Abstract
In this paper, infinite horizon optimal control problems subject to semilinear parabolic equations are investigated. A finite number of only time-dependent controls intervening at disjoint positions in the space domain are considered. The controls are subject to integral constraints and a term is included in the cost functional that promotes control sparsity. The existence of optimal controls is proven, first and second order optimality conditions are derived, and the approximation by finite horizon control problems is addressed.