American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process
  • MCRF Home
  • This Issue
  • Next Article
    A biographical note and tribute to xunjing li on his 80th birthday
September  2015, 5(3): 377-399. doi: 10.3934/mcrf.2015.5.377

Sparse initial data identification for parabolic PDE and its finite element approximations

Eduardo Casas 1, , Boris Vexler 2, and  Enrique Zuazua 3,

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. 

Centre for Mathematical Sciences, Technische Universität München, Bolzmannstrasse 3, D-85747 Garching b. München, Germany
3. 

BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao-Basque Country

Received  August 2014 Revised  November 2014 Published  July 2015

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.
Keywords: approximate controllability, Parabolic equations, Borel measures., sparse controls.
Mathematics Subject Classification: 35K15, 49K20, 93B05, 93C2.
Citation: Eduardo Casas, Boris Vexler, Enrique Zuazua. Sparse initial data identification for parabolic PDE and its finite element approximations. Mathematical Control & Related Fields, 2015, 5 (3) : 377-399. doi: 10.3934/mcrf.2015.5.377
References:
[1]

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487-535.  Google Scholar

[2]

Springer-Verlag, New York, Berlin, Heidelberg, 2008, Third edition. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[3]

SIAM J. Control Optim., 35 (1997), 1297-1327. doi: 10.1137/S0363012995283637.  Google Scholar

[4]

SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216.  Google Scholar

[5]

SIAM J. Control Optim., 51 (2013), 28-63. doi: 10.1137/120872395.  Google Scholar

[6]

SIAM J. Optim., 22 (2012), 795-820. doi: 10.1137/110834366.  Google Scholar

[7]

SIAM J. Control Optim., 52 (2014), 339-364. doi: 10.1137/13092188X.  Google Scholar

[8]

________, Parabolic control problems in space-time measure spaces,, To appear in ESAIM Control Optim. Calc. Var., ().   Google Scholar

[9]

SIAM J. Control Optim., 52 (2014), 1010-1033. doi: 10.1137/130917314.  Google Scholar

[10]

Systems Control Lett., 62 (2013), 311-318. doi: 10.1016/j.sysconle.2013.01.001.  Google Scholar

[11]

ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003.  Google Scholar

[12]

North-Holland-Elsevier, New York, 1976.  Google Scholar

[13]

Control and optimal design of distributed parameter systems (Minneapolis, MN, 1992), IMA Vol. Math. Appl., Springer, New York, 70 (1995), 73-91. doi: 10.1007/978-1-4613-8460-1_4.  Google Scholar

[14]

Advances in Differential Equations, 5 (2000), 465-514.  Google Scholar

[15]

SIAM J. Control Optim., 52 (2014), 97-119. doi: 10.1137/110840133.  Google Scholar

[16]

Adv. Differential Equations, 12 (2007), 1031-1078.  Google Scholar

[17]

Pitman, Boston-London-Melbourne, 1985.  Google Scholar

[18]

BIT, 42 (2002), 351-379. doi: 10.1023/A:1021903109720.  Google Scholar

[19]

SIAM J. Control Optim., 50 (2012), 943-963. doi: 10.1137/100815037.  Google Scholar

[20]

Proc. Amer. Math. Soc., 130 (2002), 1055-1064. doi: 10.1090/S0002-9939-01-06163-9.  Google Scholar

[21]

SIAM J. Control Optim., 52 (2014), 3078-3108. doi: 10.1137/140959055.  Google Scholar

[22]

SIAM J. Numer. Anal., 51 (2013), 2797-2821. doi: 10.1137/120885772.  Google Scholar

[23]

Inverse Probl. and Imaging, 8 (2014), 199-221. doi: 10.3934/ipi.2014.8.199.  Google Scholar

[24]

Math. Scand., 8 (1960), 277-286.  Google Scholar

[25]

SIAM J. Control Optim., 49 (2011), 1961-1997. doi: 10.1137/100793888.  Google Scholar

[26]

SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.  Google Scholar

[27]

App. Math. Optim., 39 (1999), 143-177. doi: 10.1007/s002459900102.  Google Scholar

[28]

McGraw-Hill, London, 1970. Google Scholar

[29]

Comm. Pure Appl. Math., 33 (1980), 265-304. doi: 10.1002/cpa.3160330305.  Google Scholar

[30]

Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.  Google Scholar

[31]

Second edition, Spinger-Verlag, Berlin, 2006.  Google Scholar

[32]

ESAIM Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.  Google Scholar

show all references

References:
[1]

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487-535.  Google Scholar

[2]

