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The paper deals with nonlinear differential systems with random parameters in a general form. A new method for construction of the Lyapunov functions is proposed and is used to obtain sufficient conditions for

The method of Lyapunov functions is one of the most effective methods for investigation of self-regulating systems. It is important for determining the fact of stability or instability of given systems among other purposes. A successfully constructed Lyapunov function for given nonlinear self-regulating systems makes it possible to solve all the complex problems important in practical applications such as estimation of changes of a self-regulated variable, estimation of transient processes, estimation of integral criteria of the quality of self-regulation, or estimation of what is called guaranteed domain of stability.

In [

However, the method of Lyapunov functions is often difficult to apply to the investigation of some kinds of stability of nonstationary differential systems because the concept of Lyapunov stability can make the Lyapunov functions inconvenient to use. This problem was solved by a new definition of what is called

In this paper, we deal with much more complicated investigation of the Lyapunov stability of differential systems with random parameters. We define a concept of

In this part, a new concept of semi-Markov function is proposed. It will be used later for the construction of Lyapunov functions.

Consider nonlinear

A space of solutions

Together with (

The random process

The transition from state

The function

It means that the semi-Markov function

In the paper, in addition to what was mentioned above, the following notations and assumptions are introduced:

the functions

If

The inequalities

We introduce the Lyapunov functional

The trivial solution of the differential systems (

It is easy to see that (

Let the function

Then the Lyapunov functional (

The functions

Let the functions

The relationships

If the spectral radius of the matrix

Under assumption (

Applying estimation (

Under assumptions (

If the trivial solution of the differential systems (

Let the function

Let the semi-Markov process

Next result relates to the case of the semi-Markov process

Let the nonlinear differential system (

Let us write the solution of the system (

If the solutions of the system (

Let us investigate the stability of solutions of two-dimensional system

This paper was supported by Grant P201/11/0768 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00 216 30519, and by Grant no 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).