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This paper is concerned with first- and second-order optimality conditions aswell as the stability for non-smooth semilinear optimal control problemsinvolving the $L^1$-norm of the control in the cost functional.  In addition to the appearance of the $L^1$-norm leading to thenon-differentiability of the objective and promoting the sparsity of theoptimal controls, the non-smoothness of the nonlinear coefficient in the stateequation causes the same property of the control-to-state operator. Exploitinga regularization scheme, we derive $C$-stationarity conditions for any localoptimal control. Under a structural assumption on the associated state, wedefine the curvature functional for the part not including the $L^1$-norm ofcontrols of the objective for which the second-order necessary and sufficientoptimality conditions are shown. Furthermore, under a more restrictivestructural assumption imposed on the mentioned state, an explicit formulationof the curvature is established and thus the explicit second-order optimalityconditions are stated. Finally, the Lipschitz stability of local solutions withrespect to the sparsity parameter is shown.

Optimality conditions and Lipschitz stability for non-smooth semilinear elliptic optimal control problems with sparse controls