Stability in Affine Optimal Control Problems Constrained by Semilinear Elliptic Partial Differential Equations

Domínguez Corella, Alberto; Jork, Nicolai; Veliov, Vladimir

doi:10.1051/cocv/2022075

Domínguez Corella, Alberto ; Jork, Nicolai ; Veliov, Vladimir

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 79

Résumé

This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.

MR Zbl

DOI : 10.1051/cocv/2022075

Classification : 35J60, 49J20, 49K20, 49K40
Keywords: Semilinear elliptic equations, stability analysis, metric subregularity, optimality mapping, optimality conditions, Tikhonov regularization

@article{COCV_2022__28_1_A79_0,
     author = {Dom{\'\i}nguez Corella, Alberto and Jork, Nicolai and Veliov, Vladimir},
     title = {Stability in {Affine} {Optimal} {Control} {Problems} {Constrained} by {Semilinear} {Elliptic} {Partial} {Differential} {Equations}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     year = {2022},
     publisher = {EDP-Sciences},
     volume = {28},
     doi = {10.1051/cocv/2022075},
     mrnumber = {4525177},
     zbl = {1506.35080},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2022075/}
}

TY  - JOUR
AU  - Domínguez Corella, Alberto
AU  - Jork, Nicolai
AU  - Veliov, Vladimir
TI  - Stability in Affine Optimal Control Problems Constrained by Semilinear Elliptic Partial Differential Equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2022
VL  - 28
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv/2022075/
DO  - 10.1051/cocv/2022075
LA  - en
ID  - COCV_2022__28_1_A79_0
ER  -

%0 Journal Article
%A Domínguez Corella, Alberto
%A Jork, Nicolai
%A Veliov, Vladimir
%T Stability in Affine Optimal Control Problems Constrained by Semilinear Elliptic Partial Differential Equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2022
%V 28
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv/2022075/
%R 10.1051/cocv/2022075
%G en
%F COCV_2022__28_1_A79_0

Domínguez Corella, Alberto; Jork, Nicolai; Veliov, Vladimir. Stability in Affine Optimal Control Problems Constrained by Semilinear Elliptic Partial Differential Equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 79. doi: 10.1051/cocv/2022075

Bibliographie
Cité par

[1] W. Alt, C. Schneider and M. Seydenschwanz, Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions. Appi. Math. Comput. 287/288 (2016) 104-124. | MR | Zbl | DOI

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

[3] R. S. Burachik and V. Jeyakumar, A simple closure condition for the normal cone intersection formula. Proc. Am. Math. Soc. 133 (2005) 1741-1748. | MR | Zbl | DOI

[4] E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31 (1993) 993-1006. | MR | Zbl | DOI

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

[6] E. Casas and K. Chrysafìnos, Error estimates for the approximation of the velocity tracking problem with bang-bang controls. ESAIM: COCV 23 (2017) 1267-1291. | MR | Zbl | Numdam

[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. | MR | Zbl | DOI

[8] E. Casas, M. Mateos and A. Rösch, Analysis of control problems of nonmontone semilinear elliptic equations. ESAIM: COCV 26 (2020) Paper No. 80, 21. | MR | Zbl | Numdam

[9] E. Casas and F. Tröltzsch, On optimal control problems with controls appearing nonlinearly in an elliptic state equation. SIAM J. Control Optim. 58 (2020) 1961-1983. | MR | Zbl | DOI

[10] E. Casas, D. Wachsmuth and G. Wachsmuth, Sufficient second-order conditions for bang-bang control problems. SIAM J. Control Optim. 55 (2017) 3066-3090. | MR | Zbl | DOI

[11] E. Casas, D. Wachsmuth and G. Wachsmuth, Second-order analysis and numerical approximation for bang-bang bilinear control problems. SIAM J. Control Optim. 56 (2018) 4203-4227. | MR | Zbl | DOI

[12] R. Cibulka, A. L. Dontchev and A. Y. Kruger, Strong metric subregularity of mappings in variational analysis and optimization. J. Math. Anal. Appl. 457 (2018) 1247-1282. | MR | Zbl | DOI

[13] K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls. Comput. Optim. Appl. 51 (2012) 931-939. | MR | Zbl | DOI

[14] A. Dominguez Corella, M. Quincampoix and V. M. Veliov, Strong bi-metric regularity in affine optimal control problems. Pure Appl. Funct. Anal. 6 (2021) 1119-1137. | MR | Zbl

[15] A. Dominguez Corella and V. M. Veliov, Hölder regularity in bang-bang type affine optimal control problems, in Large-scale scientific computing, volume 13127 of Lecture Notes in Comput. Sci.. Springer, Cham (2022), pp. 306-313. | MR | Zbl | DOI

[16] A. L. Dontchev, I. V. Kolmanovsky, M. I. Krastanov, V. M. Veliov and P. T. Vuong, Approximating optimal finite horizon feedback by model predictive control. Syst. Control Lett. 139 (2020) 104666. | MR | Zbl | DOI

