Numerical approach to the non-linear diofantic equations with applications to the controllability of infinite dimensional dynamical systems

Abstract

The article is devoted to the analysis of the approximate controllability of a given type of second order infinite dimensional system, defined by the hyperbolic type partial differential equation with Dirichlet-like type zero boundary conditions defined in the n-dimensional rectangular prism. Following this aim, spectral theory for linear unbounded operators is involved. New numerical algorithms for solving some kind of non-linear diofantic equations, corresponding with the spectral properties of the unbounded operators, are presented and proved. An algorithm is optimized for eliminating the symmetrical solutions, not interesting in the scope of verification of the controllability. Next the limit which shows that the direct application of the analytical methods of verifying the controllability in the numerical approach is impossible is proved. Finally so-called partial approximate controllability is defined and the numerical algorithm for its verification is presented. Finally, proven theorems are applied to one particular infinite dimensional dynamical system.