Mathematical Control and Related Fields (MCRF)

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

Pages: 377 - 399, Volume 5, Issue 3, September 2015      doi:10.3934/mcrf.2015.5.377

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Eduardo Casas - Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain (email)
Boris Vexler - Centre for Mathematical Sciences, Technische Universität München, Bolzmannstrasse 3, D-85747 Garching b. München, Germany (email)
Enrique Zuazua - BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao-Basque Country, Spain (email)

Abstract: 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:  Parabolic equations, approximate controllability, sparse controls, Borel measures.
Mathematics Subject Classification:  35K15, 49K20, 93B05, 93C20.

Received: August 2014;      Revised: November 2014;      Available Online: July 2015.