Nonlinear Stochastic 𝐻_∞ Control with Markov Jumps and (𝑥,𝑢,𝑣)-Dependent Noise: Finite and Infinite Horizon Cases

Abstract

This paper is concerned with the 𝐻_∞ control problem for nonlinear stochastic Markov jump systems with state, control, and external disturbance-dependent noise. By means of inequality techniques and coupled Hamilton-Jacobi inequalities, both finite and infinite horizon 𝐻_∞ control designs of such systems are developed. Two numerical examples are provided to illustrate the effectiveness of the proposed design method.

1. Introduction

𝐻_∞ control is one of the most important robust control approaches, which can efficiently eliminate the effect of the exogenous disturbance [1, 2]. Since Hinrichsen and Pritchard introduced 𝐻_∞ control to linear stochastic systems [3], the nonlinear stochastic 𝐻_∞ control and filtering problems have received considerable attention in both theory and practical applications [4–9]. In [4], the nonlinear stochastic 𝐻_∞ designs were first developed by solving a second-order nonlinear Hamilton-Jacobi inequality. The 𝐻_∞ filtering problems for general nonlinear continuous-time and discrete-time stochastic systems were discussed in [6] and [7], respectively. In [8], the quantized 𝐻_∞ control problem for a class of nonlinear stochastic time-delay network-based systems with probabilistic data missing is studied.

On the other hand, Itô stochastic systems with Markov jumps have attracted increasing attention due to their powerful modeling ability in many fields [10, 11]. For linear stochastic systems with Markov jumps, many important issues have been studied, such as stability and stabilization [12, 13], observability and detectability [14], optimal control [15], and 𝐻₂/𝐻_∞ control [16]. The control issues for nonlinear stochastic Markov jump systems (NSMJSs) have also been widely investigated. In [17], the notion of exponential dissipativity of NSMJSs was introduced, and it was used to estimate the possible variations of the output feedback control. In [18], the stabilization of nonlinear Markov jump systems with partly unknown transition probabilities was studied via fuzzy control. In [19], the 𝐻_∞ control problems of NSMJSs were studied, which extended the results of [4] to stochastic systems with Markovian jump parameters.

Most of the existing literature was concerned with stochastic Markov jump systems with state-dependent noise or both state and disturbance dependent noise ((𝑥,𝑣)-dependent noise for short) [16, 19]. However, for most natural phenomena described by Itô stochastic systems, not only state but also control input or external disturbance maybe corrupted by noise. By introducing three coupled Hamilton-Jacobi equations (HJEs), the finite and infinite horizon 𝐻_∞ control problems were solved for Itô stochastic systems with all system state, control, and disturbance-dependent noise ((𝑥,𝑢,𝑣)-dependent noise for short) in [20] and [21], respectively. In [22], the finite/infinite horizon 𝐻_∞ control of nonlinear stochastic systems with (𝑥,𝑢,𝑣)-dependent noise was solved by means of a Hamilton-Jacobi inequality (HJI) instead of three coupled Hamilton-Jacobi equations (HJEs). However, the control problems of nonlinear stochastic systems with Markov jumps and (𝑥,𝑢,𝑣)-dependent noise have never been tackled and deserved further research.

In this paper, the finite and infinite horizon 𝐻_∞ control problems are studied for nonlinear stochastic Markov jumps systems with (𝑥,𝑢,𝑣)-dependent noise. Firstly, a very useful elementary identity is proposed. Then, by using the technique of squares completion, a sufficient condition for finite/infinite horizon 𝐻_∞ control of NSMJSs is presented based on a set of coupled HJIs. By means of linear matrix inequalities (LMIs), a sufficient condition for infinite horizon 𝐻_∞ control of linear stochastic Markov jump systems is derived. Finally, two numerical examples are provided to show the effectiveness of our obtained results.

For conveniences, we make use of the following notations throughout this paper. ℛ^𝑛 is the 𝑛-dimensional Euclidean space. ℛ^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices. 𝐴>0(𝐴≥0): 𝐴 is a positive definite (positive semi-definite) symmetric matrix. 𝐴^′ is the transpose of a matrix 𝐴. 𝐼 is the identity matrix. ‖𝑥‖ is the Euclidean norm of a vector 𝑥. 𝐿²_ℱ([0,𝑇];ℛ^𝑙) (resp., 𝐿²_ℱ(ℛ₊;ℛ^𝑙)) is the space of nonanticipative stochastic processes 𝑦(𝑡)∈ℛ^𝑙 with respect to increasing 𝜎-algebras ℱ_𝑡(𝑡≥0) satisfying ‖𝑦(𝑡)‖_{𝐿2ℱ([0,𝑇];ℛ}^𝑙₎=(𝐸∫^𝑇₀‖𝑦(𝑡)‖²𝑑𝑡)^1/2<∞ (resp., ‖𝑦(𝑡)‖_{𝐿2ℱ(ℛ+;ℛ}^𝑙₎=(𝐸∫^∞₀‖𝑦(𝑡)‖²𝑑𝑡)^1/2<∞).

