Semi-definite Lyapunov functions in stability problem

of functional-differential equations

O.A.Peregudova

Ul'yanovsk State University

Leo Tolstoi str., 42, Ul'yanovsk,432970, Russia

 

In the work the problem on stability of the functional - differential equations with finite delay is considered. We assume that the right-hand side satisfies the uniform Lipshitz condition.љ From the assumptions one can find that the family of shifts of the right-hand side is precompact in a certain compact metric space of continuous functions,љ and the entire family of limiting equations corresponds to the initial system. We assume that there exists the non-negative Lyapunov function and the derivative of its satisfies the certain differential inequality on the segments of continuous functions which belong to hole. Hole is constructed by use the generalization of B.S.Razumikhin results which are otained in the work of K.Kato.

The development of a method of semi-definite Lyapunov functions љin a direction of use bothљ the scalar comparison equations and limiting functions and equations is offered. The theorems on stability and asymptotic stability of the zero solution are obtained. In the theorems the conditions for both comparison equation and Lyapunov function are weaken in comparison with conditions ofљ classic theorems. For example, in the theorem of asymptotic stability there is required the condition of stability of zero solution the comparison equation only, but not asymptotic stability. Convergence of solutions to the zero is archived by use the property of positive limit set.

On the base of proved theorems the problem on stabilization of stationary motion of rigid body with fixed point in mass centre is solved by use of control moments, defined with delay. The rotation is realized about the largest axis of inertia. The problem is solved by use both the method of transformations and construction of Lyapunov function in the form of Euclidean vector norm. The derivative of the function is estimated by use the method of logarithmic matrix norm.

peregudovaoa@sv.ulsu.ru