Reconstruction of Distributed Controls in Hyperbolic Systems by Dynamic Method

Abstract

In the paper an inverse dynamic problem is considered. It consists of reconstructing
a priori unknown distributed controls in dynamical systems described by boundary value
problems for partial differential equations of hyperbolic type. The source information for
solving the inverse problem is the results of approximate measurements of the states
(velocities) of the observed system’s motion. The problem is solved in the dynamic case,
i.e. to solve the problem we can use only the approximate measurements accumulated by
this moment. Unknown controls must be reconstructed in dynamics (during the process,
during the motion of the system). The problem under consideration is ill-posed. We
propose the method of dynamic regularization to solve the problem. This method was
elaborated by Yu.S. Osipov and his school. New modifications of dynamic regularizing
solution algorithms are devised in this paper. Using these algorithms in contrast to tradition
approach we can obtain stronger convergence of regularized approximations, in particular
the piecewise uniform convergence.We also demonstrate a finite-dimensional approximation
of the problem and the present results of numerical modelling. These results enable us
to assess the ability of modified algorithms to reconstruct the subtle structure of desired
controls.