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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

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

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

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

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McGraw-Hill, London, 1970. Google Scholar

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

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

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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

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