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

Casas E, Vexler B, Zuazua E (2015)

Publication Type: Journal article

Publication year: 2015

Journal

Mathematical Control and Related Fields American Institute of Mathematical Sciences (AIMS)

Book Volume: 5

Pages Range: 377-399

Journal Issue: 3

DOI: 10.3934/mcrf.2015.5.377

Abstract

We address the problem of inverse source identication 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 nite number of pointwise locations. Moreover, we discuss the nite 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.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Universidad de Cantabria (UC) ES Spain (ES) Technische Universität München (TUM) DE Germany (DE) Basque Center of Applied Mathematics (BCAM) ES Spain (ES)

How to cite

APA:

Casas, E., Vexler, B., & Zuazua, E. (2015). Sparse initial data identification for parabolic PDE and its finite element approximations. Mathematical Control and Related Fields, 5(3), 377-399. https://dx.doi.org/10.3934/mcrf.2015.5.377

MLA:

Casas, Eduardo, Boris Vexler, and Enrique Zuazua. "Sparse initial data identification for parabolic PDE and its finite element approximations." Mathematical Control and Related Fields 5.3 (2015): 377-399.

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