Abstract

The form of weak infinitesimal operator of Lyapunov type on solutions of stochastic dynamic systems of random structure with constant delay which exist under the action of Markov perturbations is obtained.

1. Introduction

The explicit view of the Lyapunov operator is necessary for solving the problem of stabilization for stochastic dynamic systems. The view of the Lyapunov operator for different stabilization problems can be found in [1–7]. In particular, in [6], the view of the Lyapunov operator for stochastic diffusion dynamical system of random structure with Markov switching is obtained, in [7], for controlled stochastic dynamical system with impulse Markov switching and parameters.

The explicit view of infinitesimal operator (of Lyapunov type) for the solution of dynamic systems of random structure with Markov switching and constant delay is obtained in this article.

2. Formulation of the Problem

On the probabilistic basis [8] consider a random process , , given by stochastic differential-difference equation (SDDE)with Markov switchingand initial conditions

Here , , is Markov process with values in dimensional space ; , , is Markov chain with values in dimensional space ; is one-dimensional standard Wiener process [8–10]; , , , and   , are -mesurable and independent of each other; is Skorokhod’s space of continuous functions affair with left sided border [11] with the norm

Suppose that measurable maps ; ; satisfy Lipschitz condition for , , , , ; and next condition hold, , , moreover for

The existence and uniqueness of the solution of problem (1)–(3) are proved in [12] under conditions (5)–(7).

3. Calculation Infinitesimal Operator

Let be continuous in the set of variables and continuously differentiable in functional.

One can prove [13] that the pair , where , is a Markov process and we can consider the weak infinitesimal operator [9]where , , and , . It is natural to assume that the functional is in the domain of the operator , if the limit (8) exists in sense of uniform convergence in a neighborhood uniformly on that .

Let us introduce the operator , which is connected with Markov switching (2) at time , ,where is transition probability of Markov chain on th step and is indicator of set .

Calculate the operator on solutions of system (1)–(3).

Consider the following cases.

Case 1. Let , , be a Markov chain with a finite number of states and generator , .
At the time of changing of a structure of parameter of system is the stepwise changing of vector with transition probability : .

Theorem 1. The weak infinitesimal operator on solutions of system (1)–(3) of functional is calculated by the formulawhere ;where is scalar product, , is Frechet derivative at th coordinate of the vector , is Frechet second derivative matrix, is trace of matrix, , is calculated by formula (9), and is differentiable in functional that has Frechet derivative of the 1st and 2nd orders by the last variable.

Proof. By definition (see (8)) Further, That, can be represented as Consider each term separately. For the first term image is obvious.
Establish an explicit form of term . Consider a complete group of disjoin events constructed as follows: denote by event, when in the time interval structure of (1) is not changing; that is, for . Then without terms [14] we obtain following equality: Then denote by event, when in the time interval there is a change on . Then without terms we will have Denote by and by increment by doing . Calculate increments and of functional when the events , , , occurrence, ignoring the terms order :Here, all Frechet partial derivatives are calculated in , where is solution of (1) with initial conditions , , and .
Further, for , in case of change of structure on the time interval we will have an incrementwith probability .
The terms which illustrate the possibility of changing of structure of parameter are not included in last equality and there is no Markov switching. After averaging these terms will be equal to , and we can ignore them.
To calculate use the law of total probability where external expectation on the right side is calculated by variable in .
Ignoring terms, from (19) and (20), we get equalityIn calculation of the third term we used property and property of Wiener process about covariance of increment [8, 9].
Dividing on and passing to the limit on we obtained the first, second, and forth terms in (11). The idea of calculating the third term can be found in [15, pp. 163-164]. Theorem 1 is proved.

Case 2. Let in the time of changing of structure phase vector change by determined law , .

Theorem 2. Let the conditions of Theorem 1 hold; then the operator is calculated by the formula The proof is carried out under the scheme proof of Theorem 1.
If at the time of change of structure phase vector changes continuously .

Corollary 3. Under the conditions of Theorem 1

Case 3. Let , , be purely discontinuous Markov processes, allowing the decomposition of probability and at the time of jump phase vector changes continuously, leading to calculations where , , and are, respectively, calculated by formulas (9), (11), and (12).

4. Model Example

Consider SDDEwith switching,and initial conditions,Here ; is Markov process with values in the space with generator ; , , is Markov chain with values in the space with matrix of transition probabilities .

We assume that the phase vector changes continuously during , and functional is defined as , , .

Then WIO based system (27)–(29) have following form: where