Sparse initial data indentification for parabolic pde and its finite element approximations
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© American Institute of Mathematical Sciences. This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Mathematical Control and Related Fields following peer review. The definitive publisher-authenticated version, Mathematical Control and Related Fields, 2015, 5(3), 377-399 is available online at: http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11431
Mathematical control and related fields, 2015, 5(3), 377-399
American Institute of Mathematical Sciences
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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.