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We are concerned with the problems of stability and stabilization for stochastic Markovian jump systems subject to partially unknown transition probabilities and multiplicative noise, including the continuous- and discrete-time cases. Sufficient conditions guaranteeing systems considered to be asymptotically stable in the mean square are presented in the form of LMIs. Furthermore, the desired state feedback controllers are designed. It is shown that, by introducing the free-weighing matrix method, the results we have obtained not only are less conservative than the existing ones but also can be regarded as extensions of the corresponding results of Markovian jump systems without noise. Numerical examples are finally provided to illustrate the effectiveness of the proposed theoretical results.

Over the past years, considerable attention has been devoted to the study of a class of stochastic systems governed by Itô’s differential equation because of their extensive applications in some practical areas such as economics, finance, biology, and fault detection [

On the other hand, Markovian jump linear systems (MJLS), which are referred to as the stochastic systems with abrupt changes, have come to play an important role in practical applications owing to the powerful modeling capability of Markov chains [

In this paper, we will investigate problems of stability and stabilization for stochastic Markovian jump systems with partially unknown transition rates and multiplicative noise, including the continuous- and discrete-time cases. With the aid of the free-weighting matrices, a new stability criterion is established, which is less conservative than that in [

The outline of this paper is listed as follows: Section

Consider the following continuous- and discrete-time stochastic systems subject to Markov jump parameters and multiplicative noise, respectively:

In this paper, the transition rates of Markovian jump process are considered to be partially accessible. For example, the transition rate matrix

Unforced system (

In this section, the problems of stability and stabilization for stochastic Markovian jump systems with multiplicative noise and partially unknown transition rates in both continuous- and discrete-time cases are investigated. The state feedback controllers guaranteeing systems to be mean square stable are designed.

Before proceeding, let us first recall the stability results for system (

System (

Selecting a stochastic Lyapunov functional candidate

System (

Following the same line as done in Lemma

This section aims to develop a new stability criterion for system (

Unforced system (

It is noted that

On the other hand, if

Below, we further discuss the stabilization problem of system (

The closed-loop system (

Substituting the state feedback controller

If

It can be seen from Theorem

In this section, we focus our attention on the stability and stabilization problems for discrete-time stochastic Markovian jump systems subject to incomplete knowledge of transition probability and multiplicative noise. Sufficient conditions for the stability and stabilization of systems under consideration are formulated as LMIs.

System (

Because of

Next, we are set about to investigate the stabilization problem of system (

The closed-loop system (

Applying the state feedback controller

In this section, two numerical examples are proposed to demonstrate the effectiveness of our presented approaches, including the continuous- and discrete-time cases.

Consider the continuous-time stochastic Markov jumping system in the form of (

Based on Theorem

It was shown that the random packet loss and channel delay in the network control system are often modeled as Markov chains and the variation of delays and packet dropouts may be random in the different period of networks [

In this paper, the stability and stabilization problems for a class of stochastic Markovian jump linear systems (MJLS) with partly unknown transition rates have been studied. The LMI-based sufficient conditions ensuring systems considered to be stable are given in the continuous- and discrete-time cases. Numerical examples are provided to show the validness and applicability of the developed results.

The authors declare that they have no conflicts of interest.

This work was supported by NSF of China (Grant nos. 61573227 and 61703248), the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant no. LAPS16011), Shandong Provincial Natural Science Foundation, China (Grant no. ZR2015FM014), and the SDUST Research Fund no. 2015TDJH105.

_{∞}control of nonlinear systems with time-delay and state-dependent noise

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_{∞}control of discrete-time Markovian jump linear systems with mode-dependent time-delays

_{∞}control for discrete-time Markovian jump linear systems with partly unknown transition probabilities

_{∞}control of discrete-time Markovian jump systems in the presence of incomplete knowledge of transition probabilities and saturating actuator

_{∞}control for discrete-time Markovian jump systems with partial information on transition probabilities

_{∞}approach to networked control