The 𝐻_∞ Control for Bilinear Systems with Poisson Jumps

Abstract

This paper discusses the state feedback 𝐻_∞ control problem for a class of bilinear stochastic systems driven by both Brownian motion and Poisson jumps. By completing square method, we obtain the 𝐻_∞ control by solutions of the corresponding Hamilton-Jacobi equations (HJE). By the tensor power series method, we also shift such HJEs into a kind of Riccati equations, and the 𝐻_∞ control is represented with the form of tensor power series.

1. Introduction

The main purpose of 𝐻_∞ control design is to find the law to efficiently eliminate the effect of the disturbance [1, 2]. Theoretically, study of 𝐻_∞ control first starts from the deterministic linear systems, and the derivation of the state-space formulation of the standard 𝐻_∞ control leads to a breakthrough, which can be found in the paper [3]. In recent years, stochastic 𝐻_∞ control systems, such as Markovian jump systems [4–6], 𝐻_∞ Gaussian control design [7], and Itô differential systems [8–13], have received a great deal of attention. However, up to now, most of the work on stochastic 𝐻_∞ control is confined to Itô type or Markovian jump systems. Yet, there are still many systems which contain Poisson jumps in economics and natural science. In 1970s, Boel and Varaiya [14] and Rishel [15] considered the optimal control problem with random Poisson jumps, and many basic results have been made. From then on, many scholars and economists also study the system and its applications; for further reference, we refer to [16–20] and their references. But those results mostly concentrate on optimal control and its application in financial market or corresponding theories. Of course, such model still can be disturbed by exogenous disturbance and its robustness is also an important problem. The objective of this paper is to develop an 𝐻_∞-type theory over infinite time horizon for the disturbance attenuation of stochastic bilinear systems with Poisson jumps by dynamic state feedback.

Generally, the key of 𝐻_∞ control design is to solve a general Hamilton-Jacobi equation (HJE). However, up to now, there is still no effective algorithm to solve such a general HJE. In order to solve the HJE given in this paper, we extend a tensor power series approach which is used in [21] and also give the simulation of the trajectory of output 𝑧 under 𝐻_∞ control. This paper will follow along the lines of [22] to study the stochastic 𝐻_∞ control with infinite horizons and finite horizon for a class of nonlinear stochastic differential systems with Poisson jumps. The paper is organized as follows.

In Section 2, we review Itô’s theories about the system driven by Brownian motion and Poisson jumps. In Section 3, we obtain the 𝐻_∞ by solving the HJE which is proved by the completing square method. In Section 4, we discuss the problem of finite horizon 𝐻_∞ control with jumps, and using the tensor power series approach, we discuss the approximating 𝐻_∞ control given in the paper. For convenience, we adopt the following notation.

𝒮_𝑛(ℛ) denotes the set of all real 𝑛×𝑛 symmetric matrices; 𝐴^′ is the transpose of the corresponding matrix 𝐴; 𝐴>0(𝐴≥0) is the positive definite (semidefinite) matrix 𝐴; 𝐼 is the identity matrix; 𝔼𝜉 is the expectation of random variable 𝜉; ‖𝑥‖ is the Euclidean norm of vector 𝑥∈ℛ^𝑛𝑥 and 𝑛_𝑥 is the dimension of 𝑥; ℒ²([0,𝑇],ℛ^𝑛𝑦) is the set of 𝑛_𝑦-dimensional stochastic process 𝑦 defined on interval [0,𝑇] (𝑇 can take ∞), taking values in ℛ^𝑛𝑦, with norm 𝑦_{ℒ2([0,𝑇],ℛ}^𝑛_𝑦)=𝐸^𝑇₀𝑦(𝑡)²𝑑𝑡^1/2<∞; (1)𝐶^1,2(ℛ₊,ℛ^𝑛𝑥) is the class of function 𝑉(𝑡,𝑥) twice continuously differential with respect to 𝑥∈ℛ^𝑛𝑥 and once continuously differential with respect to 𝑡; ⟨𝑥,𝑦⟩ is the inner product of two vectors 𝑥,𝑦∈ℛ^𝑛.