2. Definitions and Preliminaries

Consider the following time-varying nonlinear stochastic Markov jump systems with (𝑥,𝑢,𝑣)-dependent noise: 𝑑𝑥(𝑡)=𝑓𝑡,𝑥,𝑟_𝑡+𝑔𝑡,𝑥,𝑟_𝑡𝑢(𝑡)+ℎ𝑡,𝑥,𝑟_𝑡𝑣(𝑡+𝑙)𝑑𝑡𝑡,𝑥,𝑟_𝑡+𝑞𝑡,𝑥,𝑟_𝑡𝑢(𝑡)+𝑠𝑡,𝑥,𝑟_𝑡𝑣(𝑡(𝑡),𝑧(𝑡)=)𝑑𝑤col𝑚𝑡,𝑥,𝑟_𝑡(𝑡)𝑚,𝑢∶=𝑡,𝑥,𝑟_𝑡𝑢(𝑡),𝑥(0)=𝑥₀∈ℛ^𝑛, (1) where 𝑥(𝑡)∈ℛ^𝑛, 𝑢(𝑡)∈ℛ^𝑛𝑢, 𝑣(𝑡)∈ℛ^𝑛𝑣, and 𝑧(𝑡)∈ℛ^𝑛𝑧 represent the system state, control input, exogenous input, and regulated output, respectively. 𝑤(𝑡) is the one-dimensional standard Wiener process defined on the complete probability space (Ω,ℱ,𝑃), with the natural filter ℱ_𝑡 generated by 𝑤(⋅) and 𝑟(⋅) up to time 𝑡. The jumping process {𝑟_𝑡,𝑡≥0} is a continuous-time discrete-state Markov process taking values in a finite set 𝒯={1,…,𝑁}. The transition probabilities for the process 𝑟_𝑡 are defined as 𝑃𝑟_𝑡+ℎ=𝑗∣𝑟_𝑡=𝜋=𝑘^𝑘𝑙ℎ+𝑜(ℎ),if𝑘≠𝑗,1+𝜋_𝑘𝑘ℎ+𝑜(ℎ),if𝑘=𝑗, (2) where ℎ>0, lim_ℎ→0(𝑜(ℎ)/ℎ)=0, and 𝜋_𝑘𝑙≥0(𝑘,𝑗∈𝒯,𝑘≠𝑗) denotes the switching rate from mode 𝑘 at time 𝑡 to mode 𝑗 at time 𝑡+ℎ and 𝜋_𝑘𝑘=−∑_{𝑗=1,𝑗≠𝑘}𝜋_𝑘𝑗 for all 𝑘∈𝒯. In this paper, the processes 𝑟_𝑡 and 𝑤(𝑡) are assumed to be independent. For every 𝑟_𝑡=𝑘∈𝒯, 𝑓(𝑡,𝑥,𝑘), 𝑔(𝑡,𝑥,𝑘), ℎ(𝑡,𝑥,𝑘), 𝑙(𝑡,𝑥,𝑘), 𝑞(𝑡,𝑥,𝑘), 𝑠(𝑡,𝑥,𝑘), and 𝑚(𝑡,𝑥,𝑘) are Borel measurable functions of suitable dimensions, which guarantee that system (1) has a unique strong solution [23]. The finite horizon 𝐻_∞ control for system (1) is defined as follows.

Definition 1. For given 𝛾>0, the state feedback control 𝑢^∗_𝑇(𝑡)∈𝐿²_ℱ([0,𝑇];ℛ^𝑛𝑢) is called a finite horizon 𝐻_∞ control of system (1), if for the zero initial state and any nonzero 𝑣(𝑡)∈𝐿²_ℱ([0,𝑇];ℛ^𝑛𝑣), we have ‖ℒ_𝑇‖<𝛾 with ℒ_𝑇=sup_{𝑣(𝑡)∈ℒ2ℱ][0,𝑇;ℛ𝑛𝑣,𝑣(𝑡)(𝑡)≠0,𝑢=𝑢∗𝑇(𝑡),𝑥0=0}‖𝑧(𝑡)‖_𝐿²_{ℱ([0,𝑇];ℛ}^𝑛_𝑧)‖𝑣(𝑡)‖_𝐿²_{ℱ([0,𝑇];ℛ}^𝑛_𝑣)=sup_{𝑣(𝑡)∈ℒ2ℱ][0,𝑇;ℛ𝑛𝑣,𝑣(𝑡)≠0,𝑢(𝑡)=𝑢∗𝑇(𝑡),𝑥0=0}𝐸^𝑇₀𝑚𝑡,𝑥,𝑟_𝑡²+𝑢^∗_𝑇(𝑡)²𝑑𝑡∣𝑟₀=𝑘^1/2×𝐸^𝑇₀‖𝑣(𝑡)‖²𝑑𝑡∣𝑟₀=𝑘^1/2−1,𝑖∈𝒯, (3) where ℒ_𝑇 is an operator associated with system (1) which is defined as ℒ_𝑇∶ℒ²_ℱ[0,𝑇];ℛ^𝑛𝑣⟼ℒ²_ℱ[0,𝑇];ℛ^𝑛𝑧,ℒ_𝑇(𝑣(𝑡))=𝑧(𝑡)|_𝑥0=0,𝑡∈[0,𝑇]. (4)

Consider the time-invariant nonlinear stochastic Markov jump systems with (𝑥,𝑢,𝑣)-dependent noise 𝑑𝑥(𝑡)=𝑓𝑥,𝑟_𝑡+𝑔𝑥,𝑟_𝑡𝑢(𝑡)+ℎ𝑥,𝑟_𝑡𝑣(𝑡+𝑙)𝑑𝑡𝑥,𝑟_𝑡+𝑞𝑥,𝑟_𝑡𝑢(𝑡)+𝑠𝑥,𝑟_𝑡𝑣(𝑡(𝑡),𝑧(𝑡)=)𝑑𝑤col𝑚𝑥,𝑟_𝑡(𝑡)𝑚,𝑢∶=𝑥,𝑟_𝑡𝑢(𝑡),𝑥(0)=𝑥₀∈ℛ^𝑛. (5) The infinite horizon 𝐻_∞ control for system (5) is defined as follows.

