June  2018, 8(2): 411-449. doi: 10.3934/mcrf.2018017

Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients

1. 

Department of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria

2. 

Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, 85748 Garching bei München, Germany

3. 

Department of Mathematics at Faculty of Economic Sciences, National Research University Higher School of Economics, Myasnitskaya 20, 101000 Moscow, Russia

Received  February 2017 Revised  February 2018 Published  March 2018

Fund Project: The first and the second author were supported by FWF and DFG through the International Research Training Group IGDK 1754 'Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures'. The third has been funded within the framework of the Academic Fund Program at the National Research University Higher School of Economics in 2016-2017 (grant no. 16-01-0054) and by the Russian Academic Excellence Project '5-100'. He also thanks the Technical University of Munich for its hospitality in 2014-2015 years

This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L_{{{w}^{*}}}^{2}\left( I, \mathcal M\left( {\mathit \Omega } \right) \right)$ or vector measures $\mathcal M({\mathit \Omega }, L^2(I))$. The cost functional involves the standard quadratic tracking terms and the regularization term $α\|u\|_{\mathcal M_T}$ with $α>0$. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.

Citation: Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
References:
[1]

L. Bales and I. Lasiecka, Continuous finite elements in space and time for the nonhomogeneous wave equation, Computers. Mathematics with Applications, 27 (1994), 91-102. doi: 10.1016/0898-1221(94)90048-5. Google Scholar

[2]

W. BangerthM. Geiger and R. Rannacher, Adaptive Galerkin finite element methods for the wave equation, Comput. Methods Appl. Math., 10 (2010), 3-48. Google Scholar

[3]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin-New York, 1976. Google Scholar

[4]

A. BermúdezP. Gamallo and R. Rodríguez, Finite element methods in local active control of sound, SIAM J. Control Optim., 43 (2004), 437-465. doi: 10.1137/S0363012903431785. Google Scholar

[5]

V. I. Bogachev, Measure Theory. Vol. I, II, Springer, Berlin, 2007. Google Scholar

[6]

K. Bredies and H. K. Pikkarainen, Inverse problems in spaces of measures, ESAIM Control Optim. Calc. Var., 19 (2013), 190-218. doi: 10.1051/cocv/2011205. Google Scholar

[7]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, 3 ed., 2008. Google Scholar

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Google Scholar

[9]

P. BrunnerC. ClasonM. Freiberger and H. Scharfetter, A deterministic approach to the adapted optode placement for illumination of highly scattering tissue, Biomed. Opt. Express, 3 (2012), 1732-1743. doi: 10.1364/BOE.3.001732. Google Scholar

[10]

G. Butazzo, M. Giaqinta and S. Hildebrandt, One-dimensional Variational Problems. An Introduction, Clarendon Press, Oxford, 1998. Google Scholar

[11]

E. CasasC. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216. Google Scholar

[12]

_____, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28–63. doi: 10.1137/120872395. Google Scholar

[13]

E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces, SIAM J. Control Optim., 52 (2014), 339-364. doi: 10.1137/13092188X. Google Scholar

[14]

_____, Parabolic control problems in space-time measure spaces, ESAIM Control Optim. Calc. Var., 22 (2016), 355–376.Google Scholar

[15]

E. CasasB. Vexler and E. Zuazua, Sparse initial data indentification for parabolic pde and its finite element approximations, Math. Control Relat. Fields, 5 (2015), 377-399. doi: 10.3934/mcrf.2015.5.377. Google Scholar

[16]

E. Casas and E. Zuazua, Spike controls for elliptic and parabolic PDEs, Systems Control Lett., 62 (2013), 311-318. doi: 10.1016/j.sysconle.2013.01.001. Google Scholar

[17]

P. Cembranos and J. Mendoza, Banach Spaces of Vector-Valued Functions, Lect. Notes in Math., 1676, Springer, Berlin, 1997. Google Scholar

[18]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003. Google Scholar

[19]

_____, A measure space approach to optimal source placement, Comp. Optim. Appl. , 53 (2011), 155–171.Google Scholar

[20]

R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, New York, 1965. Google Scholar

[21]

C. Fabre and J.-P. Puel, Pointwise controllability as limit of internal controllability for the wave equation in one space dimension, Portugaliae Mathematica, 51 (1994), 335-350. Google Scholar

[22]

M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints, SIAM J. Numer. Anal., 46 (2008), 577-613. doi: 10.1137/0606668909. Google Scholar

