Hyperbolic equation, nonlocal characteristic velocity, stability, explicit difference scheme

Investigation of exponential stability of numerical solution of hyperbolic equation with negative nonlocal characteristic velocity and measurement error

Алоев Р.Д.,
[e-mail]
Национальный университет Узбекистана имени Мирзо Улугбека, 100174, Узбекистан, Ташкент, ул. Университетская, 4;
Алимова В.Б
[e-mail]
Национальный университет Узбекистана имени Мирзо Улугбека, 100174, Узбекистан, Ташкент, ул. Университетская, 4;

Keywords.
Hyperbolic equation, nonlocal characteristic velocity, stability, explicit difference scheme.
Abstract
In this paper, we investigate the problem of stabilization of the equilibrium state for a hyperbolic equation with negative nonlocal characteristic velocity and measurement error. The formulation of a mixed control problem is given. The definition of the weak solution, exponential stability of the equilibrium of the mixed problem and the Lyapunov function is given. A theorem on the exponential stability of the equilibrium of a mixed problem is proved. Stability in the -norm with respect to a discrete disturbance of the equilibrium state of the initial boundary difference problem is determined. A discrete Lyapunov function is constructed and a theorem of stability of the equilibrium state of the initial-boundary difference problem in the -norm with respect to a discrete perturbation is proved.