Definition 2. For given 𝛾>0, the control 𝑢^∗_∞(𝑡)∈𝐿²_ℱ(ℛ₊;ℛ^𝑛𝑢) is called an infinite horizon 𝐻_∞ control of system (1), if the following is considered.(i)For the zero initial state and any nonzero 𝑣(𝑡)∈𝐿²_ℱ(ℛ₊;ℛ^𝑛𝑣), we have ‖ℒ_∞‖<𝛾 with ℒ_∞=sup_{𝑣(𝑡)∈ℒ2ℱℛ+;ℛ𝑛𝑣,𝑣(𝑡)(𝑡)≠0,𝑢=𝑢∗∞(𝑡),𝑥0=0}‖𝑧(𝑡)‖_𝐿²_ℱ(ℛ+;ℛ^𝑛_𝑧)‖𝑣(𝑡)‖_𝐿²_ℱ(ℛ+;ℛ^𝑛_𝑣)=sup_{𝑣(𝑡)∈ℒ2ℱℛ+;ℛ𝑛𝑣,𝑣(𝑡)(𝑡)≠0,𝑢=𝑢∗∞(𝑡),𝑥0=0}𝐸^∞₀𝑚𝑥,𝑟_𝑡²+𝑢^∗_∞(𝑡)²𝑑𝑡∣𝑟₀=𝑘^1/2×𝐸^∞₀‖𝑣(𝑡)‖²𝑑𝑡∣𝑟₀=𝑘^1/2−1,𝑖∈𝒯, (6) where ℒ_∞ is an operator associated with system (5) which is defined as ℒ_∞∶ℒ²_ℱℛ₊;ℛ^𝑛𝑣⟼ℒ²_ℱℛ₊;ℛ^𝑛𝑧,ℒ_∞(𝑣(𝑡))=𝑧(𝑡)∣_𝑥0=0,𝑡∈ℛ₊. (7)(ii)System (5) is internally stable; that is, the following system 𝑑𝑥(𝑡)=𝑓𝑥,𝑟_𝑡+𝑔𝑥,𝑟_𝑡𝑢^∗_∞𝑥,𝑟_𝑡+𝑙𝑑𝑡𝑥,𝑟_𝑡+𝑞𝑥,𝑟_𝑡𝑢^∗_∞𝑥,𝑟_𝑡𝑑𝑤(𝑡), (8)is globally asymptotically stable in probability [10].

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

Lemma 3 (see [10]). (Generalized Itô formula). Let 𝛼(𝑡,𝑥,𝑘) and 𝛽(𝑡,𝑥,𝑘) be given ℛ^𝑛-valued, ℱ_𝑡-adapted process, 𝑘∈𝒯, and 𝑑𝑥(𝑡)=𝛼(𝑡,𝑥(𝑡),𝑟_𝑡)𝑑𝑡+𝛽(𝑡,𝑥(𝑡),𝑟_𝑡)𝑑𝑤(𝑡). Then for given 𝑉(𝑡,𝑥,𝑘)∈𝒞^1,2([0,𝑇);ℛ^𝑛), 𝑘∈𝒯, we have 𝐸𝑉𝑇,𝑥(𝑇),𝑟_𝑇−𝑉𝑠,𝑥(𝑠),𝑟_𝑠∣𝑟_𝑠=𝑘=𝐸^𝑇_𝑠ℒ_𝑇𝑉𝑡,𝑥(𝑡),𝑟_𝑡𝑑𝑡∣𝑟_𝑠=𝑘, (9) where ℒ_𝑇𝑉()=()𝑡,𝑥,𝑘𝜕𝑉𝑡,𝑥,𝑘+𝜕𝑡𝜕𝑉^′(𝑡,𝑥,𝑘)𝛼()+𝜕𝑥𝑡,𝑥,𝑘_𝑁𝑗=1𝜋_𝑘𝑗𝑉()+1𝑡,𝑥,𝑘2𝛽^′(𝑡,𝑥,𝑘)𝜕𝑉²(𝑡,𝑥,𝑘)𝜕𝑥²𝛽(𝑡,𝑥,𝑘). (10)

Lemma 4. If 𝑥,𝑏∈ℛ^𝑛, 𝐴∈ℛ^𝑛×𝑛 is a symmetric matrix and 𝐴⁻¹ exists, we have 𝑥^′𝐴𝑥+𝑥^′𝑏+𝑏^′𝑥=𝑥+𝐴⁻¹𝑏^′𝐴𝑥+𝐴⁻¹𝑏−𝑏^′𝐴⁻¹𝑏. (11)

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

3. Main Results

3.1. Finite Horizon Case

The following sufficient condition is presented for the finite horizon 𝐻_∞ control of system (1). For convenience, denote (⋅)_𝑘∶=(⋅)(𝑡,𝑥,𝑘) in this subsection.

Theorem 5. Assume that there exists a set of nonnegative functions 𝑉(𝑡,𝑥,𝑘)∈𝒞^1,2([0,𝑇]×ℛ^𝑛×𝒯;ℛ), 𝑉(0,0,𝑘)=0, and 𝜕²𝑉(𝑡,𝑥,𝑘)/𝜕𝑥²≥0 for all nonzero 𝑥∈ℛ^𝑛, 𝑘∈𝒯. If 𝑉(𝑡,𝑥,𝑘) solves the following coupled HJIs: Δ_𝑘=𝜕𝑉_𝑘+𝜕𝑡𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑚^′_𝑘𝑚_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗+14𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘+𝜕𝑉^′_𝑘ℎ𝜕𝑥^𝑘𝛾²𝐼−𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘⁻¹×𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+ℎ^′_𝑘𝜕𝑉_𝑘−1𝜕𝑥4𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘+𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹×𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘𝛾𝜕𝑥<0,₂𝐼−𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘>0,𝑉(𝑇,𝑥,𝑘)=0, (12) then 𝑢^∗_𝑇()1𝑡,𝑥,𝑘=−2𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘𝜕𝑥 (13) is a finite horizon 𝐻_∞ control of system (1).

