Optimal control/Numerical analysis
On a finite element method for measure-valued optimal control problems governed by the 1D generalized wave equation
Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 523-531.

The paper deals with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces of either measure-valued functions Lw2(I,M(Ω)) or vector measures M(Ω,L2(I)). Bilinear finite element discretizations are constructed and their stability and error analysis is accomplished.

Cet article traite des problèmes de contrôle optimal régis par l'équation d'onde 1D avec coefficients variables, les espaces de contrôle étant, soit des fonctions mesurées Lw2(I,M(Ω)), soit des mesures vectorielles M(Ω,L2(I)). On construit des discrétisations bilinéaires des éléments finis et on en analyse la stabilité et l'erreur.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.02.011
Vexler, Boris 1; Zlotnik, Alexander 2; Trautmann, Philip 3

1 Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, 85748 Garching bei München, Germany
2 Department of Mathematics at Faculty of Economic Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russia
3 Department of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria
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     title = {On a finite element method for measure-valued optimal control problems governed by the {1D} generalized wave equation},
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Vexler, Boris; Zlotnik, Alexander; Trautmann, Philip. On a finite element method for measure-valued optimal control problems governed by the 1D generalized wave equation. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 523-531. doi : 10.1016/j.crma.2018.02.011. http://www.numdam.org/articles/10.1016/j.crma.2018.02.011/

[1] Bergh, J.; Löfström, J. Interpolation Spaces. An Introduction, Springer, Berlin, New York, 1976

[2] Bogachev, V.I. Measure Theory. Vol. I, II, Springer, Berlin, 2007

[3] Casas, E.; Clason, C.; Kunisch, K. Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., Volume 51 (2013), pp. 28-63

[4] Casas, E.; Kunisch, K. Parabolic control problems in space–time measure spaces, ESAIM Control Optim. Calc. Var., Volume 22 (2016), pp. 355-376

[5] Casas, E.; Vexler, B.; Zuazua, E. Sparse initial data identification for parabolic PDE and its finite element approximations, Math. Control Relat. Fields, Volume 5 (2015), pp. 377-399

[6] Ciarlet, P.G. Finite Element Method for Elliptic Problems, SIAM, Philadelphia, PA, USA, 2002

[7] Fabre, C.; Puel, J.-P. Pointwise controllability as limit of internal controllability for the wave equation in one space dimension, Port. Math., Volume 51 (1994), pp. 335-350

[8] Kantorovich, L.V.; Akilov, G.P. Functional Analysis, Pergamon Press, Oxford, UK, 1982

[9] Kunisch, K.; Pieper, K.; Vexler, B. Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., Volume 52 (2014), pp. 3078-3108

[10] Kunisch, K.; Trautmann, Ph.; Vexler, B. Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., Volume 54 (2016), pp. 1212-1244

[11] Lions, J.L. Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985

[12] Lions, J.L.; Magenes, E. Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer, Berlin, 1972

[13] Nikolskii, S.M. Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin, 1975

[14] Pieper, K. Finite Element Discretization and Efficient Numerical Solution of Elliptic and Parabolic Sparse Control Problems, TU Munich, Germany, 2015 (PhD thesis)

[15] Samarskii, A.A. The Theory of Difference Schemes, Marcel Dekker, New York, Basel, 2001

[16] Scott, R. A sharp form of the Sobolev trace theorems, J. Funct. Anal., Volume 25 (1997) no. 1, pp. 70-80

[17] Trautmann, P. Sparse Measure-Valued Optimal Control Problems Governed by the Wave Equations, KFU Graz, Austria, 2015 (PhD thesis)

[18] Trautmann, P.; Vexler, B.; Zlotnik, A. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients, Math. Control Relat. Fields, Volume 8 (2018) no. 2 (accepted for publication)

[19] Zlotnik, A.A. Projective-Difference Schemes for Nonstationary Problems with Nonsmooth Data, Lomonosov Moscow State University, 1979 PhD thesis (in Russian)

[20] Zlotnik, A.A. Convergence rate estimates of finite-element methods for second order hyperbolic equations (Marchuk, G.I., ed.), Numerical Methods and Applications, CRC Press, Boca Raton, FL, USA, 1994, pp. 155-220

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