This paper considers the finite time stability of stochastic hybrid systems, which has both Markovian switching and impulsive effect. First, the concept of finite time stability is extended to stochastic hybrid systems. Then, by using common Lyapunov function and multiple Lyapunov functions theory, two sufficient conditions for finite time stability of stochastic hybrid systems are presented. Furthermore, a new notion called stochastic minimum dwell time is proposed and then, combining it with the method of multiple Lyapunov functions, a sufficient condition for finite time stability of stochastic hybrid systems is given. Finally, a numerical example is provided to illustrate the theoretical results.

Nowadays, stochastic modeling, control, and optimization have played a crucial role in many applications especially in the areas of controlling science and communication technology [

In many applications, it is desirable that the stochastic system possesses the property that trajectories which converge to a Lyapunov stable equilibrium state must do so in finite time rather than infinite time. Hence, the concept of finite time stability for stochastic systems arises naturally in stochastic control problems. For the deterministic case, finite time stability for continuous time systems was studied in [

In this paper, we will study the finite time stability of stochastic hybrid systems, which has both Markovian switching and impulsive effect. The main contributions of this paper include the following: (i) extend the concept of finite time stability to stochastic hybrid systems, (ii) present two sufficient conditions for finite time stability of stochastic hybrid systems by using common Lyapunov function and multiple Lyapunov functions theory, (iii) introduce a new notion called stochastic minimum dwell time (SMDT), and (iv) propose a sufficient condition for finite time stability of stochastic hybrid systems by combining SMDT with the method of multiple Lyapunov functions.

The organization of the paper is as follows. In Section

Throughout this paper, we let

Let

A function

In this paper, we will consider a stochastic hybrid system with

To ensure the existence and uniqueness of solutions for (

Assume that for any

Assume that, for any

Under Assumptions

It is known that the existence of a unique solution to (

In mathematics, we need that functions

Next, we will extend the concept of finite time stability to the stochastic hybrid system (

The trivial solution of (

Finite time attractiveness in probability: For every initial value

Stability in probability: for every pair of

whenever

In [

Let

For the convenience of the reader we cite the generalized Itô formula established by [

If

We will show next how finite time stability can be indirectly determined by studying the probability associated with a function

In this section, we will extend the existing results to study the finite time stability for the stochastic hybrid system (

Let

function

for any

where

Let

Let

Therefore, we have that

In Lemma

In mathematics, we allow that the function

Consider the system (

The proof follows the same line of the proof of [

If all conditions of Theorem

Theorem

Let

Assume that there exist a constant

for any

Using similar arguments of Lemma

Lemma

Consider the system (

for any

By combining Lemma

Based on Theorem

Before giving this sufficient condition, we need the following definition and lemma.

For a given Markov chain

For a given Markov chain

Using Definition

We now state another sufficient condition for finite time stability of system (

Consider the system (

for any

By combining Theorem

In Theorem

The mathematical conclusion of Theorem

In this section, we will present an example to illustrate the theoretical results.

It is not hard to verify that all conditions in Assumption

Now, by choosing

The state

The Markov chain

It is worth noting that, although each subsystem of (

The state

Our simulation results are obtained by using the Matlab (version 6.5) programming, where all the differential equations are solved through using the improved Runge-Kutta algorithm. Comparing with the existing methods (such as ode23 and Euler method), our method has higher precision and less simulation time. The accuracy and simulation time of other methods (such as ode23, Euler method) are 10^{−3}~10^{−4} and 50~70 minutes, respectively, while the accuracy and simulation time of our algorithm are 10^{−5} and 36 minutes, respectively.

The issues of finite time stability for stochastic hybrid systems have been studied and corresponding results have been presented. Based on common Lyapunov function and multiple Lyapunov functions theory, two sufficient conditions for finite time stability of systems have been derived. Furthermore, a new notion called

Future research directions include the research for more relaxed conditions for finite time stability and the applications of the results presented here to packet-dropping problems in network control systems and time-delayed systems.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Natural Science Foundation of Guangdong Province, China (S2011040003733), National Natural Science Foundation of China (60974139), and Science and Technology Project of Huizhou (2012P10).