Optimal control of semilinear parabolic equations with non-smooth pointwise-integral control constraints in time-space

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form llu(t)ll L1(Ω) ≤ γ for t ∈ (0, T ). This limits the total control that can be applied to the system at any instant of time. The L1-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to γ is investigated on the basis of different second order conditions.Eduardo Casas was supported by MCIN/ AEI/10.13039/501100011033/ under research projects MTM2017-83185-P and PID2020-114837GB-I00. Karl Kunisch was supported by the ERC advanced Grant 668998 (OCLOC) under the EU’s H2020 research program