Optimal control of quasilinear parabolic PDEs with state-constraints

Abstract

We discuss state-constrained optimal control of a quasilinear parabolic PDE. Existence of optimal controls and first-order necessary optimality conditions are derived for a rather general setting including pointwise in time and space constraints on the state. Second-order sufficient optimality conditions are obtained for averaged-in-time and pointwise in space state-constraints under general regularity assumptions for the equation, and for pointwise in time and space state-constraints when restricting in return to a more regular setting for the state equation.