A note on source term representation for control-and-state-constrained parabolic control problems with purely time-dependent control

Abstract

We revisit the concept of source term representation, originally developed for boundary control problems subject to pointwise state constraints in the domain, and apply it to linear-quadratic parabolic control problems with purely time-dependent control subject to pointwise state constraints and control bounds. Since control and state are defied on difierent domains, classical Lavrentiev regularization cannot be applied. Instead, in the source term representation ansatz, the adjoint operator is applied to an auxiliary control that is defined in the same domain as the state. By this ansatz, additional pure control bounds would be transformed into artificial state constraint and were therefore previously excluded from the discussion. For purely time dependent control we observe that control constraints will become averaged in space and pointwise in time state constraints, and can apply available theory for the existence of Lagrange multipliers. We briefly visit elliptic problems with finitely many control parameters, where this regularization of pointwise state constraints transforms additional simple bounds on the control parameters into easy-to-handle integral-state constraints.