Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations∗
1 Departamento de Matemática Aplicada y
Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad
de Cantabria, Av. Los Castros
2 Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Methods (Partial Differential Equations), 09107 Chemnitz, Germany.
Revised: 24 June 2015
Accepted: 10 September 2015
Optimal control problems with semilinear parabolic state equations are considered. The objective features one out of three different terms promoting various spatio-temporal sparsity patterns of the control variable. For each problem, first-order necessary optimality conditions, as well as second-order necessary and sufficient optimality conditions are proved. The analysis includes the case in which the objective does not contain the squared norm of the control.
Mathematics Subject Classification: 49K20 / 49J52 / 35K58 / 65K10
Key words: Optimal control / directional sparsity / second-order optimality conditions / semilinear parabolic equations
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