Proof. For any 𝑇>0 and the initial state 𝑥₀=0, 𝑟₀=𝑘, applying Lemma 3, we have 𝐸𝑉𝑥(𝑇),𝑟_𝑇−𝑉0,𝑟₀∣𝑟₀=𝑘=𝐸^𝑇₀𝜕𝑉_𝑟𝑡+𝜕𝑡𝜕𝑉^′_𝑟𝑡𝑓𝜕𝑥^𝑟𝑡+𝑔_𝑟𝑡𝑢+ℎ_𝑟𝑡𝑣+12𝑙_𝑟𝑡+𝑞_𝑟𝑡𝑢+𝑠_𝑟𝑡𝑣^′𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+𝑞_𝑟𝑡𝑢+𝑠_𝑟𝑡𝑣+𝑚_𝑟𝑡²+‖𝑢‖²−𝛾²‖𝑣‖²−‖𝑧‖²+𝛾²‖𝑣‖²+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗𝑑𝑡∣𝑟₀=𝑘=𝐸^𝑇₀𝜕𝑉_𝑟𝑡−‖𝑧‖𝜕𝑡²+𝛾²‖𝑣‖²+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗+Θ₁𝑡,𝑣,𝑥,𝑟_𝑡+Θ₂𝑡,𝑥,𝑟_𝑡+Θ₃𝑡,𝑢,𝑥,𝑟_𝑡+12𝑢^′𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑣+𝑣^′𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡𝑢𝑑𝑡∣𝑟₀=𝑘, (14) where (⋅)_𝑟𝑡 denotes (⋅)(𝑡,𝑥,𝑟_𝑡) and Θ₁𝑡,𝑣,𝑥,𝑟_𝑡=𝑣^′−𝛾²1𝐼+2𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑣+12𝑙^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡+𝜕𝑉^′_𝑟𝑡ℎ𝜕𝑥^𝑟𝑡𝑣+12𝑣^′𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+ℎ^′_𝑟𝑡𝜕𝑉_𝑟𝑡,Θ𝜕𝑥₂𝑡,𝑥,𝑟_𝑡=𝜕𝑉_𝑟𝑡+𝜕𝑡𝜕𝑉^′_𝑟𝑡𝑓𝜕𝑥^𝑟𝑡+12𝑙^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+𝑚^′_𝑟𝑡𝑚_𝑟𝑡,Θ₃𝑡,𝑢,𝑥,𝑟_𝑡=𝑢^′1𝐼+2𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡𝑢+12𝑙^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡+𝜕𝑉^′_𝑟𝑡𝑔𝜕𝑥^𝑟𝑡𝑢+12𝑢^′𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+𝑔^′_𝑟𝑡𝜕𝑉_𝑟𝑡.𝜕𝑥 (15) Since 𝜕²𝑉_𝑘/𝜕𝑥²≥0, 𝑘∈𝒯, we have 12−𝑢^′𝑞^′_𝑟𝑡+𝑣^′𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²−𝑞_𝑟𝑡𝑢+𝑠_𝑟𝑡𝑣≥0, (16) which means that 12𝑢^′𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑣+𝑣^′𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡𝑢≤12𝑢^′𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡1𝑢+2𝑣^′𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑣. (17) Considering the above inequality and (14), we have 𝐸𝑉𝑥(𝑇),𝑟_𝑇−𝑉0,𝑟₀∣𝑟₀=𝑘≤𝐸^𝑇₀Θ₁𝑡,𝑣,𝑥,𝑟_𝑡+Θ₂𝑡,𝑥,𝑟_𝑡+Θ₃𝑡,𝑢,𝑥,𝑟_𝑡+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉𝑥,𝑗−‖𝑧‖²+𝛾²‖𝑣‖²+12𝑢^′𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡𝑣+𝑣^′𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑢𝑑𝑡∣𝑟₀=𝑘=𝐸^𝑇₀Θ₁𝑡,𝑣,𝑥,𝑟_𝑡+Θ₂𝑡,𝑥,𝑟_𝑡+Θ₃𝑡,𝑢,𝑥,𝑟_𝑡+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉𝑥,𝑗−‖𝑧‖²+𝛾²‖𝑣‖²𝑑𝑡∣𝑟₀=𝑘, (18) where Θ₁𝑣,𝑥,𝑟_𝑡=𝑣^′−𝛾²𝐼+𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑣+12𝑙^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡+𝜕𝑉^′_𝑟𝑡ℎ𝜕𝑥^𝑟𝑡𝑣+12𝑣^′𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+ℎ^′_𝑟𝑡𝜕𝑉_𝑟𝑡.𝜕𝑥Θ₃𝑡,𝑢,𝑥,𝑟_𝑡=𝑢^′𝐼+𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡𝑢+12𝑙^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡+𝜕𝑉^′_𝑟𝑡𝑔𝜕𝑥^𝑟𝑡𝑢+12𝑢^′𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+𝑔^′_𝑟𝑡𝜕𝑉_𝑟𝑡.𝜕𝑥 (19)
Applying Lemma 4 to Θ₁(𝑡,𝑣,𝑥,𝑟_𝑡) and Θ₃(𝑡,𝑢,𝑥,𝑟_𝑡), we have Θ₁𝑡,𝑣,𝑥,𝑟_𝑡=𝑣+Γ₁^′−𝛾²𝐼+𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑣+Γ₁−14𝑙^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡+𝜕𝑉^′_𝑟𝑡ℎ𝜕𝑥^𝑟𝑡×−𝛾²𝐼+𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡⁻¹×𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+ℎ^′_𝑟𝑡𝜕𝑉_𝑟𝑡,𝜕𝑥Θ₃𝑡,𝑢,𝑥,𝑟_𝑡=𝑢+Γ₂^′𝐼+𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡𝑢+Γ₂−14𝑙^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡+𝜕𝑉^′_𝑟𝑡𝑔𝜕𝑥^𝑟𝑡×𝐼+𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡⁻¹×𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+𝑔^′_𝑟𝑡𝜕𝑉_𝑟𝑡,𝜕𝑥 (20) where Γ₁=12𝛾²𝐼+𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡⁻¹𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+ℎ^′_𝑟𝑡𝜕𝑉_𝑟𝑡,Γ𝜕𝑥₂=12𝐼+𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡⁻¹𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑙_𝑟𝑡+𝑔^′_𝑟𝑡𝜕𝑉_𝑟𝑡.𝜕𝑥 (21)
Substituting (20) into (18), and considering (12), it yields 𝐸𝑉𝑥(𝑇),𝑟_𝑇−𝑉0,𝑟₀∣𝑟₀=𝑘<𝐸^𝑇₀𝑣+Γ₁^′−𝛾²𝐼+𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡𝑣+Γ₁+𝑢+Γ₂^′𝐼+𝑞^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑞_𝑟𝑡𝑢+Γ₂+𝛾²‖𝑣‖²−‖𝑧‖²𝑑𝑡∣𝑟₀=𝑘. (22) Taking 𝑢=𝑢^∗_𝑇=−Γ₂ and considering the second item of (12), (22) leads to 𝐸^𝑇₀‖𝑧(𝑡)‖²𝑑𝑡∣𝑟₀=𝑘<𝛾²𝐸^𝑇₀‖𝑣(𝑡)‖²𝑑𝑡∣𝑟₀=𝑘−𝐸^𝑇₀𝑣(𝑡)+Γ₁^′×𝛾²𝐼−𝑠^′_𝑟𝑡𝜕²𝑉_𝑟𝑡𝜕𝑥²𝑠_𝑟𝑡×𝑣(𝑡)+Γ₁𝑑𝑡∣𝑟₀=𝑘, (23) which means ‖ℒ_𝑇‖<𝛾 in Definition 1. The theorem is proved.

