Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. OPTIM. c  2012 Society for Industrial and Applied Mathematics Vol. 22, No. 3, pp. 795–820 OPTIMALITY CONDITIONS AND ERROR ANALYSIS OF SEMILINEAR ELLIPTIC CONTROL PROBLEMS WITH L 1 COST FUNCTIONAL * EDUARDO CASAS † , ROLAND HERZOG ‡ , AND GERD WACHSMUTH ‡ Abstract. Semilinear elliptic optimal control problems involving the L 1 norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piece- wise linear discretizations of the state are shown. Error estimates for the variational discretization of the problem in the sense of [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45–61] are also obtained. Numerical experiments confirm the convergence rates. Key words. optimal control of partial differential equations, nondifferentiable objective, sparse controls, finite element discretization, a priori error estimates AMS subject classifications. 35J61, 49K20, 49M25 DOI. 10.1137/110834366 1. Introduction. In this paper we consider an optimal control problem subject to a semilinear elliptic state equation. The objective functional contains the L 1 norm of the control and it is therefore nondifferentiable. Problems of this type are of interest for two reasons. First, the L 1 norm of the control is often a natural measure of the control cost. Second, this term leads to sparsely supported optimal controls, which are desirable, for instance, in actuator placement problems [17]. In optimal control of distributed parameter systems, it may be impossible or undesirable to put the controllers at every point of the domain. Instead, we can decide to control the system by localizing the controls in small regions. The big issue is to determine the most effective location of the controls. An answer to this question is given by solving the control problem with an L 1 norm of the control. However, the nondifferentiability of the objective leads to some difficulties. While first-order necessary optimality conditions can be derived in a standard way via Clarke’s calculus of generalized derivatives, second-order conditions require additional effort. From the first-order optimality conditions, we deduce a representation formula (see (3.5c)) for the subdifferential ¯ λ of the nondifferentiable term at the optimal control ¯ u, i.e., ¯ λ ∈ ∂ ‖ ¯ u‖ L 1 (Ω) . This formula is new and it has some important consequences. First, it proves the uniqueness of ¯ λ, which is not usually obtained for a nondifferen- tiable optimization problem. Second, it proves that ¯ λ is not only an L ∞ (Ω) function, but it is a Lipschitz function in ¯ Ω, which implies, with formula (3.5a) for the optimal control, that ¯ u is also Lipschitz in ¯ Ω. This extra regularity for the optimal control is essential in deriving the error estimates. We should emphasize that there are no error estimates if we do not have extra regularity of the optimal control. Moreover, ∗ Received by the editors May 17, 2011; accepted for publication (in revised form) March 16, 2012; published electronically July 12, 2012. http://www.siam.org/journals/siopt/22-3/83436.html † Departmento de Matem´atica Aplicada y Ciencias de la Computaci´ on, E.T.S.I. Industriales y de Telecomunicaci´ on, Universidad de Cantabria, 39005 Santander, Spain (eduardo.casas@unican.es). This author was partially supported by the Spanish Ministerio de Econom´ ıa y Competitividad under the project MTM2011-22711. ‡ Faculty of Mathematics, Chemnitz University of Technology, 09126 Chemnitz, Germany (roland.herzog@mathematik.tu-chemnitz.de, gerd.wachsmuth@mathematik.tu-chemnitz.de). 795 Downloaded 08/20/13 to 193.144.185.39. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php