# American Institute of Mathematical Sciences

September  2020, 10(3): 527-546. doi: 10.3934/mcrf.2020009

## State-constrained semilinear elliptic optimization problems with unrestricted sparse controls

 1 Departamento de Matemática Aplicada y Ciencias de la Computació, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain 2 Institut für Mathematik, Technische Universität Berlin, D-10623 Berlin, Germany

* Corresponding author: Eduardo Casas

Dedicated to Prof. Dr. Fréderic Bonnans on the occasion of his 60th birthday

Received  February 2018 Revised  December 2018 Published  September 2020 Early access  December 2019

Fund Project: The first author was partially supported by Spanish Ministerio de Economía, Industria y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P

In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.

Citation: Eduardo Casas, Fredi Tröltzsch. State-constrained semilinear elliptic optimization problems with unrestricted sparse controls. Mathematical Control and Related Fields, 2020, 10 (3) : 527-546. doi: 10.3934/mcrf.2020009
##### References:
 [1] T. Bayen, J. F. Bonnans and F. J. Silva, Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations, Trans. Amer. Math. Soc., 366 (2014), 2063-2087.  doi: 10.1090/S0002-9947-2013-05961-2. [2] T. Bayen and F. J. Silva, Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM J. Control Optim., 54 (2016), 819-844.  doi: 10.1137/141000415. [3] J. F. Bonnans and E. Casas, Contrôle de systèmes elliptiques semilinéaires comportant des contraintes sur l'état, in Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 166 (1988), 69–86. [4] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9. [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [6] E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM J. Control Optim., 24 (1986), 1309-1318.  doi: 10.1137/0324078. [7] E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643.  doi: 10.1137/07068240X. [8] 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.  doi: 10.1137/110834366. [9] E. Casas and M. Mateos, Optimal control of partial differential equations, Computational Mathematics, Numerical Analysis and Applications, SEMA SIMAI Springer Ser., Springer, Cham, 13 (2017), 3-59. [10] E. Casas, M. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM Control Optim. Calc. Var., 20 (2014), 803-822.  doi: 10.1051/cocv/2013084. [11] E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 1010-1033.  doi: 10.1137/130917314. [12] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [13] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808. [14] K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal., 45 (2007), 1937-1953.  doi: 10.1137/060652361. [15] M. Degiovanni and M. Scaglia, A variational approach to semilinear elliptic equations with measure data, Discrete Contin. Dyn. Syst., 31 (2011), 1233-1248.  doi: 10.3934/dcds.2011.31.1233. [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [17] C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control Cybernet., 37 (2008), 51-83. [18] K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM J. Control Optim., 51 (2013), 2788-2808.  doi: 10.1137/120889137. [19] J.-C. Saut and B. Scheurer, Sur l'unicité du problème de Cauchy et le prolongement unique pour des équations elliptiques à coefficients non localement bornés, J. Differential Equations, 43 (1982), 28-43.  doi: 10.1016/0022-0396(82)90072-9. [20] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.  doi: 10.5802/aif.204. [21] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

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##### References:
 [1] T. Bayen, J. F. Bonnans and F. J. Silva, Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations, Trans. Amer. Math. Soc., 366 (2014), 2063-2087.  doi: 10.1090/S0002-9947-2013-05961-2. [2] T. Bayen and F. J. Silva, Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM J. Control Optim., 54 (2016), 819-844.  doi: 10.1137/141000415. [3] J. F. Bonnans and E. Casas, Contrôle de systèmes elliptiques semilinéaires comportant des contraintes sur l'état, in Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 166 (1988), 69–86. [4] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9. [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. [6] E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM J. Control Optim., 24 (1986), 1309-1318.  doi: 10.1137/0324078. [7] E. Casas, J. C. de los Reyes and F. Tröltzsch, Sufficient second order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643.  doi: 10.1137/07068240X. [8] 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.  doi: 10.1137/110834366. [9] E. Casas and M. Mateos, Optimal control of partial differential equations, Computational Mathematics, Numerical Analysis and Applications, SEMA SIMAI Springer Ser., Springer, Cham, 13 (2017), 3-59. [10] E. Casas, M. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM Control Optim. Calc. Var., 20 (2014), 803-822.  doi: 10.1051/cocv/2013084. [11] E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 1010-1033.  doi: 10.1137/130917314. [12] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [13] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808. [14] K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM J. Numer. Anal., 45 (2007), 1937-1953.  doi: 10.1137/060652361. [15] M. Degiovanni and M. Scaglia, A variational approach to semilinear elliptic equations with measure data, Discrete Contin. Dyn. Syst., 31 (2011), 1233-1248.  doi: 10.3934/dcds.2011.31.1233. [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [17] C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control Cybernet., 37 (2008), 51-83. [18] K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM J. Control Optim., 51 (2013), 2788-2808.  doi: 10.1137/120889137. [19] J.-C. Saut and B. Scheurer, Sur l'unicité du problème de Cauchy et le prolongement unique pour des équations elliptiques à coefficients non localement bornés, J. Differential Equations, 43 (1982), 28-43.  doi: 10.1016/0022-0396(82)90072-9. [20] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258.  doi: 10.5802/aif.204. [21] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.
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