Remark 6. The proof of Theorem 5 is based on an elementary identity (11), which avoids using stochastic dissipative theory as done in [4, 5]. We believe that the identity (11) will have many other applications in system analysis and synthesis.

3.2. Infinite Horizon Case

In contrast to the finite horizon case, the infinite horizon 𝐻_∞ control exhibits more complexity due to the additional requirement of stabilizing the closed-loop system internally. The following sufficient condition is derived for the infinite horizon 𝐻_∞ control of system (5). In this subsection, denote (⋅)_𝑘∶=(⋅)(𝑥,𝑘).

Theorem 7. Assume that there exists a set of nonnegative functions 𝑉(𝑥,𝑘)∈𝒞²(ℛ^𝑛×𝒯;ℛ), 𝑉(0,𝑘)=0, and 𝜕²𝑉(𝑥,𝑘)/𝜕𝑥²≥0 for all nonzero 𝑥∈ℛ^𝑛, 𝑘∈𝒯. If 𝑉(𝑥,𝑘) solves the following coupled HJIs: Λ_𝑘=𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑚^′_𝑘𝑚_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗+14𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘+𝜕𝑉^′_𝑘ℎ𝜕𝑥^𝑘𝛾²𝐼−𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘⁻¹×𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+ℎ^′_𝑘𝜕𝑉_𝑘−1𝜕𝑥4𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘+𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹×𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘𝛾𝜕𝑥<0,₂𝐼−𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘>0, (24) then 𝑢^∗_∞()1𝑥,𝑘=−2𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘𝜕𝑥 (25) is an infinite horizon 𝐻_∞ control of system (5).

Proof. Similar to the proof of Theorem 5, it is easy to show ‖ℒ_∞‖<𝛾 under condition (24). Next, we need to prove system (8) to be globally asymptotically stable in probability. Let 𝑢^∗_𝑘=𝑢^∗_∞(𝑥,𝑘) and let ℒ_∞ be the infinitesimal generator of the system (8) which is similar to ℒ_𝑇 in Lemma 3; then ℒ_∞𝑉_𝑘=𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+𝑔_𝑘𝑢^∗_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗+12𝑙_𝑘+𝑞_𝑘𝑢^∗_𝑘^′𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑞_𝑘𝑢^∗_𝑘=𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘𝑢^∗_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘𝑢^∗_𝑘+12𝑞_𝑘𝑢^∗_𝑘^′𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+12𝑞_𝑘𝑢^∗_𝑘^′𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘𝑢^∗_𝑘=𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗+Υ_1𝑘+Υ_2𝑘, (26) where Υ_1𝑘=𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘𝑢^∗_𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘𝑢^∗_𝑘+12𝑞_𝑘𝑢^∗_𝑘^′𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘1=−2𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹×𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘−1𝜕𝑥4𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘×𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘−1𝜕𝑥4𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘+𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘×𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘,Υ_2𝑘=12𝑞_𝑘𝑢^∗_𝑘^′𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘𝑢^∗_𝑘=18𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘+𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹×𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹×𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘.𝜕𝑥 (27) It can be checked that Υ_1𝑘1=−2𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘+𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘×𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘,𝜕𝑥Υ_2𝑘≤18𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘+𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘×𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘.𝜕𝑥 (28)(29) The following inequality is used during the calculation of (29): (𝐼+𝑋)⁻¹𝑋(𝐼+𝑋)⁻¹≤(𝐼+𝑋)⁻¹(𝐼+𝑋)(𝐼+𝑋)⁻¹=(𝐼+𝑋)⁻¹,𝑋>0. (30)
Implementing (28) and (29) into (26) and considering (24), it yields ℒ_∞𝑉_𝑘≤𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗−38𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘+𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘×𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘≤𝜕𝑥𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗−14𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘+𝜕𝑉^′_𝑘𝑔𝜕𝑥^𝑘×𝐼+𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑞_𝑘⁻¹𝑞^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑔^′_𝑘𝜕𝑉_𝑘𝜕𝑥<−𝑚^′_𝑘𝑚_𝑘−14𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘+𝜕𝑉^′_𝑘ℎ𝜕𝑥^𝑘×𝛾²𝐼−𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘⁻¹𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+ℎ^′_𝑘𝜕𝑉_𝑘𝜕𝑥≤0 (31) which implies that (8) is globally asymptotically stable in probability from [10]. This theorem is completed.

