  This paper is concerned with first- and second-order optimality conditions as
well as the stability for non-smooth semilinear optimal control problems
involving the $L^1$-norm of the control in the cost functional.
  In addition to the appearance of the $L^1$-norm leading to the
non-differentiability of the objective and promoting the sparsity of the
optimal controls, the non-smoothness of the nonlinear coefficient in the state
equation causes the same property of the control-to-state operator. Exploiting
a regularization scheme, we derive $C$-stationarity conditions for any local
optimal control. Under a structural assumption on the associated state, we
define the curvature functional for the part not including the $L^1$-norm of
controls of the objective for which the second-order necessary and sufficient
optimality conditions are shown. Furthermore, under a more restrictive
structural assumption imposed on the mentioned state, an explicit formulation
of the curvature is established and thus the explicit second-order optimality
conditions are stated. Finally, the Lipschitz stability of local solutions with
respect to the sparsity parameter is shown.