[23]

D. A. French and T.E. Peterson, A continuous space-time finite element method for the wave equation, Math. Comp., 65 (1996), 491-506. doi: 10.1090/S0025-5718-96-00685-0. Google Scholar

[24]

M. GugatA. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, ZAMM Z. Angew. Math. Mech., 89 (2009), 420-444. doi: 10.1002/zamm.200800196. Google Scholar

[25]

M. GugatE. Trélat and E. Zuazua, Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property, Systems Control Lett., 90 (2016), 61-70. doi: 10.1016/j.sysconle.2016.02.001. Google Scholar

[26]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963. doi: 10.1137/100815037. Google Scholar

[27]

M. Hinze, A variational discretization concept in control constrained optimization: The linear quadratic case, Comput. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5. Google Scholar

[28]

L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. Google Scholar

[29]

A. Kröner, Adaptive finite element methods for optimal control of second order hyperbolic equations, Comput. Methods Appl. Math., 11 (2011), 214-240. Google Scholar

[30]

_____, Semi-smooth Newton methods for optimal control of the dynamical Lamé system with control constraints, Numer. Funct. Anal. Optim., 34 (2013), 741–769. doi: 10.1080/01630563.2013.772423. Google Scholar

[31]

A. Kröner and K. Kunisch, A minimum effort optimal control problem for the wave equation, Comput. Optim. Appl., 57 (2014), 241-270. doi: 10.1007/s10589-013-9587-y. Google Scholar

[32]

A. KrönerK. Kunisch and B. Vexler, Semismooth Newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49 (2011), 830-858. doi: 10.1137/090766541. Google Scholar

[33]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108. doi: 10.1137/140959055. Google Scholar

[34]

K. KunischP. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244. doi: 10.1137/141001366. Google Scholar

[35]

K. Kunisch and D. Wachsmuth, On time optimal control of the wave equation and its numerical realization as parametric optimization problem, SIAM J. Control Optim., 51 (2013), 1232-1262. doi: 10.1137/120877520. Google Scholar

[36]

I. Lasiecka and J. Sokolowski, Sensitivity analysis of optimal control problems for wave equations, SIAM J. Control Optim., 29 (1991), 1128-1149. doi: 10.1137/0329060. Google Scholar

[37]

J. Lions, Control of Distributed Singular Systems, 13. Gauthier-Villars, Montrouge, 1983. Google Scholar

[38]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. 1, Springer Berlin, 1972. Google Scholar

[39]

A. Milzarek and M. Ulbrich, A semismooth Newton method with multidimensional filter globalization for l1-optimization, SIAM J. Optim., 24 (2014), 298-333. doi: 10.1137/120892167. Google Scholar

[40]

B.S. Mordukhovich and J.-P. Raymond, Dirichlet boundary control of hyperbolic equations in the presence of state constraints, Appl. Math. Optim., 49 (2004), 145-157. doi: 10.1007/BF02638149. Google Scholar

[41]

_____, Neumann boundary control of hyperbolic equations with pointwise state constraints, SIAM J. Control Optim. , 43 (2004/05), 1354–1372. doi: 10.1137/S0363012903431177. Google Scholar

[42]

S. M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin, 1975. Google Scholar

[43]

K. Pieper, Finite Element Discretization and Efficient Numerical Solution of Elliptic and Parabolic Sparse Control Problems, PhD thesis, TU Munich, 2015.Google Scholar

[44]

K. Pieper, P. Trautmann, B. Tang Quoc and D. Walter, Inverse point source location for the helmholtz equation, Submitted.Google Scholar

[45]

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. Google Scholar

[46]

A. R. Raymond, Introduction to Tensor Products of Banach Spaces, Springer, New York, 2002.Google Scholar

[47]

A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York-Basel, 2001. Google Scholar

[48]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of NorthHolland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978. Google Scholar

[49]

A. A. Zlotnik, Projective-difference Schemes for Nonstationary Problems with Nonsmooth Data, PhD thesis, Lomonosov Moscow State University, 1979 (in Russian).Google Scholar

[50]

_____, Lower error estimates for three-layer difference methods of solving the wave equation with data from hölder spaces, Math. Notes, 51 (1992), 321–323. Google Scholar

[51]

_____, Convergence Rate Estimates of Finite-Element Methods for Second Order Hyperbolic Equations, in Numerical methods and applications, G. I. Marchuk, ed., CRC Press, Boca Raton, FL, 1994,155–220. Google Scholar

