A Priori Error Analysis for Discretization of Sparse Elliptic Optimal Control Problems in Measure Space

Konstantin Pieper, Boris Vexler
2013 SIAM Journal of Control and Optimization  
In this paper an optimal control problem is considered, where the control variable lies in a measure space and the state variable fulfills an elliptic equation. This formulation leads to a sparse structure of the optimal control. In this setting we prove a new regularity result for the optimal state and the optimal control. Moreover, a finite element discretization based on [E. Casas, C. Clason, and K. Kunisch, SIAM J. Control Optim., 50 (2012), pp. 1735-1752] is discussed and a priori error
more » ... d a priori error estimates are derived, which significantly improve the estimates from that paper. Numerical examples for problems in two and three space dimensions illustrate our results. ). Optimality system and regularity. As the first step we recall the weak formulation of the state equation (1.2). For a given q ∈ M(Ω) the solution u = u(q) is determined by u ∈ L 2 (Ω) : (u, −Δϕ) = q, ϕ for all ϕ ∈ H 2 (Ω) ∩ H 1 0 (Ω).
doi:10.1137/120889137 fatcat:cgzwke26o5edfavkf5bv3xe4gy