We study first-order necessary optimality conditions and finite element error estimates for a class of distributed parabolic optimal control problems with pointwise state constraints. It is demonstrated that, if the bound in the state constraint and the differential operator in the governing PDE fulfil a certain compatibility assumption, then locally optimal controls satisfy a stationarity system that allows to significantly improve known regularity results for adjoint states and Lagrange multipliers in the parabolic setting. In contrast to classical approaches to first-order necessary optimality conditions for state-constrained problems, the main arguments of our analysis require neither a Slater point, nor uniform control constraints, nor differentiability of the objective function, nor a restriction of the spatial dimension. As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0) − cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation. The paper concludes with numerical experiments that confirm our theoretical findings.
Keywords: Optimal control, parabolic partial differential equation, state constraints, first-order necessary optimality condition, regularity result, finite element method, $$ error estimate
@article{COCV_2021__27_1_A6_0,
author = {Christof, Constantin and Vexler, Boris},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020059},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020059/}
}
TY - JOUR AU - Christof, Constantin AU - Vexler, Boris ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020059/ DO - 10.1051/cocv/2020059 LA - en ID - COCV_2021__27_1_A6_0 ER -
%0 Journal Article %A Christof, Constantin %A Vexler, Boris %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020059/ %R 10.1051/cocv/2020059 %G en %F COCV_2021__27_1_A6_0
Christof, Constantin; Vexler, Boris. New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 4. doi: 10.1051/cocv/2020059
[1] , and , Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford & New York (2000).
[2] , and , State-constrained control-affine parabolic problems I: first and second order necessary optimality conditions. Preprint (2019). | arXiv
[3] , and , Variational Analysis in Sobolev and BV Spaces. SIAM, Philadelphia (2006).
[4] , Optimal Control of Variational Inequalities. Research Notes in Mathematics. Pitman (1984).
[5] , Optimal control of parabolic problems with state constraints: a penalization method for optimality conditions. Appl. Math. Optim. 29 (1994) 285–307. | MR | Zbl | DOI
[6] and , Primal-dual strategy for state-constrained optimal control problems. Comput. Optim. Appl. 22 (2002) 193–224.
[7] , Numerical Analysis of Parabolic Time-optimal Control Problems. Ph.D. thesis, Technische Universität München (2018).
[8] and , Optimal control of a parabolic equation with time-dependent state constraints. SIAM J. Control Optim. 48 (2010) 4550–4571.
[9] and , Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000).
[10] and , The Mathematical Theory of Finite Element Methods, 3rd edn. Springer (2008).
[11] , Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35 (1997) 1297–1327.
[12] , and , New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM: COCV 20 (2014) 803–822. | MR | Numdam | Zbl
[13] and , Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems. ESAIM Control Optim. Calc. Var. 16 (2010) 581–600.
[14] , Sensitivity Analysis of Elliptic Variational Inequalities of the First and the Second Kind. Ph.D. thesis, Technische Universität Dortmund (2018).
[15] , Sensitivity analysis and optimal control of obstacle-type evolution variational inequalities. SIAM J. Control Optim. 57 (2019) 192–218.
[16] and , A note on a priori L$$-error estimatesfor the obstacle problem. Numer. Math. 139 (2018) 27–45. | MR | DOI
[17] and Sensitivity analysis for a class of H01-elliptic variational inequalities of the second kind. Set-Valued Var. Anal. 27 (2018) 469–502. | MR | DOI
[18] and , On second-order optimality conditions for optimal control problems governed by the obstacle problem. To appear in: Optimization (2020) 1–41. | DOI | MR
[19] , Optimization and Nonsmooth Analysis. SIAM’s Classics in Applied Mathematics. SIAM, Philadelphia, PA (1990). | MR | Zbl
[20] , , and , Optimality conditions for state-constrained PDE control problems with time-dependent controls. Control Cybernet. 37 (2008) 5–38.
[21] and , Strong stationarity conditions for a class of optimization problems governed by variational inequalities of the second kind. J. Optim. Theory Appl. 168 (2016) 375–409.
[22] and , Variational discretization of parabolic control problems in the presence of pointwise state constraints. J. Comput. Math. 29 (2011) 1–15. | MR | Zbl | DOI
[23] , and , Hölder estimates for parabolic operators on domains with rough boundary. Ann. Sc. Norm. Super. Pisa Cl. Sci. XVII (2017) 65–79. | MR
[24] and , Convex Analysis and Variational Problems. North-Holland (1976). | MR | Zbl
[25] , and , Optimal regularity for elliptic transmission problems including C1 -interfaces. Interfaces Free Bound. 9 (2007) 233–252.
