Infinite Horizon Optimal Control Problems for a Class of Semilinear Parabolic Equations

Infinite horizon open loop optimal control problems for semilinear parabolic equations are investigated.The controls are subject to a cost functional which promotes sparsity in time.The focus is put on deriving first order optimality conditions.This is achieved without relying on a welldefined control-to-state mapping in a neighborhood of minimizers.The technique of proof is based on the approximation of the original problem by a family of finite horizon problems.The optimality conditions allow deduction of sparsity properties of the optimal controls in time.