Geodesic
Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations
Casas, Eduardo1  ; Herzog, Roland2 ; Wachsmuth, Gerd2
1 Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, Av. Los Castros s/n, 39005 Santander, Spain.
2 Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Methods (Partial Differential Equations), 09107 Chemnitz, Germany.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 263-295.

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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.

DOI : 10.1051/cocv/2015048
Classification : 49K20, 49J52, 35K58, 65K10
Keywords: Optimal control, directional sparsity, second-order optimality conditions, semilinear parabolic equations

Casas, Eduardo 1 ; Herzog, Roland 2 ; Wachsmuth, Gerd 2

1 Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, Av. Los Castros s/n, 39005 Santander, Spain.
2 Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Methods (Partial Differential Equations), 09107 Chemnitz, Germany. 
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     author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd},
     title = {Analysis of {Spatio-Temporally} {Sparse} {Optimal} {Control} {Problems} of {Semilinear} {Parabolic} {Equations}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {263--295},
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Casas, Eduardo; Herzog, Roland; Wachsmuth, Gerd. Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 263-295. doi : 10.1051/cocv/2015048. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2015048/

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