Remark 8. The methods proposed in [19, 24] cannot be applied to study the 𝐻_∞ problem of NSMJSs with (𝑥,𝑢,𝑣)-dependent noise, although they are suitable for NSMJSs with (𝑥,𝑣)-dependent noise. One reason for this is that 𝑢 and 𝑣 are no longer separable in the conditions, when they enter the diffusion term simultaneously. In the proofs of Theorems 5 and 7, we resort to Lemma 4 to solve this problem.

Remark 9. When there is no Markov jump parameters, system (1)/(5) will reduce to general nonlinear stochastic systems with (𝑥,𝑢,𝑣)-dependent noise, which was studied in [20, 21]. However, all the conditions of 𝐻_∞ control design for nonlinear stochastic systems in [20, 21] were given in terms of three coupled HJEs, which were difficult to be solved. According to this paper, the sufficient condition for 𝐻_∞ control of nonlinear stochastic systems can be derived by means of a single set of coupled HJIs, which is easier to be verified than three coupled HJEs in [20, 21].
From Theorem 7, the following nonlinear stochastic bounded real lemma will be derived for Markov jump system: 𝑑𝑥(𝑡)=𝑓𝑥,𝑟_𝑡+ℎ𝑥,𝑟_𝑡𝑣+𝑙𝑑𝑡𝑥,𝑟_𝑡+𝑠𝑥,𝑟_𝑡𝑣𝑧(𝑡)𝑑𝑤,=𝑚𝑥,𝑟_𝑡,𝑥(0)=𝑥₀∈ℛ^𝑛. (32)

Lemma 10. For a prescribed 𝛾>0, system (32) is internally stable and ‖ℒ_∞‖_{𝑢(𝑡)≡0}<𝛾, if there exists a set of nonnegative functions 𝑉(𝑥,𝑘)∈𝒞²(ℛ^𝑛×𝒯;ℛ), 𝑉(0,𝑘)=0, and 𝜕²𝑉(𝑥,𝑘)/𝜕𝑥²≥0 for all nonzero 𝑥∈ℛ^𝑛, 𝑘∈𝒯, satisfying the following coupled HJIs: 𝜕𝑉^′_𝑘𝑓𝜕𝑥^𝑘+12𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+𝑚^′_𝑘𝑚_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑉_𝑗+14𝑙^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘+𝜕𝑉^′_𝑘ℎ𝜕𝑥^𝑘×𝛾²𝐼−𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘⁻¹𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑙_𝑘+ℎ^′_𝑘𝜕𝑉_𝑘𝛾𝜕𝑥<0,₂𝐼−𝑠^′_𝑘𝜕²𝑉_𝑘𝜕𝑥²𝑠_𝑘>0,𝑘∈𝒯. (33)

Proof. Letting 𝑔_𝑘≡0, 𝑞_𝑘≡0, and 𝑢≡0 in Theorem 7, we obtain (33) easily.

Next, we present a sufficient condition for the following linear stochastic Markov jump systems with (𝑥,𝑢,𝑣)-dependent noise: 𝑑𝑥(𝑡)=𝐴𝑟_𝑡𝑥+𝐵𝑟_𝑡𝑢+𝐶𝑟_𝑡𝑣+𝐴𝑑𝑡₁𝑟_𝑡𝑥+𝐵₁𝑟_𝑡𝑢+𝐶₁𝑟_𝑡𝑣𝑧(𝑡)=𝑑𝑤,col𝐷𝑟_𝑡𝑥,𝑢,𝑥(0)=𝑥₀∈ℛ^𝑛. (34)

Corollary 11. System (34) is internally stable and ‖ℒ_∞‖<𝛾 for given 𝛾>0, if there exist matrices 𝑋_𝑘>0, 𝑌_𝑘𝑘∈𝒯 satisfying the following LMIs: Σ_𝑘𝐶_𝑘𝑋_𝑘𝐷^′_𝑘𝑋_𝑘𝐴^′_1𝑘+𝑌^′_𝑘𝐵^′_1𝑘0𝜓^′_𝑘(𝑋)∗−𝛾²𝐼0𝐶^′_1𝑘𝐶^′_1𝑘0∗∗−𝐼000∗∗∗−𝑋_𝑘00∗∗∗∗−𝑋_𝑘0∗∗∗∗∗−𝜙_𝑘(𝑋)<0, (35) where Σ_𝑘=𝐴_𝑘𝑋_𝑘+𝐵_𝑘𝑌_𝑘+𝑋_𝑘𝐴^′_𝑘+𝑌^′_𝑘𝐵^′_𝑘+𝜋_𝑘𝑘𝑋_𝑘,𝜓_𝑘(𝑋)=√𝜋_𝑘1𝑋_𝑘,…,√𝜋_𝑘𝑘−1𝑋_𝑘,√𝜋_𝑘𝑘+1𝑋_𝑘,…,√𝜋_𝑘𝑁𝑋_𝑘^′,𝜙_𝑘(𝑋)=diag𝑋₁,…,𝑋_𝑘−1,𝑋_𝑘+1,…,𝑋_𝑁. (36) Moreover, the state feedback gain matrices are given by 𝐾_𝑘=𝑌_𝑘𝑋_𝑘⁻¹.

