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This paper is concerned with the

On the other hand, Itô stochastic systems with Markov jumps have attracted increasing attention due to their powerful modeling ability in many fields [

Most of the existing literature was concerned with stochastic Markov jump systems with state-dependent noise or both state and disturbance dependent noise (

In this paper, the finite and infinite horizon

For conveniences, we make use of the following notations throughout this paper.

Consider the following time-varying nonlinear stochastic Markov jump systems with

For given

Consider the time-invariant nonlinear stochastic Markov jump systems with

For given

For the zero initial state and any nonzero

System (

To give our main results, we need the following lemmas.

(Generalized Itô formula). Let

If

This lemma is very easily proved by using completing squares technique, so the proof is omitted.

The following sufficient condition is presented for the finite horizon

Assume that there exists a set of nonnegative functions

For any

Applying Lemma

Substituting (

The proof of Theorem

In contrast to the finite horizon case, the infinite horizon

Assume that there exists a set of nonnegative functions

Similar to the proof of Theorem

Implementing (

The methods proposed in [

When there is no Markov jump parameters, system (

From Theorem

For a prescribed

Letting

Next, we present a sufficient condition for the following linear stochastic Markov jump systems with

System (

Firstly, consider the following unforced linear stochastic Markov jump systems:

Let

By Schur complement, (

Considering closed-loop system (

Although HJIs (

In this section, two numerical examples are provided to illustrate the effectiveness of the developed results.

Consider one-dimensional two-mode time-invariant nonlinear stochastic Markov jump systems with generator matrix

Consider two-dimensional two-mode linear stochastic Markov jump systems (

Figure

Result of the changing between modes.

The state responses of unforced system (

The state responses of controlled system (

In this paper, we have studied the

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

This work is supported by National Natural Science Foundation of China (nos. 61203053 and 61174078), China Postdoctoral Science Foundation (no. 2013M531635), Research Fund for the Doctoral Program of Higher Education of China (no. 20120133120014), Special Funds for Postdoctoral Innovative Projects of Shandong Province (no. 201203096), Fundamental Research Fund for the Central Universities (nos. 11CX04042A, 12CX02010A, and 14CX02093A), Research Fund for the Taishan Scholar Project of Shandong Province of China, and SDUST Research Fund (no. 2011KYTD105).