Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

TL;DR

This paper studies optimal control problems for semilinear parabolic equations with non-smooth, pointwise-integral control constraints, establishing existence, optimality conditions, regularity, and stability of solutions.

Contribution

It introduces new analytical techniques to handle non-smooth $L^1$-norm constraints, proving existence, optimality conditions, and stability results for the control problem.

Findings

Existence of optimal controls is established.

First and second order optimality conditions are derived.

Stability of controls with respect to the constraint parameter $oldsymbol{oldsymbol{ extit{ extgamma}}}$ is demonstrated.

Abstract

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form $\|u(t)\|_{L^1(\Omega)} \le \gamma$ for $t \in (0,T)$. This limits the total control that can be applied to the system at any instant of time. The $L^1$-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to $\gamma$ is investigated on the basis of different second order conditions.