Proof. Firstly, consider the following unforced linear stochastic Markov jump systems: 𝑑𝑥(𝑡)=𝐴𝑟_𝑡𝑥+𝐶𝑟_𝑡𝑣𝑑𝑡+𝐴₁𝑟_𝑡𝑥+𝐶₁𝑟_𝑡𝑣𝑧(𝑡)𝑟𝑑𝑤,=𝐷_𝑡𝑥,𝑥(0)=𝑥₀∈ℛ^𝑛. (37)
Let 𝑉_𝑘=𝑥^′𝑃_𝑘𝑥,𝑃_𝑘>0,𝑘∈𝒯; (33) in Lemma 10 can be written as follows: 2𝑥^′𝑃_𝑘𝐴_𝑘𝑥+𝑥^′𝐴^′_1𝑘𝑃_𝑘𝐴_1𝑘𝑥+𝑥^′𝐷^′_𝑘𝐷_𝑘𝑥+𝑥^′𝑁_𝑗=1𝜋_𝑘𝑗𝑃_𝑗𝑥+𝑥^′𝐴^′_1𝑘𝑃_𝑘𝐶_1𝑘+𝑥^′𝑃_𝑘𝐶_𝑘×𝛾²𝐼−2𝐶^′_1𝑘𝑃_𝑘𝐶_1𝑘⁻¹𝐶^′_1𝑘𝑃_𝑘𝐴_1𝑘𝑥+𝐶^′_𝑘𝑃_𝑘𝑥=𝑥^′𝑃_𝑘𝐴_𝑘+𝐴^′_𝑘𝑃_𝑘+𝐴^′_1𝑘𝑃_𝑘𝐴_1𝑘+𝐷^′_𝑘𝐷_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑃_𝑗+𝐴^′_1𝑘𝑃_𝑘𝐶_1𝑘+𝑃_𝑘𝐶_𝑘𝛾²𝐼−2𝐶^′_1𝑘𝑃_𝑘𝐶_1𝑘⁻¹×𝐶^′_1𝑘𝑃_𝑘𝐴_1𝑘+𝐶^′_𝑘𝑃_𝑘𝑥<0,𝛾²𝐼−2𝐶^′_1𝑘𝑃_𝑘𝐶_1𝑘>0. (38)(39)
By Schur complement, (38) is equivalent to 𝑃_𝑘𝐴_𝑘+𝐴^′_𝑘𝑃_𝑘+𝐴^′_1𝑘𝑃_𝑘𝐴_1𝑘𝐴^′_1𝑘𝑃_𝑘𝐶_1𝑘+𝑃_𝑘𝐶_𝑘𝐷^′_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑃_𝑗∗−𝛾²𝐼+2𝐶^′_1𝑘𝑃_𝑘𝐶_1𝑘0∗∗−𝐼<0, (40) which also implies (39). Moreover, (40) can be rewritten as 𝑃_𝑘𝐴_𝑘+𝐴^′_𝑘𝑃_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑃_𝑗𝑃_𝑘𝐶_𝑘𝐷^′_𝑘∗−𝛾²+𝐴𝐼0∗∗−𝐼_′1𝑘𝐶^′_1𝑘0𝑃_𝑘𝐴_1𝑘𝐶_1𝑘0+0𝐶^′_1𝑘0𝑃_𝑘0𝐶_1𝑘0<0, (41) which yields 𝑃_𝑘𝐴_𝑘+𝐴^′_𝑘𝑃_𝑘+^𝑁_𝑗=1𝜋_𝑘𝑗𝑃_𝑗𝑃_𝑘𝐶_𝑘𝐷^′_𝑘𝐴^′_1𝑘0∗−𝛾²𝐼0𝐶^′_1𝑘𝐶^′_1𝑘∗∗−𝐼00∗∗∗−𝑃_𝑘⁻¹0∗∗∗∗−𝑃_𝑘⁻¹<0, (42) according to Schur complement. Pre- and postmultiplying (42) by diag{𝑃_𝑘⁻¹𝐼𝐼𝐼𝐼} and denoting 𝑋_𝑘=𝑃_𝑘⁻¹, we have 𝐴_𝑘𝑋_𝑘+𝑋_𝑘𝐴^′_𝑘+𝜋_𝑘𝑘𝑋_𝑘𝐶_𝑘𝑋_𝑘𝐷^′_𝑘𝑋_𝑘𝐴^′_1𝑘0𝜓^′_𝑘(𝑋)∗−𝛾²𝐼0𝐶^′_1𝑘𝐶^′_1𝑘0∗∗−𝐼000∗∗∗−𝑋_𝑘00∗∗∗∗−𝑋_𝑘0∗∗∗∗∗−𝜙_𝑘(𝑋)<0, (43) where 𝜓_𝑘(𝑋) and 𝜙_𝑘(𝑋) are defined as in (35).
Considering closed-loop system (34) with state feedback control 𝑢(𝑡)=𝐾_𝑘𝑥(𝑡), 𝑘∈𝒯. Replacing 𝐴_𝑘,𝐴_1𝑘 by 𝐴_𝑘+𝐵_𝑘𝐾_𝑘, 𝐴_1𝑘+𝐵_1𝑘𝐾_𝑘, respectively, and setting 𝑌_𝑘=𝐾_𝑘𝑋_𝑘 in (43) yield (35). Therefore, the state feedback gain matrices can be obtained by 𝐾_𝑘=𝑌_𝑘𝑋_𝑘⁻¹.

