September  2017, 7(3): 393-417. doi: 10.3934/mcrf.2017014

Finite element approximation of sparse parabolic control problems

1. 

Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain

2. 

Departamento de Matemáticas, Campus de Gijón, Universidad de Oviedo, 33203, Gijón, Spain

3. 

Fakultät für Mathematik, Universtät Duisburg-Essen, D-45127 Essen, Germany

* Corresponding author: Mariano Mateos

Received  October 2016 Revised  May 2017 Published  July 2017

Fund Project: The first two authors were partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2014-57531-P

We study the finite element approximation of an optimal control problem governed by a semilinear partial differential equation and whose objective function includes a term promoting space sparsity of the solutions. We prove existence of solution in the absence of control bound constraints and provide the adequate second order sufficient conditions to obtain error estimates. Full discretization of the problem is carried out, and the sparsity properties of the discrete solutions, as well as error estimates, are obtained.

Citation: Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014
References:
[1]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-3658-8.

[2]

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.

[3]

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.

[4]

E. CasasC. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63. doi: 10.1137/120872395.

[5]

E. CasasR. Herzog and G. Wachsmuth, Approximation of sparse controls in semilinear equations by piecewise linear functions, Numer. Math., 122 (2012), 645-669. doi: 10.1007/s00211-012-0475-7.

[6]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820. doi: 10.1137/110834366.

[7]

E. CasasR. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations, ESAIM:COCV, 23 (2017), 263-295. doi: 10.1051/cocv/2015048.

[8]

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.

[9]

E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces, ESAIM Control Optim. Calc. Var., 22 (2016), 355-370. doi: 10.1051/cocv/2015008.

[10]

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

[11]

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.

[12]

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.

[13]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1968.

[14]

I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numer. Math., 120 (2012), 345-386. doi: 10.1007/s00211-011-0409-9.

[15]

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.

[16]

P. -A. Raviart and J. -M. Thomas, Introduction á L'analyse Numérique des Équations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maî trise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.

[17]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.

[18]

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

show all references

References:
[1]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-3658-8.

[2]

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.

[3]

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.

[4]

E. CasasC. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63. doi: 10.1137/120872395.

[5]

E. CasasR. Herzog and G. Wachsmuth, Approximation of sparse controls in semilinear equations by piecewise linear functions, Numer. Math., 122 (2012), 645-669. doi: 10.1007/s00211-012-0475-7.

[6]

E. CasasR. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820. doi: 10.1137/110834366.

[7]

E. CasasR. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations, ESAIM:COCV, 23 (2017), 263-295. doi: 10.1051/cocv/2015048.

[8]

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.

[9]

E. Casas and K. Kunisch, Parabolic control problems in space-time measure spaces, ESAIM Control Optim. Calc. Var., 22 (2016), 355-370. doi: 10.1051/cocv/2015008.

[10]

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

[11]

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.

[12]

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.

[13]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1968.

[14]

I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numer. Math., 120 (2012), 345-386. doi: 10.1007/s00211-011-0409-9.

[15]

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.

[16]

P. -A. Raviart and J. -M. Thomas, Introduction á L'analyse Numérique des Équations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maî trise. [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.

[17]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.

[18]

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

Figure 1.  Desired state (left) and Optimal control (right)
Figure 2.  Experiment 2. Support of the optimal control for different values of $\mu$
Table 1.  Results for $h_i=\tau_i=2^{-i}$
ieiEOCi
64:37E-3
72:22E-30:98
81:12E-30:99
95:60E-40:99
102:81E-41:00
111:40E-41:00
127:03E-51:00
133:51E-51:00
ieiEOCi
64:37E-3
72:22E-30:98
81:12E-30:99
95:60E-40:99
102:81E-41:00
111:40E-41:00
127:03E-51:00
133:51E-51:00
Table 2.  Results for fixed $\tau=2^{-13}$ and decreasing $h_i=2^{-i}$
ieiEOCi
61:71E-3
78:84E-40:95
84:57E-40:95
92:40E-40:93
101:30E-40:88
117:54E-50:79
124:83E-50:64
133:51E-50:46
ieiEOCi
61:71E-3
78:84E-40:95
84:57E-40:95
92:40E-40:93
101:30E-40:88
117:54E-50:79
124:83E-50:64
133:51E-50:46
Table 3.  Results for fixed $h=2^{-13}$ and $\tau_j=2^{-j}$
ieiEOCi
61:71E-3
78:84E-40:95
84:57E-40:95
92:40E-40:93
101:30E-40:88
117:54E-50:79
124:83E-50:64
133:51E-50:46
ieiEOCi
61:71E-3
78:84E-40:95
84:57E-40:95
92:40E-40:93
101:30E-40:88
117:54E-50:79
124:83E-50:64
133:51E-50:46
Table 4.  Experiment 2. Value of the objective functional as the parameter $\mu$ increases
µ0 µ02µ03µ04µ0
Jσ(uσ)0:009350:034650:048790:057380:06273
µ5µ06µ07µ08µ0
Jσ(uσ)0:067050:068030:068960:06906
µ0 µ02µ03µ04µ0
Jσ(uσ)0:009350:034650:048790:057380:06273
µ5µ06µ07µ08µ0
Jσ(uσ)0:067050:068030:068960:06906
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