Equivalent a posteriori error estimates for elliptic optimal control problems with L1-control cost

An elliptic optimal control problem involving the $L^1$ norm of the control in the cost functional is considered in this paper. We use the full discretization and the variational discretization to approximate the control problem and the efficient and reliable a posteriori error estimates are obtained for the two cases. For the variational discretization, we also analyze the convergence of adaptive finite element methods. In the end, some examples are provided to validate our analysis.