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
Mots-clés : 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. 
@article{COCV_2017__23_1_263_0,
     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},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {1},
     year = {2017},
     doi = {10.1051/cocv/2015048},
     mrnumber = {3601024},
     zbl = {1479.49047},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2015048/}
}
TY  - JOUR
AU  - Casas, Eduardo
AU  - Herzog, Roland
AU  - Wachsmuth, Gerd
TI  - Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2017
SP  - 263
EP  - 295
VL  - 23
IS  - 1
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv/2015048/
DO  - 10.1051/cocv/2015048
LA  - en
ID  - COCV_2017__23_1_263_0
ER  - 
%0 Journal Article
%A Casas, Eduardo
%A Herzog, Roland
%A Wachsmuth, Gerd
%T Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2017
%P 263-295
%V 23
%N 1
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv/2015048/
%R 10.1051/cocv/2015048
%G en
%F COCV_2017__23_1_263_0
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. https://www.numdam.org/articles/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. | DOI | MR | Zbl

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

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

E. Casas, R. Herzog, and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional. SIAM J. Optim. 22 (2012) 795–820. | DOI | MR | Zbl

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

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. | DOI | MR | Zbl

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

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. | DOI | MR | Zbl

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

G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159–181. | DOI | MR | Zbl

  • Robert, Maria; Nadupuri, Suresh Kumar; Chamakuri, Nagaiah Analysis and simulation of sparse optimal control of the monodomain model, Computers Mathematics with Applications, Volume 184 (2025), p. 29 | DOI:10.1016/j.camwa.2025.02.008
  • Casas, Eduardo; Kunisch, Karl Temporally sparse controls for infinite horizon semilinear parabolic equations with norm constraints, Control and Cybernetics, Volume 53 (2025) no. 1, p. 11 | DOI:10.2478/candc-2024-003
  • Casas, Eduardo; Kunisch, Karl Temporally sparse controls for infinite horizon semilinear parabolic equations with norm constraints, Control and Cybernetics, Volume 53 (2025) no. 1, p. 11 | DOI:10.2478/candc-2024-0003
  • Gong, Wei; Liang, Dongdong Analysis and approximation to parabolic optimal control problems with measure-valued controls in time, ESAIM: Control, Optimisation and Calculus of Variations, Volume 31 (2025), p. 2 | DOI:10.1051/cocv/2024085
  • Hou, Li-Feng; Li, Li; Chang, Lili; Wang, Zhen; Sun, Gui-Quan Pattern dynamics of vegetation based on optimal control theory, Nonlinear Dynamics, Volume 113 (2025) no. 1, p. 1 | DOI:10.1007/s11071-024-10241-6
  • Colli, Pierluigi; Sprekels, Jürgen; Tröltzsch, Fredi Optimality Conditions for Sparse Optimal Control of Viscous Cahn–Hilliard Systems with Logarithmic Potential, Applied Mathematics Optimization, Volume 90 (2024) no. 2 | DOI:10.1007/s00245-024-10187-6
  • Sprekels, Jürgen; Tröltzsch, Fredi Second-Order Sufficient Conditions in the Sparse Optimal Control of a Phase Field Tumor Growth Model with Logarithmic Potential, ESAIM: Control, Optimisation and Calculus of Variations, Volume 30 (2024), p. 13 | DOI:10.1051/cocv/2023084
  • Hernández, Erwin; Merino, Pedro Sparse optimal control of Timoshenko's beam using a locking‐free finite element approximation, Optimal Control Applications and Methods, Volume 45 (2024) no. 3, p. 1007 | DOI:10.1002/oca.3085
  • Casas, Eduardo; Kunisch, Karl First- and second-order optimality conditions for the control of infinite horizon Navier–Stokes equations, Optimization (2024), p. 1 | DOI:10.1080/02331934.2024.2358406
  • Salfenmoser, Lena; Obermayer, Klaus Optimal control of a Wilson–Cowan model of neural population dynamics, Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 33 (2023) no. 4 | DOI:10.1063/5.0144682
  • Casas, Eduardo; Mateos, Mariano Error Estimates for the Numerical Approximation of Unregularized Sparse Parabolic Control Problems, Computational Methods in Applied Mathematics, Volume 23 (2023) no. 4, p. 877 | DOI:10.1515/cmam-2022-0130
  • Casas, Eduardo; Kunisch, Karl; Mateos, Mariano Error estimates for the numerical approximation of optimal control problems with nonsmooth pointwise-integral control constraints, IMA Journal of Numerical Analysis, Volume 43 (2023) no. 3, p. 1485 | DOI:10.1093/imanum/drac027
