Exponential stability of a numerical solution of a hyperbolic system with negative nonlocal characteristic velocities and measurement error

Abstract

In this work, the problem of stabilizing the equilibrium state for a hyperbolic system with negative nonlocal characteristic velocities and measurement error is investigated. A mixed problem is considered for such systems, when a limited perturbation of measurement errors is taken into account in the boundary conditions. The study is based on the use of the adequacy between the stability for a mixed problem for the original hyperbolic system of linear differential equations and the stability of the initial-boundary difference problem for it. When analyzing the initial-boundary difference problems constructed in this way, the properties of logarithmic norms are used. Algorithms are proposed that make it possible to obtain sufficient conditions for the exponential stability of a numerical solution of an initial-boundary difference problem with nonlocal coefficients and limited perturbation of measurement errors in boundary conditions. Sufficient conditions are presented in the form of matrix inequalities, which involve matrices of boundary conditions. The results are presented in the form of an a priori estimate of the numerical solution in the norm through the norms of the functions of the initial data and the norms of perturbation of measurement errors.