[52]

E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation, Rend. Mat. Appl., 24 (2004), 201-237. Google Scholar

show all references

References:
[1]

L. Bales and I. Lasiecka, Continuous finite elements in space and time for the nonhomogeneous wave equation, Computers. Mathematics with Applications, 27 (1994), 91-102. doi: 10.1016/0898-1221(94)90048-5. Google Scholar

[2]

W. BangerthM. Geiger and R. Rannacher, Adaptive Galerkin finite element methods for the wave equation, Comput. Methods Appl. Math., 10 (2010), 3-48. Google Scholar

[3]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin-New York, 1976. Google Scholar

[4]

A. BermúdezP. Gamallo and R. Rodríguez, Finite element methods in local active control of sound, SIAM J. Control Optim., 43 (2004), 437-465. doi: 10.1137/S0363012903431785. Google Scholar

[5]

V. I. Bogachev, Measure Theory. Vol. I, II, Springer, Berlin, 2007. Google Scholar

[6]

K. Bredies and H. K. Pikkarainen, Inverse problems in spaces of measures, ESAIM Control Optim. Calc. Var., 19 (2013), 190-218. doi: 10.1051/cocv/2011205. Google Scholar

[7]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, 3 ed., 2008. Google Scholar

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Google Scholar

[9]

P. BrunnerC. ClasonM. Freiberger and H. Scharfetter, A deterministic approach to the adapted optode placement for illumination of highly scattering tissue, Biomed. Opt. Express, 3 (2012), 1732-1743. doi: 10.1364/BOE.3.001732. Google Scholar

[10]

G. Butazzo, M. Giaqinta and S. Hildebrandt, One-dimensional Variational Problems. An Introduction, Clarendon Press, Oxford, 1998. Google Scholar

[11]

E. CasasC. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216. Google Scholar

[12]

_____, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28–63. doi: 10.1137/120872395. Google Scholar

[13]

E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces, SIAM J. Control Optim., 52 (2014), 339-364. doi: 10.1137/13092188X. Google Scholar

[14]

_____, Parabolic control problems in space-time measure spaces, ESAIM Control Optim. Calc. Var., 22 (2016), 355–376.Google Scholar

[15]

E. CasasB. Vexler and E. Zuazua, Sparse initial data indentification for parabolic pde and its finite element approximations, Math. Control Relat. Fields, 5 (2015), 377-399. doi: 10.3934/mcrf.2015.5.377. Google Scholar

[16]

E. Casas and E. Zuazua, Spike controls for elliptic and parabolic PDEs, Systems Control Lett., 62 (2013), 311-318. doi: 10.1016/j.sysconle.2013.01.001. Google Scholar

[17]

P. Cembranos and J. Mendoza, Banach Spaces of Vector-Valued Functions, Lect. Notes in Math., 1676, Springer, Berlin, 1997. Google Scholar

[18]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003. Google Scholar

[19]

_____, A measure space approach to optimal source placement, Comp. Optim. Appl. , 53 (2011), 155–171.Google Scholar

[20]

R. E. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, New York, 1965. Google Scholar

[21]

C. Fabre and J.-P. Puel, Pointwise controllability as limit of internal controllability for the wave equation in one space dimension, Portugaliae Mathematica, 51 (1994), 335-350. Google Scholar

[22]

M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints, SIAM J. Numer. Anal., 46 (2008), 577-613. doi: 10.1137/0606668909. Google Scholar

[23]

D. A. French and T.E. Peterson, A continuous space-time finite element method for the wave equation, Math. Comp., 65 (1996), 491-506. doi: 10.1090/S0025-5718-96-00685-0. Google Scholar

[24]

M. GugatA. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, ZAMM Z. Angew. Math. Mech., 89 (2009), 420-444. doi: 10.1002/zamm.200800196. Google Scholar

[25]

M. GugatE. Trélat and E. Zuazua, Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property, Systems Control Lett., 90 (2016), 61-70. doi: 10.1016/j.sysconle.2016.02.001. Google Scholar

[26]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963. doi: 10.1137/100815037. Google Scholar

[27]

M. Hinze, A variational discretization concept in control constrained optimization: The linear quadratic case, Comput. Optim. Appl., 30 (2005), 45-61. doi: 10.1007/s10589-005-4559-5. Google Scholar

[28]

L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. Google Scholar

[29]