Remark 12. Although HJIs (12) in Theorem 5 or (24) in Theorem 7 can be solved by trial and error in some simple cases (see Example 13 in Section 4), they are difficult to be dealt with for high-dimensional systems. In order to avoid solving the HJIs, the Taylor series approach [25] or fuzzy approach based on Takagi-Sugeno model [24, 26] can be considered to design the nonlinear stochastic 𝐻_∞ controller.

4. Numerical Example

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

Example 13. Consider one-dimensional two-mode time-invariant nonlinear stochastic Markov jump systems with generator matrix Π=⁻¹¹₁₋₁, and the two subsystems are as follows: I∶𝑑𝑥=𝑥³+4𝑥𝑢+𝑥𝑣𝑑𝑡+𝑥²+𝑢+𝑣𝑑𝑤,𝑧=col𝑥²,𝑢,𝑥(0)=𝑥₀∈ℛ^𝑛,II∶1𝑑𝑥=2𝑥³1+3𝑥𝑢+2𝑥𝑣𝑑𝑡+𝑥²+𝑢+𝑣𝑑𝑤,𝑧=col12𝑥²,𝑢,𝑥(0)=𝑥₀∈ℛ^𝑛. (44) Set 𝑉(𝑥)=𝑝_𝑖𝑥², 𝑖=1,2, with 𝑝₁>0 and 𝑝₂>0 to be determined; then HJIs (24) become 2𝑝₁𝑥⋅𝑥³+12𝑥²⋅2𝑝₁⋅𝑥²+𝑥²⋅𝑥²+14𝑥²⋅2𝑝₁+2𝑝₁𝑥⋅𝑥²𝛾²−2𝑝₁⁻¹−14𝑥²⋅2𝑝₁+2𝑝₁𝑥⋅4𝑥²1+2𝑝₁⁻¹−𝑝₁𝑥²+𝑝₂𝑥²𝛾<0,₂−2𝑝₁>0,2𝑝₂1𝑥⋅2𝑥³+12𝑥²⋅2𝑝₂⋅𝑥²+12𝑥²⋅12𝑥²+14𝑥²⋅2𝑝₂+2𝑝₂1𝑥⋅2𝑥²𝛾²−2𝑝₂⁻¹−14𝑥²⋅2𝑝₂+2𝑝₂𝑥⋅3𝑥²1+2𝑝₂⁻¹+𝑝₁𝑥²−𝑝₂𝑥²𝛾<0,₂−2𝑝₂>0. (45) For given 𝛾=√3, the above inequalities have solutions 𝑝₁=1 and 𝑝₂=1. According to Theorem 7, the infinite 𝐻_∞ controllers of system (44) are 𝑢^∗_∞(𝑥,1)=−(5/3)𝑥² and 𝑢^∗_∞(𝑥,2)=−(4/3)𝑥².

Example 14. Consider two-dimensional two-mode linear stochastic Markov jump systems (34) with the following parameters: 𝐴₁=110−2,𝐵₁=𝐶₁=10,𝐴₁₁=1101,𝐵₁₁=𝐶₁₁=10,𝐴₂=210−2,𝐵₂=𝐶₂=11,𝐴₁₂=20111,𝐵₁₂=𝐶₁₂=01,𝐷₁=10,𝐷₂=,.11Π=−0.50.50.5−0.5 (46) With the choice of 𝛾=√2, a possible solution of LMIs (35) in Corollary 11 can be found by using Matlab LMI control toolbox: 𝑋₁=0.8879−0.4074−0.40743.6974,𝑋₂=,𝑌0.50260.04640.04645.4520₁=−3.1690−5.3082,𝑌₂=−4.5025−8.5671. (47) Then, the 𝐻_∞ control gain matrices of system (34) are as follows: 𝐾₁=𝑌₁𝑋₁⁻¹=,𝐾−4.4531−1.9263₂=𝑌₂𝑋₂⁻¹=−8.8209−1.4963. (48)

Figure 1 shows the result of the changing between modes during the simulation with the initial mode at mode 1. The initial condition is chosen as 𝑥₀=[−0.20.3]^′ and exogenous input 𝑣(𝑡)=𝑒^−𝑡. By means of Euler-Maruyama method [27], the state responses of unforced system (37) and controlled system (34) are shown in Figures 2 and 3, respectively. From Figure 3, one can find that the controlled system (34) can achieve stability and attenuation performance in the sense of mean square by using the proposed 𝐻_∞ controller.

Figure 1 

Result of the changing between modes.

Figure 2 

The state responses of unforced system (37).

Figure 3 

The state responses of controlled system (34).

5. Conclusions

In this paper, we have studied the 𝐻_∞ control problem for NSMJSs with (𝑥,𝑢,𝑣)-dependent noise. A sufficient condition for finite/infinite horizon 𝐻_∞ control has been derived in terms of a set of coupled HJIs. It can be found that Lemma 4 plays an essential role during the proof of Theorems 5 and 7. The validity of the obtained results has been verified by two examples.

Conflict of Interests

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

Acknowledgments

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).