Boundary Control of Semilinear Parabolic Equations with Non-Smooth Point-wise-Integral Control Constraints in Time-Space

This work concentrates on a class of Neumann optimal control problems for semilinear parabolic equations on a bounded domain Q = Ω × (0,T) subject to a control constraint of the form $||u(t)|{|_{{L^1}(\Gamma )}} \leq \gamma $ for t ∈ (0,T), where Γ is the boundary of Ω. Thus the total amount of control that can be exercised at any instant of time is limited. Constraining by the L
¹
-norm leads to sparsity of the control in space, for time instances when the constraint is active. The non-smoothness of the constraint, necessitates new techniques for the analysis of the optimal control problem. Existence of a solution, first order optimality conditions, and regularity of the optimal control are established.