A. Kröner, Adaptive finite element methods for optimal control of second order hyperbolic equations, Comput. Methods Appl. Math., 11 (2011), 214-240. Google Scholar

[30]

_____, Semi-smooth Newton methods for optimal control of the dynamical Lamé system with control constraints, Numer. Funct. Anal. Optim., 34 (2013), 741–769. doi: 10.1080/01630563.2013.772423. Google Scholar

[31]

A. Kröner and K. Kunisch, A minimum effort optimal control problem for the wave equation, Comput. Optim. Appl., 57 (2014), 241-270. doi: 10.1007/s10589-013-9587-y. Google Scholar

[32]

A. KrönerK. Kunisch and B. Vexler, Semismooth Newton methods for optimal control of the wave equation with control constraints, SIAM J. Control Optim., 49 (2011), 830-858. doi: 10.1137/090766541. Google Scholar

[33]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108. doi: 10.1137/140959055. Google Scholar

[34]

K. KunischP. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244. doi: 10.1137/141001366. Google Scholar

[35]

K. Kunisch and D. Wachsmuth, On time optimal control of the wave equation and its numerical realization as parametric optimization problem, SIAM J. Control Optim., 51 (2013), 1232-1262. doi: 10.1137/120877520. Google Scholar

[36]

I. Lasiecka and J. Sokolowski, Sensitivity analysis of optimal control problems for wave equations, SIAM J. Control Optim., 29 (1991), 1128-1149. doi: 10.1137/0329060. Google Scholar

[37]

J. Lions, Control of Distributed Singular Systems, 13. Gauthier-Villars, Montrouge, 1983. Google Scholar

[38]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. 1, Springer Berlin, 1972. Google Scholar

[39]

A. Milzarek and M. Ulbrich, A semismooth Newton method with multidimensional filter globalization for l1-optimization, SIAM J. Optim., 24 (2014), 298-333. doi: 10.1137/120892167. Google Scholar

[40]

B.S. Mordukhovich and J.-P. Raymond, Dirichlet boundary control of hyperbolic equations in the presence of state constraints, Appl. Math. Optim., 49 (2004), 145-157. doi: 10.1007/BF02638149. Google Scholar

[41]

_____, Neumann boundary control of hyperbolic equations with pointwise state constraints, SIAM J. Control Optim. , 43 (2004/05), 1354–1372. doi: 10.1137/S0363012903431177. Google Scholar

[42]

S. M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin, 1975. Google Scholar

[43]

K. Pieper, Finite Element Discretization and Efficient Numerical Solution of Elliptic and Parabolic Sparse Control Problems, PhD thesis, TU Munich, 2015.Google Scholar

[44]

K. Pieper, P. Trautmann, B. Tang Quoc and D. Walter, Inverse point source location for the helmholtz equation, Submitted.Google Scholar

[45]

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. Google Scholar

[46]

A. R. Raymond, Introduction to Tensor Products of Banach Spaces, Springer, New York, 2002.Google Scholar

[47]

A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York-Basel, 2001. Google Scholar

[48]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of NorthHolland Mathematical Library, North-Holland Publishing Co., Amsterdam-New York, 1978. Google Scholar

[49]

A. A. Zlotnik, Projective-difference Schemes for Nonstationary Problems with Nonsmooth Data, PhD thesis, Lomonosov Moscow State University, 1979 (in Russian).Google Scholar

[50]

_____, Lower error estimates for three-layer difference methods of solving the wave equation with data from hölder spaces, Math. Notes, 51 (1992), 321–323. Google Scholar

[51]

_____, Convergence Rate Estimates of Finite-Element Methods for Second Order Hyperbolic Equations, in Numerical methods and applications, G. I. Marchuk, ed., CRC Press, Boca Raton, FL, 1994,155–220. Google Scholar

[52]

E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation, Rend. Mat. Appl., 24 (2004), 201-237. Google Scholar

Figure 1.  Example 1: the reference solution $(\hat u, \hat y)$
Figure 2.  Example 1: errors as $h$ refines and $M = 2^{10}$
Figure 3.  Example 1: errors as $\tau$ refines and $N = 2^{10}$
Figure 4.  Example 2: the reference solution $(\hat u, \hat y)$
Figure 5.  Example 2: errors as $h$ refines and $M = 2^{10}$
Figure 6.  Example 2: errors as $\tau$ refines and $N = 2^{10}$
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