New results for exponential stability in probability of a composite stochastic control system are established. The main results of this paper enable us to derive sufficient conditions for exponential stability in

The aim of this paper is to study the exponential stability in probability in

We consider the composite system (

In recent years, the stablizability of various types of stochastic systems has been studied for different concepts of stochastic stability (see, for instance, [

The structure of the paper is as follows. In Section

In this section, we introduce the class of stochastic systems and recall some definitions and results concerning exponential stability in probability of these systems. For a complete presentation of exponential stability, we refer the reader to the book of Khasminiskii [

Let

We consider the SDS

Under restriction on growth (

The origin of the SDS (

Suppose that the origin of the SDS (

Let us denote by

Now, we consider the SCS

The origin of the SCS (

The SCS (

Suppose that the origin of the SCS (

Note that Definition

The origin of the SCS (

We now derive the Florchinger's decomposition [

Let

Then, the feedback law

Suppose that there exists a positive function

We will now turn the attention to a general composite stochastic system and provide some results related to the rESP of this system.

Let

Consider the pair of stochastic processes solution

for any

We say that an input measurable function

Suppose that there exists functionals

Our aims of this section are twofold. On one hand, we study the problem of finding state feedback law that guarantees that the CSCS (

In the following theorem, we suppose that the function

Let

Suppose that the origin of the stochastic system (

In the following theorem, we show that existence of a Lyapunov function satisfying the exponential Lyapunov condition implies rESP and asESP of the CSCS (

Assume that there exists a

the origin of the CSCS (

the origin of the CSCS (

Part (i) Assume that there exists a positive function

Part (ii) Suppose that there exists a positive composite function

Comparing the existing results in [

Theorem

The rESP and asESP results for stochastic system proved in Spiliotis and Tsinias [

This section illustrates applicability of our results by designing the following two numerical examples.

Consider the SDS

Next, consider the SCS

Consider the CSCS

In this paper, we have provided the new results for rESP and asESP of the CSCS