[17] A. L. Dontchev and R. T. Rockafellar, Implicit functions and solution mappings. Springer Monographs in Mathematics. Springer, Dordrecht (2009) A view from variational analysis. | MR | Zbl

[18] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control. Springer-Verlag, Berlin-New York (1975). | MR | Zbl | DOI

[19] R. Griesse, Lipschitz stability of solutions to some state-constrained elliptic optimal control problems. Z. Anal. Anwend. 25 (2006) 435-455. | MR | Zbl | DOI

[20] P. Grisvard, Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985). | MR | Zbl

[21] R. Haller-Dintelmann, C. Meyer, J. Rehberg and A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60 (2009) 397-428. | MR | Zbl | DOI

[22] M. Hinze and C. Meyer, Stability of semilinear elliptic optimal control problems with pointwise state constraints. Comput. Optim. Appl. 52 (2012) 87-114. | MR | Zbl | DOI

[23] B. T. Kien, N. Q. Tuan, C.-F. Wen and J.-C. Yao. $L^{\infty}$ -stability of a parametric optimal control problem governed by semilinear elliptic equations. Appl. Math. Optim. 84 (2021) 849-876. | MR | Zbl | DOI

[24] D. Luenberger, Optimization by Vector Space Methods. Wiley-Interscience (1969). | MR | Zbl

[25] K. Malanowski and F. Tröoltzsch, Lipschitz stability of solutions to parametric optimal control for elliptic equations. Control Cybernet. 29 (2000) 237-256. | MR | Zbl

[26] B. S. Mordukhovich and T. T. A. Nghia, Full Lipschitzian and Höolderian stability in optimization with applications to mathematical programming and optimal control. SIAM J. Optim. 24 (2014) 1344-1381. | MR | Zbl | DOI

[27] J. R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1975). | MR | Zbl

[28] R. Nittka, Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differ. Equ. 251 (2011) 860-880. | MR | Zbl | DOI

[29] N. P. Osmolovskii and V. M. Veliov, Metric sub-regularity in optimal control of affine problems with free end state. ESAIM: COCV 26 (2020) Paper No. 47, 19. | MR | Zbl | Numdam

[30] N. P. Osmolovskii and V. M. Veliov, On the regularity of Mayer-type affine optimal control problems, In Large-scale scientific computing, volume 11958 of Lecture Notes in Comput. Sci.. Springer, Cham (2020), pp. 56-63. | MR | Zbl | DOI

[31] F. Pöorner and D. Wachsmuth, An iterative Bregman regularization method for optimal control problems with inequality constraints. Optimization 65 (2016) 2195-2215. | MR | Zbl | DOI

[32] F. Pöorner and D. Wachsmuth, Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Math. Control Relat. Fields 8 (2018) 315-335. | MR | Zbl | DOI

[33] F. Pöorner and D. Wachsmuth, Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Math. Control Relat. Fields 8 (2018) 315-335. | MR | Zbl | DOI

[34] J. Preininger, T. Scarinci and V. M. Veliov, On the regularity of linear-quadratic optimal control problems with bang-bang solutions, In Large-scale scientific computing, volume 10665 of Lecture Notes in Comput. Sci.. Springer, Cham (2018), pp. 237-245. | MR | Zbl | DOI

[35] N. T. Qui and D. Wachsmuth, Stability for bang-bang control problems of partial differential equations. Optimization 67 (2018) 2157-2177. | MR | Zbl | DOI

[36] N. T. Qui and D. Wachsmuth, Full stability for a class of control problems of semilinear elliptic partial differential equations. SIAM J. Control Optim. 57 (2019) 3021-3045. | MR | Zbl | DOI

[37] S. M. Robinson, Generalized equations and their solutions. I. Basic theory. Math. Programming Stud., (10) (1979) 128-141. Point-to-set maps and mathematical programming. | MR | Zbl | DOI

[38] M. Seydenschwanz, Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions. Comput. Optim. Appl. 61 (2015) 731-760. | MR | Zbl | DOI

[39] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | MR | Zbl | Numdam | DOI

[40] F. Tröltzsch, Optimal control of partial differential equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). Theory, methods and applications, Translated from the 2005 German original by JUrgen Sprekels. | MR | Zbl

[41] A. Visintin, Strong convergence results related to strict convexity. Comm. Partial Differ. Equ. 9 (1984) 439-466. | MR | Zbl | DOI

[42] N. Von Daniels, Tikhonov regularization of control-constrained optimal control problems. Comput. Optim. Appl. 70 (2018) 295-320. | MR | Zbl | DOI

[43] D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints. Control Cybernet. 40 (2011) 1125-1158. | MR | Zbl

[44] G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: COCV 17 (2011) 858-886. | MR | Zbl | Numdam

Cité par Sources :

This research was supported by the Austrian Science Foundation (FWF) under grant I 4571-N.