Finite element approximation of sparse parabolic control problems

Eduardo Casas; Mariano Mateos; Arnd Rösch

doi:10.3934/mcrf.2017014

Article Contents

2017, Volume 7, Issue 3: 393-417. Doi: 10.3934/mcrf.2017014

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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
* Corresponding author: Mariano Mateos

Received: September 30, 2016

Revised: April 30, 2017

Published: October 2017

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 49K20, 35K58, 65M15; Secondary: 49J52.

Citation:

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Figure 1. Desired state (left) and Optimal control (right)

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Figure 2. Experiment 2. Support of the optimal control for different values of $\mu$

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Table 1. Results for $h_i=\tau_i=2^{-i}$

i e_i EOC_i

6 4:37E-3 −

7 2:22E-3 0:98

8 1:12E-3 0:99

9 5:60E-4 0:99

10 2:81E-4 1:00

11 1:40E-4 1:00

12 7:03E-5 1:00

13 3:51E-5 1:00

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Table 2. Results for fixed $\tau=2^{-13}$ and decreasing $h_i=2^{-i}$

i e_i EOC_i

6 1:71E-3 −

7 8:84E-4 0:95

8 4:57E-4 0:95 ∗

9 2:40E-4 0:93

10 1:30E-4 0:88

11 7:54E-5 0:79

12 4:83E-5 0:64

13 3:51E-5 0:46

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Table 3. Results for fixed $h=2^{-13}$ and $\tau_j=2^{-j}$

i e_i EOC_i

6 1:71E-3 −

7 8:84E-4 0:95

8 4:57E-4 0:95 ∗

9 2:40E-4 0:93

10 1:30E-4 0:88

11 7:54E-5 0:79

12 4:83E-5 0:64

13 3:51E-5 0:46

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Table 4. Experiment 2. Value of the objective functional as the parameter $\mu$ increases

µ 0 µ₀ 2µ₀ 3µ₀ 4µ₀

J_σ(u_σ) 0:00935 0:03465 0:04879 0:05738 0:06273

µ 5µ₀ 6µ₀ 7µ₀ 8µ₀

J_σ(u_σ) 0:06705 0:06803 0:06896 0:06906

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References

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