  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 well-defined 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 to deduce sparsity properties of the optimal
controls in time.
