Finite element approximation of sparse parabolic control problems

We study the finite element approximation of an optimal control problem governed by a semilinear partial differential equation and whose objective function includes a term promoting space sparsity of the solutions. We prove existence of solution in the absence of control bound constraints and provide the adequate second order sufficient conditions to obtain error estimates. Full discretization of the problem is carried out, and the sparsity properties of the discrete solutions, as well as error estimates, are obtained.