Reconstruction of boundary controls in parabolic systems

A. I. Korotkii ; D. O. Mikhailova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 1, pp. 178-197

Résumé

In this paper an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamic systems described by boundary problems for parabolic partial differential equations. The source information for solving the inverse problem is results of approximate measurements of a states of the observed system's motion. The problem is solved in static case, i.e. we can use all the accumulated during the definite observation period data of measurements to solve the problem. The problem under consideration is ill-posed. We propose the Tikhonov method with stabilizer containing sum of mean-square norm and total variation of control in time to solve the problem. The usage of such non-differentiable stabilizer lets obtain more precise results in some cases than approximation of the desired control in Lebesgue spaces does. In particular, this way provides the pointwise and piecewise uniform convergences of regularized approximations and permits numerical reconstruction of desired control's subtle structure. In this paper we describe and validate the gradient projection technique of receiving minimizing sequence for the Tikhonov functional. Also we demonstrate two-stage finite-dimensional approximation of the problem and present results of computational modeling.

Détail

Keywords: dynamical system, control, measurement, regularization, inverse problem, Tikhonov's method
Mots-clés : reconstruction, observation, variation, piecewise uniform convergence.