinear polyhedral constrained optimization solver called the Polyhedral Active Set Algorithm (PASA) for solving a general singular control problem. We present methods of discretizing a general optimal control problem that involves the use of the gradient of the Lagrangian for computing the gradient of the cost functional so that PASA can be applied. When a numerical solution contains artifacts that resemble "chattering'', a phenomenon where the control oscillates wildly along the singular region, we recommend a method of regularizing the singular control problem by adding a term to the cost functional that measures a scalar multiple of the total variation of the control, where the scalar is viewed as a tuning parameter. We then demonstrate PASA's performance on three singular control problems that give rise to different applications of mathematical biology. We also provide some exposition on the heuristics that we use in determining an appropriate size for the tuning parameter.