[26] , and , Time discretization of parabolic problems by the discontinuous Galerkin method. ESAIM Math. Model. Numer. Anal. 19 (1985) 611–643.
[27] , Partial Differential Equations, 2nd edn. AMS, Providence, RI (2010).
[28] , State Constrained Optimal Control for Some Quasilinear Parabolic Equations, edited by , , and . Optimal Control of Partial Differential Equations. Birkhäuser, Basel, (1999) 145–156.
[29] , Integral state-constrained optimal control problems for some quasilinear parabolic equations. J. Nonlinear Anal. Optim. 39 (2000) 977–996.
[30] and , Elliptic Partial Differential Equations of Second Order, reprint ofthe 1998 edn. Springer 2001.
[31] , and , An ADMM numerical approach to linear parabolic state constrained optimal control problems. Numer. Math. 144 (2020) 931–966.
[32] and , Error estimates for parabolic optimal control problems with control and state constraints. Comput. Optim. Appl. 56 (2013) 131–151. | MR | Zbl | DOI
[33] , Elliptic Problems in Nonsmooth Domains. Pitman (1985).
[34] and Comparison of optimality systems for the optimal control of the obstacle problem. GAMM-Mitt. 40 (2018) 312–338. | MR | DOI
[35] , , and , Sobolev Spaces on Metric Measure Spaces. Vol. 27 of New Mathematical Monographs. Cambridge University Press (2015).
[36] A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–61.
[37] and , Semi-smooth Newton methods for state-constrained optimal control problems. Systems Control Lett. 50 (2003) 221–228.
[38] and , An Introduction to Variational Inequalities and Their Applications. Vol. 31 of Classics in Applied Mathematics. SIAM (2000).
[39] and , Pointwise best approximation results for Galerkin finite element solutions of parabolic problems. SIAM J. Numer. Anal. 54 (2016) 1365–1384. | MR | DOI
[40] and , Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math. 135 (2017) 923–952.
[41] , and , A priori error estimates for state-constrained semilinear parabolic optimal control problems. J. Optim. Theory Appl. 178 (2018) 317–348.
[42] , and , A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time. SIAM J. Control Optim. 49 (2011) 1961–1997. | MR | Zbl | DOI
[43] and , Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations. ESAIM: M2AN 52 (2018) 2307–2325. | MR | Numdam | DOI
[44] , and , Optimal control of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33 (2006) 209–228.
[45] , Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130–185.
[46] and , Optimal control of state-constrained parabolic systems with nonregular boundary controllers, in Proceedings of the 36th IEEE Conference on Decision and Control, Vol. 1 (1997) 527–528. | DOI
[47] , and , Topics in Nonsmooth Mechanics. Birkhäuser, Basel (1988).
[48] and , On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Control Cybernet. 37 (2008) 1013–1043.
[49] and , On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. ESAIM: COCV 15 (2009) 426–453. | MR | Numdam | Zbl
[50] and , Numerical analysis of state-constrained optimal control problems for PDEs, edited by , , , , , , and Constrained Optimization and Optimal Control for Partial Differential Equations. Springer, Basel (2012) 467–482.
[51] and , Interior maximum-norm estimates for finite element methods, part II. Math. Comput. 64 (1995) 907–928.
[52] , State constrained optimal control problems with states of low regularity. SIAM J. Control Optim. 48 (2009) 2407–2432.
[53] , Partielle Differentialgleichungen. Springer, Berlin/Heidelberg (2013).
[54] , Optimale Steuerung partieller Differentialgleichungen. Vieweg und Teubner, Wiesbaden, 2nd edn. (2009).
[55] , The regularity of the positive part of functions in L2 (I ; H1 (Ω)) ∩ H1 (I ; H1 (Ω) *) with applications to parabolic equations. Comment. Math. Univ. Carolin. 57 (2016) 327–332.
[56] , Towards M-stationarity for optimal control of the obstacle problem with control constraints. SIAM J. Control Optim. 54 (2016) 964–986.
Cité par Sources :
This research was conducted within the International Research Training Group IGDK 1754, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF) under project number 188264188/GRK1754.