2. Preliminaries

For a given complete probability space (Ω,ℱ,ℙ), let 𝑊_𝑡 and 𝜇 be the Brownian motion and the Poisson random measure, respectively, which are mutually independent:(i)a 1-dimensional standard Brownian motion {𝑊_𝑡}_𝑡≥0;(ii)a Poisson random measure 𝜇 on ℝ₊×𝐸, where 𝐸⊂ℝ^𝑙 is a nonempty open set equipped with its Borel field ℬ(𝐸), with the compensator 𝜇(𝑑𝑒,𝑑𝑡)=𝜆(𝑑𝑒)𝑑𝑡, such that {𝜇((0,𝑡]×𝐴)=(𝜇−𝜇)((0,𝑡]×𝐴)}_𝑡≥0 is a martingale for all 𝐴∈ℬ(𝐸) satisfying 𝜆(𝐴)<∞. Here 𝜆 is an arbitrarily given 𝜎-finite Lévy measure on (𝐸,ℬ(𝐸)), that is, a measure on (𝐸,ℬ(𝐸)) with the property that ∫_𝐸(1∧|𝑒|²)𝜆(𝑑𝑒)<∞. We also let ℱ_𝑡=𝜎_{(0,𝑠]×𝐴}𝜇()(𝐸×𝑊𝑑𝑒,𝑑𝑠∶𝑠≤𝑡,𝐴∈ℬ)∨𝜎_𝑠∶𝑠≤𝑡∨𝒩, (2) where 𝒩 denotes the totality of 𝑃-null sets.

In order to discuss the systems driven by Brownian motion and Poisson jumps, we first review the theorem about Itô’s formula for such stochastic processes.

Theorem 1. Let 𝑀_𝑡 be a square integral continuous martingale; 𝐴_𝑡 is a continuous adapted process with finite variance. 𝛾(𝑠,𝑒) is locally square integral due to 𝜇 and 𝑃; 𝑥(𝑡) satisfies the following Itô type stochastic process: 𝑥(𝑡)=𝑥(0)+𝑀_𝑡+𝐴_𝑡+^𝑡_0𝐸𝛾(𝑠,𝑒)𝜇(𝑑𝑒,𝑑𝑠). (3) Then for 𝐹(𝑡,𝑥)∈𝐶^1,2(ℛ₊,ℛ^𝑛𝑥), we have (see [23] Chapter I, §3, Theorem 11) 𝑑𝐹𝑡,𝑥_𝑡=𝐹_𝑡𝑡,𝑥_𝑡𝑑𝑡+𝐹_𝑥𝑡,𝑥_𝑡𝑑𝑀_𝑡+𝐴_𝑡+12𝐹_𝑥𝑥𝑡,𝑥_𝑡𝑑⟨𝑀⟩_𝑡+_𝐸𝐹𝑡,𝑥_𝑡+𝛾(𝑡,𝑒)−𝐹𝑡,𝑥_𝑡−𝐹_𝑥𝑡,𝑥_𝑡𝛾(𝑡,𝑒)𝜆(𝑑𝑒)𝑑𝑡+_𝐸𝐹𝑡,𝑥_𝑡+𝛾(𝑡,𝑒)−𝐹𝑡,𝑥_𝑡𝜇(𝑑𝑒,𝑑𝑡), (4) where ⟨𝑀⟩ denotes the predictable compensator of martingale 𝑀.
In the paper, for convenience, 𝑥_𝑡 is shorten as 𝑥. Furthermore, for 𝐹∈𝐶^1,2(ℛ₊,ℛ^𝑛𝑥), if using Itô formula to 𝐹(𝑡,𝑥_𝑡) and integrating from 𝑠 to 𝑡 (0≤𝑠<𝑡), then taking expectation with both sides 𝔼𝐹𝑡,𝑥_𝑡−𝔼𝐹𝑠,𝑥_𝑠=^𝑡_𝑠𝔼𝐹_𝑡𝑟,𝑥_𝑟+1𝑑𝑟2^𝑡_𝑠𝔼𝜎𝑟,𝑥_𝑟^′𝐹_𝑥𝑥𝑟,𝑥_𝑟𝜎𝑟,𝑥_𝑟𝑑𝑟+^𝑡_𝑠𝔼_𝐸𝐹𝑟,𝑥_𝑟+𝛾𝑟,𝑥_𝑟−,𝑒−𝐹𝑟,𝑥_𝑟−𝐹_𝑥𝑟,𝑥_𝑟((,𝛾𝑟,𝑒)×𝜆𝑑𝑒)𝑑𝑟, (5) we can see that 𝔼𝐹(𝑡,𝑥_𝑡) is continuous with respect to time 𝑡. Since we mainly use the results of expectations of some well functions on 𝑥_𝑡 and those expectations are continuous with respect to time 𝑡, so, for briefness, in the rest of this paper the sign 𝑥_𝑡− under integration ∫_𝐸 is also shortened as 𝑥.

