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 imposed and cost functional does not involve itself, except possibly a non-differentiable sparsity-promoting term. Under appropriate second order sufficient optimality conditions, first estimate difference between discrete continuous states. Next, under an additional assumption on adjoint state, prove error estimates improve