Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

Abstract

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form u(t) L 1 (Ω) ≤ γ for t ∈ (0, T ). This limits the total control that can be applied to the system at any instant of time. The L 1 -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.