Reconstruction of boundary controls in hyperbolic systems

A. I. Korotkii   ; E. I. Gribanova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 2, pp. 154-169

Résumé

The inverse problem of dynamics consisting in reconstructing of a priori unknown controls generating the observed motion of dynamic system is considered. The dynamic system is defined by a boundary value problem for the equation with private derivatives of the hyperbolic type, controls are on a boundary of the object. The source information for solving the inverse problem is results of approximate measurements of the current phase positions of the observed systems motion. The problem is solved in static case, when to solve the problem, all the cumulative during the definite observation period totality of results of measurements is used. This problem is ill-posed and to solve this problem, Tikhonovs method with a stabilizer containing the sum of root mean square norm and total variation of control in time is used. This way provides not only the convergence of regularized approximations in Lebesgue spaces but piecewise uniform convergence. This permits numerical reconstruction of desired controls subtle structure. In this paper subgradient projection method of the obtaining minimizing sequence for the Tikhonovs functional is justified, two-stage finite-dimensional approximation of the problem is described. Results of numerical experiments are presented.

Détail

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

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