Springer-Verlag, New York, Berlin, Heidelberg, 2008, Third edition. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[3]

SIAM J. Control Optim., 35 (1997), 1297-1327. doi: 10.1137/S0363012995283637.  Google Scholar

[4]

SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216.  Google Scholar

[5]

SIAM J. Control Optim., 51 (2013), 28-63. doi: 10.1137/120872395.  Google Scholar

[6]

SIAM J. Optim., 22 (2012), 795-820. doi: 10.1137/110834366.  Google Scholar

[7]

SIAM J. Control Optim., 52 (2014), 339-364. doi: 10.1137/13092188X.  Google Scholar

[8]

________, Parabolic control problems in space-time measure spaces,, To appear in ESAIM Control Optim. Calc. Var., ().   Google Scholar

[9]

SIAM J. Control Optim., 52 (2014), 1010-1033. doi: 10.1137/130917314.  Google Scholar

[10]

Systems Control Lett., 62 (2013), 311-318. doi: 10.1016/j.sysconle.2013.01.001.  Google Scholar

[11]

ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003.  Google Scholar

[12]

North-Holland-Elsevier, New York, 1976.  Google Scholar

[13]

Control and optimal design of distributed parameter systems (Minneapolis, MN, 1992), IMA Vol. Math. Appl., Springer, New York, 70 (1995), 73-91. doi: 10.1007/978-1-4613-8460-1_4.  Google Scholar

[14]

Advances in Differential Equations, 5 (2000), 465-514.  Google Scholar

[15]

SIAM J. Control Optim., 52 (2014), 97-119. doi: 10.1137/110840133.  Google Scholar

[16]

Adv. Differential Equations, 12 (2007), 1031-1078.  Google Scholar

[17]

Pitman, Boston-London-Melbourne, 1985.  Google Scholar

[18]

BIT, 42 (2002), 351-379. doi: 10.1023/A:1021903109720.  Google Scholar

[19]

SIAM J. Control Optim., 50 (2012), 943-963. doi: 10.1137/100815037.  Google Scholar

[20]

Proc. Amer. Math. Soc., 130 (2002), 1055-1064. doi: 10.1090/S0002-9939-01-06163-9.  Google Scholar

[21]

SIAM J. Control Optim., 52 (2014), 3078-3108. doi: 10.1137/140959055.  Google Scholar

[22]

SIAM J. Numer. Anal., 51 (2013), 2797-2821. doi: 10.1137/120885772.  Google Scholar

[23]

Inverse Probl. and Imaging, 8 (2014), 199-221. doi: 10.3934/ipi.2014.8.199.  Google Scholar

[24]

Math. Scand., 8 (1960), 277-286.  Google Scholar

[25]

SIAM J. Control Optim., 49 (2011), 1961-1997. doi: 10.1137/100793888.  Google Scholar

[26]

SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.  Google Scholar

[27]

App. Math. Optim., 39 (1999), 143-177. doi: 10.1007/s002459900102.  Google Scholar

[28]

McGraw-Hill, London, 1970. Google Scholar

[29]

Comm. Pure Appl. Math., 33 (1980), 265-304. doi: 10.1002/cpa.3160330305.  Google Scholar

[30]

Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.  Google Scholar

[31]

Second edition, Spinger-Verlag, Berlin, 2006.  Google Scholar

[32]

ESAIM Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.  Google Scholar

[1]

Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167
[2]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719
[3]

Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control & Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171
[4]

Vahagn Nersesyan. Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension. Mathematical Control & Related Fields, 2021, 11 (2) : 237-251. doi: 10.3934/mcrf.2020035
[5]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1
[6]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953
[7]

Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations & Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025
[8]

Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020076
[9]

Yassine El Gantouh, Said Hadd, Abdelaziz Rhandi. Approximate controllability of network systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020091
[10]

Óscar Vega-Amaya, Joaquín López-Borbón. A perturbation approach to a class of discounted approximate value iteration algorithms with borel spaces. Journal of Dynamics & Games, 2016, 3 (3) : 261-278. doi: 10.3934/jdg.2016014
[11]

Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263
[12]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[13]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031
[14]

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. The cost of controlling weakly degenerate parabolic equations by boundary controls. Mathematical Control & Related Fields, 2017, 7 (2) : 171-211. doi: 10.3934/mcrf.2017006
[15]

Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations & Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741
[16]

J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136
[17]

Morteza Fotouhi, Leila Salimi. Controllability results for a class of one dimensional degenerate/singular parabolic equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1415-1430. doi: 10.3934/cpaa.2013.12.1415
[18]

El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control & Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013
[19]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699
[20]

Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences