#### Abstract

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 -stability of the trivial solution of the considered systems.

#### 1. Introduction

##### 1.1. The Aim of the Contribution

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 [1] it is explained why not every positive definite function can serve as a Lyapunov function for a system of differential equations. As experience shows, the most suitable Lyapunov functions have physical meaning. The Lyapunov function method is an effective method for the investigation of stability of linear or nonlinear differential systems that are explicitly independent of time (see, e.g., [1–9]). But there are no universal methods for constructing appropriate Lyapunov functions because, as well-known, in nonlinear differential systems, each case considered requires an individual method for constructing a Lyapunov function.

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 -stability of the trivial solution of the nonstationary differential (or difference) systems [10, 11], which is compatible with the method of Lyapunov functions.

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 -stability of the trivial solution of the differential systems with semi-Markov coefficients and give an analogy between the -stability and the stability obtained by Lyapunov functions. A new method of constructing Lyapunov functions is proposed for the study of stability of systems, and Lyapunov functions are derived for systems of differential equations with coefficients depending on a semi-Markov process. Sufficient conditions of stability are given, and it is proved that the condition of -stability implies the existence of Lyapunov functions. In addition to this, the case of the coefficients of the considered systems depending on Markov process is analyzed.

##### 1.2. Systems Considered

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 -dimensional differential system on the probability space . A vector-function , , is called a solution of (1.1) if is a random vector-function from the set of random vector-functions defined on , there exists mathematical expectation of , and (1.1) is satisfied for . The derivative is understood in the meaning of differentiability of a random process [12].

A space of solutions can be interpreted as a phase space of states of a random environment. Measurable subsets of a random environment form a collection of its states. As a phase space of states serves a complete metric separable space (as a rule the Euclidean space or a finite space equipped with -algebra of all subsets of ). Under assumptions of our problem (and in similar problems as well), solutions are defined in the meaning of a strong solution of the Cauchy problem [13].

Together with (1.1), we consider the initial condition In fact, any solution of (1.1) depends on the random variable , that is, .

The random process , , is a semi-Markov process with the states We assume where , and moments of jumps , , of the process are such that and , if , .

The transition from state to state is characterized by the intensity , , and the semi-Markov process is defined by the intensity matrix whose elements satisfy the relationships Let mutually different functions , , be defined for , .

*Definition 1.1. *The function is called a semi-Markov function if the equalities
hold for .

It means that the semi-Markov function is a functional of a random process . The value of is determined by the values at the time and also by the value of the jump of the process at time , which precedes time . In fact, the system (1.1) means different differential systems in the form where We assume that there exists a unique solution of (1.7) for every point such that , ( stands for Euclidean norm), continuable on .

##### 1.3. Auxiliaries

In the paper, in addition to what was mentioned above, the following notations and assumptions are introduced:(1) the functions , , are Lipschitz functions with the Lipschitz constants , that is, the inequalities hold.(2) If , then (3) The inequalities are valid. Here , are the Lipschitz constants, , are positive constants, and , is the solution of (1.7) in the Cauchy form, that is, (4) We introduce the Lyapunov functional where denotes mathematical expectation, and we assume that the integral is convergent.

*Definition 1.2. * The trivial solution of the differential systems (1.1) is said to be -stable if, for any solution with bounded initial values of the mathematical expectation
the integral
converges.

*Remark 1.3. *It is easy to see that (1.15) converges if and only if the matrix integral
is convergent.

Lemma 1.4. *Let the function be bounded, that is, there exists a constant such that the inequalities
**
or the inequalities
**
hold where is a positive definite and differentiable function satisfying the inequality
**
Let, moreover, the Lyapunov functional (1.13) exist for the system (1.7) with an -stable trivial solution.**Then the Lyapunov functional (1.13) can be expressed in the form
**
if the particular Lyapunov functions
**
are known. *

*Proof. *The functions , , will be defined using auxiliary functions
The mathematical expectation in (1.22) can be calculated by the transition intensities , ,
whence the system
is obtained. Integrating the system of equations (1.24) with respect to , we get the system of functional equations
for
The system (1.25) thus obtained can be solved by successive approximations

#### 2. Main Results

##### 2.1. The Case of a Semi-Markovian Random Process

Theorem 2.1. *Let the functions , , in the system (1.7) satisfy conditions (1.9), (1.11), let the semi-Markov process be determined by the transition intensities ,, satisfying (1.5), and let the functions satisfy (1.17). Then the following statements are true. *(1)* The relationships
imply that, for the system (1.7), the particular Lyapunov functions can be established in the form
*(2)* If the spectral radius of the matrix is less than one, then the particular Lyapunov functions , , can be found by the method of successive approximations (1.27). *(3)* Under assumption (1.11), the method of successive approximations (1.27) converges and the inequalities
hold. Then the sequence of functions , , is monotone increasing and bounded from above by the functions . *

* Proof. *Applying estimation (2.1) and assumption (2.2) to the successive approximations (1.27), we get
It is sufficient to assume the existence of a bounded solution of the system of inequalities (2.1) whence the existence follows of a positive solution of the system of linear algebraic equations (2.3). Moreover, assumption (2.2) guarantees the convergence of the improper integrals in the system (1.27) and so, for the existence of a positive solution of the system (2.3), it is sufficient that the spectral radius of the matrix
is less than one. For this, it is sufficient that
The convergence of the sequence , , can be determined by the system
If there exist the inequalities
where
hold, then estimation (2.4) is true for all , , .

Under assumptions (2.8), it follows
which implies a uniform convergence of the sequence , , .

Corollary 2.2. *If the trivial solution of the differential systems (1.1) is -stable, then there exist particular Lyapunov functions , that satisfy (2.3). *

Corollary 2.3. *Let the function satisfy the inequality:
**
If there exist the Lyapunov functions , for the system (1.21), then the trivial solution of the differential systems (1.1) is -stable. *

Corollary 2.4. *Let the semi-Markov process in the system (1.1) have jumps at the times , , , in the transition from state to state , and let the jumps satisfy the equation
**
where are any continuous Lipschitz vector functions. Then the system (2.3) has the form
**
and its solution can be found by the method of successive approximations. *

##### 2.2. The Case of a Markovian Random Process

Next result relates to the case of the semi-Markov process being transformed into a Markov process described by the system of ordinary differential equations: under the influence of which the considered system takes the form We also assume that, if , , then Then the system of equations (2.14) has the form

Theorem 2.5. *Let the nonlinear differential system (1.1), depending on the Markov process , be described by (2.15). Then the particular Lyapunov functions satisfy the linear differential system:
*

*Proof. * Let us write the solution of the system (2.17) in the Cauchy form:
Differentiating (2.21) with respect to , we get
which, for , , takes the form:
Then
which implies (2.20) if
and the functions , , are differentiable.

Corollary 2.6. *If the solutions of the system (2.16) have the same jumps as the solution of the system (2.14) and converge to the jumps of the Markov process such that , , then the system (2.19) takes the form
**
and the system (2.20) takes the form
*

*Example 2.7. *Let us investigate the stability of solutions of two-dimensional system
where is a random Markov process having two states , with probabilities , , that satisfy the equations
where . The random matrix function is known:
Taking the positive definite functions
we can verify that the positive definite particular Lyapunov functions
are the solutions to (2.20). Consequently, since the integral (1.13)
is convergent, the zero solution of the considered system is -stable.

#### Acknowledgments

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).