State Error Estimates for the Numerical Approximation of Sparse Distributed Control Problems in the Absence of Tikhonov Regularization

In this paper, we analyze optimal control problems of semilinear elliptic equations, where the controls are distributed. Box constraints for the controls are imposed and the cost functional does not involve the control itself, except possibly for a non-differentiable sparsity-promoting term. Under appropriate second order sufficient optimality conditions, first we estimate the difference between the discrete and continuous optimal states. Next, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states.