New Assumptions for Stability Analysis in Elliptic Optimal Control Problems
Abstract.
This paper is dedicated to the stability analysis of the optimal solutions of a control problem associated with a semilinear elliptic equation. The linear differential operator of the equation is neither monotone nor coercive due to the presence of a convection term. The control appears only linearly, or may not even appear explicitly in the objective functional. Under new assumptions, we prove Lipschitz stability of the optimal controls and associated states with respect to not only perturbations in the equation and the objective functional but also the Tikhonov regularization parameter.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
T. Bayen, F. Bonnans, and F. Silva, Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations, Trans. Amer. Math. Soc., 366 (2014), pp. 2063–2087.
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), pp. 819–844, https://doi.org/10.1137/141000415.
3.
E. Casas and M. Mateos, Critical cones for sufficient second order conditions in PDE constrained optimization, SIAM J. Optim., 30 (2020), pp. 585–603, https://doi.org/10.1137/19M1258244.
4.
E. Casas and M. Mateos, State error estimates for the numerical approximation of sparse distributed control problems in the absence of Tikhonov regularization, Vietnam J. Math., 49 (2021), pp. 713–738.
5.
E. Casas, M. Mateos, and A. Rösch, Error estimates for semilinear parabolic control problems in the absence of Tikhonov term, SIAM J. Control Optim., 57 (2019), pp. 2515–2540, https://doi.org/10.1137/18M117220X.
6.
E. Casas, M. Mateos, and A. Rösch, Analysis of control problems of nonmonotone semilinear elliptic equations, ESAIM Control Optim. Calc. Var., 26 (2020), 80.
7.
E. Casas, C. Ryll, and F. Tröltzsch, Second order and stability analysis for optimal sparse control of the FitzHugh–Nagumo equation, SIAM J. Control Optim., 53 (2015), pp. 2168–2202, https://doi.org/10.1137/140978855.
8.
E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: Improving results expected from abstract theory, SIAM J. Optim., 22 (2012), pp. 261–279, https://doi.org/10.1137/110840406.
9.
A. D. Corella, N. Jork, and V. Veliov, Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations, ESAIM Control Optim. Calc. Var., 28 (2022), 79.
10.
A. Dontchev, W. Hager, A. Poore, and B. Yang, Optimality, stability, and convergence in nonlinear control, Appl. Math. Optim., 31 (1995), pp. 297–326.
11.
J. Dunn, Second-order optimality conditions in sets of \({\rm L}^\infty\) functions with range in a polyhedron, SIAM J. Control Optim., 33 (1995), pp. 1603–1635, https://doi.org/10.1137/S036301299224031X.
12.
J. Dunn, On second order sufficient conditions for structured nonlinear programs in infinite-dimensional function spaces, in Mathematical Programming with Data Perturbations, A. Fiacco, ed., Marcel Dekker, New York, 1998, pp. 83–107.
13.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, 1983.
14.
R. Griesse, Lipschitz stability of solutions to some state-constrained elliptic optimal control problems, Z Anal. Anwend., 25 (2006), pp. 435–455.
15.
M. Hinze and C. Meyer, Stability of semilinear elliptic optimal control problems with pointwise state constraints, Comput. Optim. Appl., 52 (2012), pp. 87–114.
16.
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), pp. 849–876.
17.
K. Malanowski and F. Tröltzsch, Lipschitz stability of solutions to parametric optimal control for elliptic equations, Control Cybernet., 29 (2000), pp. 237–256.
18.
H. Maurer, First and second-order sufficient optimality conditions in mathematical programming and optimal control, Math. Program. Study, 14 (1981), pp. 163–177.
19.
H. Maurer and J. Zowe, First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems, Math. Program., 16 (1979), pp. 98–110.
20.
F. Pörner and D. Wachsmuth, Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations, Math. Control Relat. Fields, 8 (2018), pp. 315–335.
21.
N. T. Qui and D. Wachsmuth, Stability for bang-bang control problems of partial differential equations, Optimization, 67 (2018), pp. 2157–2177.
22.
N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 27 (1973), pp. 265–308.
23.
N. von Daniels, Tikhonov regularization of control-constrained optimal control problems, Comput. Optim. Appl., 70 (2018), pp. 295–320.
24.
D. Wachsmuth and G. Wachsmuth, Convergence and regularisation results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 858–886.
25.
D. Wachsmuth and G. Wachsmuth, Regularization, error estimates and discrepancy principle for optimal control problems with inequality constraints, Control Cybernet., 40 (2011), pp. 1125–1158.
Information & Authors
Information
Published In
SIAM Journal on Control and Optimization
Pages: 1394 - 1414
DOI: 10.1137/22M149199X
ISSN (online): 1095-7138
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 21 April 2022
Accepted: 20 December 2022
Published online: 12 June 2023
Keywords
MSC codes
Authors
Funding Information
Ministerio de Ciencia e Innovacion (MCIN): PID2020-114837GB-I00
Austrian Science Fund (FWF): I4571
Funding: The first author was supported by MCIN/AEI/10.13039/501100011033/ under research project PID2020-114837GB-I00. The second and third authors were supported by the Austrian Science Foundation (FWF) under grant I4571.
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- A Posteriori Error Estimates for a Bang–Bang Optimal Control ProblemApplied Mathematics & Optimization, Vol. 91, No. 3 | 7 June 2025
- Bang–bang optimal control problems for the viscous Camassa–Holm equationsOptimization, Vol. 39 | 23 April 2025
- Stability and Genericity of Bang-Bang Controls in Affine ProblemsSIAM Journal on Control and Optimization, Vol. 62, No. 3 | 11 June 2024
- Stability for Parametric Control Problems of PDEs via Generalized DifferentiationSet-Valued and Variational Analysis, Vol. 32, No. 2 | 3 May 2024
- Stability analysis of the Navier–Stokes velocity tracking problem with bang-bang controlsJournal of Optimization Theory and Applications, Vol. 201, No. 2 | 7 April 2024
- Finite element error analysis of affine optimal control problemsESAIM: Control, Optimisation and Calculus of Variations, Vol. 30 | 28 August 2024
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].