Reconstruction of boundary controls in parabolic systems

Abstract

In the paper, an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamical systems described by boundary value problems for partial differential equations of parabolic type. The source information for solving the inverse problem is the results of approximate measurements of the states of the observed system's motion. The problem is solved in the static case; i. e., to solve it, we use all the measurement data accumulated during some specified observation interval. The problem under consideration is ill-posed. To solve it, we propose the Tikhonov method with a stabilizer containing the sum of the mean-square norm and total time variation of the control. The use of such nondifferentiable stabilizer allows us to obtain more precise results than the approximation of the desired control in the Lebesgue spaces. In particular, this method provides the pointwise and piecewise uniform convergences of regularized approximations and makes possible the numerical reconstruction of the subtle structure of the desired control. In the paper, the subgradient projection method for obtaining a minimizing sequence for the Tikhonov functional is described and substantiated. Also, we demonstrate the two-stage finitedimensional approximation of the problem and present the results of numerical simulation. © 2013 Pleiades Publishing, Ltd.