3. The 𝐻_∞ Control for Bilinear Systems with Jumps

We consider the following bilinear system driven by Poisson jumps: 𝑑𝑥=(𝐴𝑥+𝐵𝑥𝑢+𝐾𝑣)𝑑𝑡+𝐶𝑥𝑑𝑊+_𝐸𝐺(𝑒)𝑥𝜇(),𝑢,𝑑𝑒,𝑑𝑡𝑧=𝑀𝑥 (6) where 𝑣∈ℒ²([0,𝑇],ℛ^𝑛𝑑) represents the exogenous disturbance, 𝐴, 𝐵, and 𝐶 are constant 𝑛_𝑥×𝑛_𝑥 matrices, 𝐾∈ℛ 𝐾∈ℛ^{𝑛𝑥×𝑛𝑑}, and 𝐺(𝑒)∈ℛ^{𝑛𝑥×𝑛𝑥} only depends on 𝑒. If there exists an 𝑢^∗_𝑇∈ℒ²([0,𝑇],ℛ^𝑛𝑢) such that for any given 𝛾>0 and all 𝑣∈ℒ²([0,𝑇],ℛ^𝑛𝑑), 𝑥(0)=0, the closed-loop system satisfies ‖𝑧‖_ℒ²_{([0,𝑇],ℛ}^𝑛_𝑧)≤𝛾‖𝑣‖_ℒ²_{([0,𝑇],ℛ}^𝑛_𝑑), (7) we call 𝑢^∗_𝑇 the 𝐻_∞ control of (6).

Theorem 2. Suppose there exists a nonnegative solution 𝑉∈𝐶^1,2([0,𝑇],ℛ^𝑛𝑥) to the HJE ℋ¹_𝑇𝑉_𝑇(𝑡,𝑥)∶=𝜕𝑉_𝑇+𝜕𝑡𝜕𝑉_𝑇𝜕𝑥^′1𝐴𝑥+2𝑥^′𝑀^′+1𝑀𝑥2𝛾²𝜕𝑉_𝑇𝜕𝑥^′𝐾𝐾^′𝜕𝑉_𝑇+1𝜕𝑥2𝑥^′𝐶^′𝜕²𝑉_𝑇𝜕𝑥²1𝐶𝑥−2𝜕𝑉_𝑇𝜕𝑥^′𝐵𝑥𝑥^′𝐵^′𝜕𝑉_𝑇+𝜕𝑥_𝐸𝑉_𝑇(𝑡,𝑥+𝐺(𝑒)𝑥)−𝑉_𝑇()−𝑡,𝑥𝜕𝑉𝜕𝑥^′𝐺(𝑒)𝑥𝜆()𝑉𝑑𝑒=0,_𝑇(𝑇,𝑥)=0,𝑉_𝑇(𝑡,0)=0,∀(𝑡,𝑥)∈[0,𝑇]×ℛ^𝑛𝑥. (8) Then 𝑢^∗_𝑇=−𝑥^′𝐵^′(𝜕𝑉_𝑇/𝜕𝑥) is an 𝐻_∞ control for system (6).

