State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions

Abstract

We consider a control- and state-constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The nonlocal radiation interface condition and the pointwise state-constraints represent the particular features of this problem. To deal with the state-constraints, continuity of the state is shown which allows to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an $L^2$-neighborhood.