A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations

TL;DR

This paper investigates the existence of solutions for control problems involving semilinear elliptic and parabolic PDEs, showing that optimal controls in L^2 are actually bounded, enabling standard optimality conditions.

Contribution

It proves that all L^2 optimal controls for these PDE control problems are bounded in L^, facilitating the application of classical optimality conditions.

Findings

L^2 controls are proven to belong to L^.

Standard optimality conditions are valid for these controls.

Well-posedness of state equations in L^2 framework is established.

Abstract

In this paper, we study optimal control problems of semilinear elliptic and parabolic equations. A tracking cost functional, quadratic in the control and state variables, is considered. No control constraints are imposed. We prove that the corresponding state equations are well-posed for controls in $L^2$. However, it is well-known that in the $L^2$ framework the mappings involved in the control problem are not Frechet differentiable in general, which makes any analysis of the optimality conditions challenging. Nevertheless, we prove that every $L^2$ optimal control belongs to $L^\infty$, and consequently standard optimality conditions are available.