  • Chamakuri, Nagaiah; Bendahmane, Mostafa; J., Manimaran Optimal sparse boundary control of cardiac defibrillation, Nonlinear Analysis: Real World Applications, Volume 74 (2023), p. 103945 | DOI:10.1016/j.nonrwa.2023.103945
  • Chrysafinos, Konstantinos; Plaka, Dimitra Analysis and approximations of an optimal control problem for the Allen–Cahn equation, Numerische Mathematik, Volume 155 (2023) no. 1-2, p. 35 | DOI:10.1007/s00211-023-01374-8
  • Casas, Eduardo; Wachsmuth, Daniel A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations, SIAM Journal on Control and Optimization, Volume 61 (2023) no. 3, p. 1095 | DOI:10.1137/22m1486418
  • Casas, Eduardo; Kunisch, Karl Infinite Horizon Optimal Control Problems with Discount Factor on the State, Part II: Analysis of the Control Problem, SIAM Journal on Control and Optimization, Volume 61 (2023) no. 3, p. 1438 | DOI:10.1137/22m1490296
  • Casas, Eduardo; Kunisch, Karl, 2022 American Control Conference (ACC) (2022), p. 284 | DOI:10.23919/acc53348.2022.9867749
  • Casas, Eduardo; Kunisch, Karl Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space, Applied Mathematics Optimization, Volume 85 (2022) no. 1 | DOI:10.1007/s00245-022-09850-7
  • Colli, Pierluigi; Signori, Andrea; Sprekels, Jürgen Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities, Journal of Optimization Theory and Applications, Volume 194 (2022) no. 1, p. 25 | DOI:10.1007/s10957-022-02000-7
  • Chang, Lili; Gong, Wei; Jin, Zhen; Sun, Gui-Quan Sparse Optimal Control of Pattern Formations for an SIR Reaction-Diffusion Epidemic Model, SIAM Journal on Applied Mathematics, Volume 82 (2022) no. 5, p. 1764 | DOI:10.1137/22m1472127
  • Casas, Eduardo; Kunisch, Karl Infinite Horizon Optimal Control Problems for a Class of Semilinear Parabolic Equations, SIAM Journal on Control and Optimization, Volume 60 (2022) no. 4, p. 2070 | DOI:10.1137/21m1464816
  • Chouzouris, Teresa; Roth, Nicolas; Cakan, Caglar; Obermayer, Klaus Applications of optimal nonlinear control to a whole-brain network of FitzHugh-Nagumo oscillators, Physical Review E, Volume 104 (2021) no. 2 | DOI:10.1103/physreve.104.024213
  • Casas, Eduardo; Kunisch, Karl Optimal Control of the Two-Dimensional Evolutionary Navier–Stokes Equations with Measure Valued Controls, SIAM Journal on Control and Optimization, Volume 59 (2021) no. 3, p. 2223 | DOI:10.1137/20m1351400
  • Allendes, A; Fuica, F; Otárola, E Adaptive finite element methods for sparse PDE-constrained optimization, IMA Journal of Numerical Analysis, Volume 40 (2020) no. 3, p. 2106 | DOI:10.1093/imanum/drz025
  • Kunisch, Karl; Meinlschmidt, Hannes Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints, Journal de Mathématiques Pures et Appliquées, Volume 138 (2020), p. 46 | DOI:10.1016/j.matpur.2020.03.006
  • Casas, Eduardo; Wachsmuth, Daniel First and Second Order Conditions for Optimal Control Problems with an L0 Term in the Cost Functional, SIAM Journal on Control and Optimization, Volume 58 (2020) no. 6, p. 3486 | DOI:10.1137/20m1318377
  • Casas, Eduardo; Mateos, Mariano Critical Cones for Sufficient Second Order Conditions in PDE Constrained Optimization, SIAM Journal on Optimization, Volume 30 (2020) no. 1, p. 585 | DOI:10.1137/19m1258244
  • Fuica, Francisco; Otárola, Enrique; Salgado, Abner J. An a posteriori error analysis of an elliptic optimal control problem in measure space, Computers Mathematics with Applications, Volume 77 (2019) no. 10, p. 2659 | DOI:10.1016/j.camwa.2018.12.043
  • Azmi, Behzad; Kunisch, Karl A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations, SIAM Journal on Control and Optimization, Volume 57 (2019) no. 5, p. 3496 | DOI:10.1137/19m1239787
  • Casas, Eduardo; Mateos, Mariano; Rösch, Arnd Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Computational Optimization and Applications, Volume 70 (2018) no. 1, p. 239 | DOI:10.1007/s10589-018-9979-0
  • Casas, Eduardo; Mateos, Mariano; Rösch, Arnd Finite element approximation of sparse parabolic control problems, Mathematical Control Related Fields, Volume 7 (2017) no. 3, p. 393 | DOI:10.3934/mcrf.2017014
  • Casas, Eduardo; Kunisch, Karl Stabilization by Sparse Controls for a Class of Semilinear Parabolic Equations, SIAM Journal on Control and Optimization, Volume 55 (2017) no. 1, p. 512 | DOI:10.1137/16m1084298
  • Casas, Eduardo A review on sparse solutions in optimal control of partial differential equations, SeMA Journal, Volume 74 (2017) no. 3, p. 319 | DOI:10.1007/s40324-017-0121-5

Cité par 33 documents. Sources : Crossref