Regularity of multipliers and second-order optimality conditions of KKT-type for semilinear parabolic control problems

TL;DR

This paper investigates the regularity and second-order optimality conditions for semilinear parabolic control problems with mixed and box constraints, establishing conditions for the fulfillment of KKT-type optimality and multiplier regularity.

Contribution

It provides new regularity results for multipliers and solutions under separation and geometric density conditions, advancing the theoretical understanding of such control problems.

Findings

KKT-type optimality conditions are satisfied under the separation condition.

Multipliers and solutions are Hölder continuous if initial data and boundary conditions meet specific criteria.

Regularity of multipliers is linked to geometric properties of the domain boundary.

Abstract

A class of optimal control problems governed by semilinear parabolic equations with mixed constraints and a box constraint for control variable is considered. We show that if the separation condition is satisfied, then both optimality conditions of KKT-type and regularity of multipliers are fulfilled. Moreover, we show that if the initial value is good enough and boundary $\partial\Omega$ has a property of positive geometric density, then multipliers and optimal solutions are H\"{o}lder continuous.