Quenching Time Optimal Control for Some Ordinary Differential Equations

TL;DR

This paper investigates the problem of controlling certain 2D differential equations to minimize the finite quenching time, establishing existence and optimality conditions for such controls, and suggesting extensions to more general systems.

Contribution

It introduces the first study of quenching time optimal control problems for specific ODEs, providing existence results and Pontryagin maximum principles.

Findings

Proved existence of optimal controls for quenching time minimization.

Derived Pontryagin maximum principles for the control problems.

Suggested potential extensions to higher dimensions and parabolic equations.

Abstract

This paper concerns some time optimal control problems of three different ordinary differential equations in $\mathbb{R}^2$. Corresponding to certain initial data and controls, the solutions of the systems quench at finite time. The goal to control the systems is to minimize the quenching time. To our best knowledge, the study on quenching time optimal control problems has not been touched upon. The purpose of this study is to obtain the existence and the Pontryagin maximum principles of optimal controls. We hope that our methods could hint people to study the same problems with more general vector fields in $\mathbb{R}^d$ with $d\in \mathbb{N}$. We also wish that our results could be extended to the same issue for parabolic equations.