Optimal control problems for some ordinary differential equations with behavior of blowup or quenching

Ping Lin,Weihan Wang

doi:10.3934/mcrf.2018036

Abstract

This paper is concerned with some optimal control problems for equations with blowup or quenching property. We first study the existence and Pontryagin's maximum principle for optimal controls which have the minimal energy among all the controls whose corresponding solutions blow up at the right-hand time end-point of a given functional. Then, the same problem for quenching case is discussed. Finally, we establish Pontryagin's maximum principle for optimal controls of extended problems after quenching.

Highlights

It is well known that solutions to some evolution equations have the behavior of blowup or quenching
Equations with the property of blowup can describe the dramatic increase in temperature leading to the ignition of a chemical reaction, and those with the property of quenching can be used to represent the potential differences of the polarization field in ionic conductors achieving the balance
Quenching of a solution means that the derivative in time t of the solution goes to infinity in finite time, while it keeps bounded itself

Summary

Introduction

It is well known that solutions to some evolution equations have the behavior of blowup or quenching. We shall first study the existence and Pontryagin’s maximum principle for optimal controls which have the minimal energy among all the controls whose corresponding solutions blow up at the right-hand time end-point of a given functional. Among all the extended controls whose corresponding extended solutions quench for the second time at the right-hand time end-point of a cost functional, Pontryagin’s maximum principle will be established for optimal controls which have the minimal energy. It is different with the problem considered in [14], where every admissive state does not quench for the second time at any point of the closed interval in which the cost functional is defined. For any T > 0, the relaxed control set RT (U ) is convex and sequentially compact

Consider the following relaxed system

From Lemma

The set

For each ε