Proof . Applying Itô’s formula to 𝑉(𝑡,𝑥), we have 𝑉𝑇,𝑥_𝑇()=−𝑉0,0_𝑇0𝑉(𝑡,𝑥)+𝑉^′_𝑥()+1𝑡,𝑥)(𝐴𝑥+𝐵𝑥𝑢+𝐾𝑣2𝑥^′𝐶^′𝑉_𝑥𝑥𝐶𝑥𝑠𝑠+_𝐸[𝑉(𝑡,𝑥+𝐺(𝑒)𝑥)−𝑉(𝑡,𝑥)−𝑉_𝑥()𝐺(𝑒)𝑥𝜆(+𝑡,𝑥𝑑𝑒)𝑑𝑡_𝑇0𝑉^′_𝑥(𝑡,𝑥)𝐶𝑥𝑑𝑊_𝑡+^𝑇₀[𝑉(𝑡,𝑥+𝐺(𝑒)𝑥)−𝑉(𝑡,𝑥)]𝜇(𝑑𝑒,𝑑𝑡). (9)Taking expectation with both sides and applying 𝑉(𝑇,𝑥)=0 and 𝑉(𝑡,0)=0, we obtain 0=^𝑇₀𝔼𝑉(𝑡,𝑥)+𝑉^′_𝑥()+1𝑡,𝑥)(𝐴𝑥+𝐵𝑥𝑢+𝐾𝑣2𝑥^′𝐶^′𝑉_𝑥𝑥+𝐶𝑥_𝐸[𝑉(𝑡,𝑥+𝐺(𝑒)𝑥)−𝑉(𝑡,𝑥)−𝑉_𝑥(𝑡,𝑥)𝐺(𝑒)𝑥𝜆(𝑑𝑒)+𝑥^′_𝑡𝑀^′𝑀𝑥+𝑢^′_𝑡𝑢−𝛾²𝑣^′_𝑡𝑣−‖𝑧‖𝑑𝑡_{2ℒ2][0,𝑇,ℛ𝑛𝑧}+𝛾²‖𝑣‖²_{ℒ2][0,𝑇,ℛ𝑛𝑑}. (10) Completing square for 𝑢 and 𝑣, respectively, we have 0=^𝑇₀𝔼ℋ¹_𝑇(𝑉(𝑡,𝑥))+𝑢−𝑢^∗_𝑇²−1𝛾²𝑣−𝑣^∗_𝑇²−‖𝑧‖𝑑𝑡_{2ℒ2][0,𝑇,ℛ𝑛𝑧}+𝛾²‖𝑣‖²_{ℒ2][0,𝑇,ℛ𝑛𝑑}, (11) where 𝑢^∗_𝑇=−𝑥^′𝐵^′𝜕𝑉_𝑇𝜕𝑥,𝑣^∗_𝑇=1𝛾²𝐾^′𝜕𝑉_𝑇𝜕𝑥+𝐾^′𝜕²𝑉_𝑇𝜕𝑥²𝐶𝑥. (12) By HJE (8) and let 𝑢=𝑢^∗_𝑇, we have ‖𝑧‖²_{ℒ2[]0,𝑇,ℛ𝑛𝑧}−𝛾²‖𝑣‖²_{ℒ2[]0,𝑇,ℛ𝑛𝑑}1=−𝛾^2𝑇₀𝔼𝑣−𝑣^∗_𝑇²𝑑𝑡. (13) So, the following inequality is true: ‖𝑧‖²_{ℒ2[]0,𝑇,ℛ𝑛𝑧}≤𝛾²‖𝑣‖²_{ℒ2[]0,𝑇,ℛ𝑛𝑑}. (14) This proves that 𝑢^∗_𝑇 is an 𝐻_∞ control for system (6).

Remark 3. From the proof of Theorems 2 and (13), we can see that (𝑢^∗_𝑇,𝑣^∗_𝑇) given by (12) is a saddle point for the following stochastic game problem: min_{𝑢∈ℒ2[]0,𝑇,ℛ𝑛𝑢}max_{𝑑∈ℒ2[]0,𝑇,ℛ𝑛𝑑}𝔼^𝑇₀‖𝑧‖²−𝛾²‖𝑑‖²𝑑𝑡. (15)

4. The Tensor Power Series Representation of 𝐻_∞ Control

Generally speaking, it is very hard to solve HJE (8). Now we use an approximation algorithm which is called tensor power series approach to solve a special case of HJE (8). In what follows, suppose 𝑉_𝑇(𝑡,𝑥) satisfying (8) has the following form: 𝑉_𝑇(𝑡,𝑥)=^∞_𝑖=1⊗_𝑖𝑥,𝑃_𝑖(𝑡)⊗_𝑖𝑥, (16) where 𝑃_𝑖(𝑡), 𝑖≥1, are symmetrically and continuously differential matrix-valued functions on [0,𝑇], ⊗ is the Kronecker product of matrix (or vectors), and ⊗_𝑖𝑥=𝑥⊗⋯⊗𝑥 is 𝑖 times Kronecker product of 𝑥.

