Second-order Optimality Conditions for Time-Optimal Control Problems Governed by Semilinear Parabolic Equations

TL;DR

This paper establishes first and second-order optimality conditions for time-optimal control problems governed by semilinear parabolic equations, addressing constraints and local optimality.

Contribution

It introduces locally optimal solutions and derives new second-order necessary and sufficient optimality conditions for such control problems.

Findings

First and second-order optimality conditions are formulated.

Conditions are applicable to problems with mixed pointwise and final constraints.

Theoretical framework enhances understanding of optimality in semilinear parabolic control problems.

Abstract

A class of time-optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints and final point constraints is considered. By introducing the so-called locally optimal solution to time-optimal control problems, we establish first and second-order necessary optimality conditions of KKT-type and second-order sufficient conditions for locally optimal solutions to the problem.