no. 1
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

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},
     year = {2017},
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     url = {https://www.numdam.org/articles/10.1051/cocv/2015048/}
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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

J.F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer-Verlag, New York (2000). | MR | Zbl

E. Casas, Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35 (1997) 1297–1327. | MR | Zbl | DOI

E. Casas, Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50 (2012) 2355–2372. | MR | Zbl | DOI

E. Casas and K. Kunisch, Optimal Control of Semilinear Elliptic Equations in Measure Spaces. SIAM J. Control Optim. 52 (2014) 339–364. | MR | Zbl | DOI

E. Casas, R. Herzog, and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with L 1 cost functional. SIAM J. Optim. 22 (2012) 795–820. | MR | Zbl | DOI

E. Casas, C. Clason, and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51 (2013) 28–63. | MR | Zbl | DOI

J. Dunn, On second order sufficient optimality conditions for structured nonlinear programs in infinite-dimensional function spaces, in Mathematical Programming with Data Perturbations, edited by A. Fiacco. Marcel Dekker (1998) 83–107. | MR | Zbl

R.E. Edwards, Functional analysis. Theory and applications, Corrected reprint of the 1965 original. Dover Publications Inc., New York (1995). | MR

I. Ekeland and R. Temam, Convex Analysis and Variational Problems. In vol. 28 of Classics in Applied Mathematics. SIAM, Philadelphia (1999). | MR | Zbl

R. Herzog, J. Obermeier, and G. Wachsmuth, Annular and sectorial sparsity in optimal control of elliptic equations. Comput. Optim. Appl. 62 (2015) 157–180. | MR | Zbl | DOI

R. Herzog, G. Stadler, and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim. 50 (2012) 943–963. | MR | Zbl | DOI

A.D. Ioffe and V.M. Tichomirov, Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979). | MR | Zbl

K. Kunisch, K. Pieper, and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems. SIAM J. Control Optim. 52 (2014) 3078–3108. | MR | Zbl | DOI

J. Nečas, Les Méthodes Directes en Théorie des Equations Elliptiques. Editeurs Academia, Prague (1967). | MR | Zbl

G. Stadler, Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159–181. | MR | Zbl | DOI

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