Theorem 4. For given 𝛾>0, suppose 𝑃_𝑖(𝑡)(𝑖=1,2,…) satisfy the following Riccati equations: 𝑃̇₁+𝐴⁽¹⁾𝑃₁+𝑃₁𝐴^(1)′+12𝑀^′2𝑀+𝛾²𝑄⁽¹⁾𝑃₁⊗𝑃₁𝑄^(1)′+𝑅⁽¹⁾𝑃₁+_𝐸𝐼_𝑛𝑥+𝐺(𝑒)^′𝑃₁𝐼_𝑛𝑥+𝐺(𝑒)−𝑃₁−𝐺⁽¹⁾(𝑒)𝑃₁−𝑃₁𝐺^(1)′(𝑒𝜆()𝑃̇)𝑑𝑒=0,_𝑖+𝐴^(𝑖)𝑃_𝑖+𝑃_𝑖𝐴^(𝑖)′+2𝛾²_{𝑟+𝑗=𝑖+1}𝐾^(𝑟)𝑃_𝑟⊗𝑃_𝑗𝐾^(𝑗)′+𝑅^(𝑖)𝑃_𝑖−2_{𝑟+𝑗=𝑖}𝐵^(𝑟)𝑃_𝑟⊗𝑃_𝑗𝐵^(𝑗)′+_𝐸⊗_𝑖𝐼_𝑛𝑥+𝐺(𝑒)^′𝑃_𝑖⊗_𝑖𝐼_𝑛𝑥+𝐺(𝑒)−𝑃_𝑖−𝐺^(𝑖)(𝑒)𝑃_𝑖−𝑃_𝑖𝐺^(𝑖)′(𝑒𝜆()𝑃)𝑑𝑒=0,_𝑖(𝑇)=0,𝑖=1,2,…. (4) Then the 𝐻_∞ control 𝑢^∗_𝑇 for system (6) can be given by 𝑢^∗_𝑇=−2^∞_𝑖=1⊗_𝑖𝑥,𝐵^(𝑖)𝑃_𝑖⊗_𝑖𝑥, (18) where 𝐵^(𝑖)=∑^𝑛𝑥_𝑗=1𝐵^′_𝑗⊗𝐷^(𝑖,𝑗) and 𝐷^(𝑖,𝑗) is given by following Lemma 6.
In order to prove Theorem 4, we need the following lemmas, and Lemmas 5–8 are given without proofs.

Lemma 5. For any 𝑥∈ℛ^𝑛𝑥, 𝑦∈ℛ^𝑛𝑦, 𝑢∈ℛ^𝑛𝑢, 𝑣∈ℛ^𝑛𝑣, 𝑃∈ℛ^{𝑛𝑥×𝑛𝑦}, and 𝑄∈ℛ^{𝑛𝑢×𝑛𝑣} we have 𝑥,𝑃𝑦⟨𝑢,𝑄𝑣⟩=𝑥⊗𝑢,(𝑃⊗𝑄)𝑦⊗𝑣. (19)

Lemma 6. For any matrix 𝑃∈𝒮_𝑛𝑖𝑥(ℛ), 𝐾∈ℛ^𝑛𝑘, and integer 𝑖, we have 𝜕⊗_𝑖𝑥,𝑃⊗_𝑖𝑥^′⊗𝜕𝑥𝐾=2_𝑖−1𝑥,𝐾^(𝑖)𝑃⊗_𝑖𝑥, (20) where 𝐾^(𝑖)=∑^𝑛𝑥_𝑗=1𝑘_𝑗𝐷^(𝑖,𝑗), 𝐷^(𝑖,𝑗)=∑^𝑖_𝑙=1𝐷_𝑙^(𝑖,𝑗), and 𝐷_𝑙^(𝑖,𝑗)=𝐼_{𝑛𝑥𝑙−1}⊗𝑒^′_𝑗⊗𝐼_{𝑛𝑥𝑖−𝑙}.

Lemma 7. Let 𝑉_𝑇(𝑡,𝑥)=∑^∞_𝑖=1⟨⊗_𝑖𝑥,𝑃_𝑖(𝑡)⊗_𝑖𝑥⟩ exist. We have 𝜕𝑉_𝑇𝜕𝑥^′𝐾𝐾^′𝜕𝑉_𝑇𝜕𝑥=4^∞_𝑚=1⊗_𝑚𝑥,_{𝑖+𝑗=𝑚+1}𝐾^(𝑖)𝑃_𝑖⊗𝑃_𝑗𝐾^(𝑗)′⊗_𝑚𝑥. (21)

Lemma 8. For any matrix 𝑃∈𝒮_𝑛𝑖𝑥(ℛ), 𝑥∈ℛ^𝑛𝑥, and integer 𝑖, we have 𝜕⊗_𝑖𝑥,𝑃⊗_𝑖𝑥𝜕𝑥^′𝐴𝑥=2⊗_𝑖𝑥,𝐴^(𝑖)𝑃⊗_𝑖𝑥, (22) where 𝐴^(𝑖)=∑^𝑛𝑥_𝑗=1𝐴^′_𝑗⊗𝐷^(𝑖,𝑗), and 𝐴_𝑗 is the 𝑗th row vector of matrix 𝐴.

Lemma 9. For any matrix 𝑃∈𝒮_𝑛𝑖𝑥(ℛ) and integer 𝑖, we have 𝑥^′𝐶^′𝜕²⊗_𝑖𝑥,𝑃⊗_𝑖𝑥𝜕𝑥²𝐶𝑥=2⊗_𝑖𝑥,𝑅^(𝑖)(𝑃)⊗_𝑖𝑥, (23) where 𝑅^(𝑖)(𝑃)=𝐶^(𝑖)𝑃𝐶^(𝑖)′+𝑄^(𝑖)𝑃, 𝑄^(𝑖)=∑^𝑛𝑥_𝑠=1∑^𝑛𝑥_𝑡=1𝐶^′_𝑠⊗𝐶^′_𝑡⊗(𝐷^{(𝑖−1,𝑠)}𝐷^(𝑖,𝑡)), 𝐶_𝑠 is the 𝑠th row vector of matrix C, and 𝐶^(𝑖) is determined as 𝐴^(𝑖) in Lemma 8.

Proof. Let 𝐾=𝐶𝑥=(𝑘₁,𝑘₂,…,𝑘_𝑛𝑥)^′; then we have 𝑥^′𝐶^′𝜕²⊗_𝑖𝑥,𝑃⊗_𝑖𝑥𝜕𝑥²=𝐶𝑥_{𝑛𝑥𝑛𝑠=1𝑥𝑡=1}𝜕²⊗_𝑖𝑥,𝑃⊗_𝑖𝑥𝜕𝑥_𝑠𝜕𝑥_𝑡𝑘_𝑠𝑘_𝑡=2^𝑛𝑥𝑛_𝑠=1^𝑥_𝑡=1𝑘_𝑠𝑘_𝑡𝜕𝜕𝑥_𝑠⊗_𝑖−1𝑥,^𝑖_𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃=2^𝑛𝑥𝑛_𝑠=1^𝑥_𝑡=1𝑘_𝑠𝑘_𝑡^𝑖−1_𝑚=1⊗_𝑚−1𝑥⊗𝑒_𝑠⊗⊗_{𝑖−𝑚−1}𝑥,_𝑖𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃⊗_𝑖𝑥+^𝑖_𝑚=1⊗_𝑖−1𝑥,^𝑖_𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃⊗_𝑚−1𝑥⊗𝑒_𝑠⊗⊗_𝑖−𝑚𝑥=2^𝑛𝑥𝑛_𝑠=1^𝑥_𝑡=1𝑘_𝑠𝑘_𝑡⊗_𝑖−2𝑥,^𝑖−1_𝑚=1𝐷_𝑚^{𝑖𝑖−1,𝑠}_𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃⊗_𝑖𝑥+⊗_𝑖−1𝑥,^𝑖𝑖_𝑚=1𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃𝐷^{(𝑖,𝑡)′}_𝑚⊗_𝑖−1=2^𝑛𝑥𝑛_𝑠=1^𝑥_𝑡=1𝑥,𝐶^′_𝑠𝑥,𝐶^′_𝑡×⊗_𝑖−2𝑥,^𝑖−1_𝑚=1𝐷_𝑚^{𝑖𝑖−1,𝑠}_𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃⊗_𝑖𝑥+𝑥,𝐶^′_𝑠⊗_𝑖−1𝑥,^𝑖𝑖_𝑚=1𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃𝐷^{(𝑖,𝑡)′}_𝑚⊗_𝑖−1×𝐶^′_𝑡,𝑥. (24) By Lemma 5, 𝑥^′𝐶^′𝜕²⊗_𝑖𝑥,𝑃⊗_𝑖𝑥𝜕𝑥²𝐶𝑥=2^𝑛𝑥𝑛_𝑠=1^𝑥_𝑡=1⊗_𝑖𝑥,𝐶^′_𝑠⊗𝐶^′_𝑡⊗^𝑖−1_𝑚=1𝐷_𝑚^{𝑖−1,𝑠}×^𝑖_𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃⊗_𝑖𝑥+⊗_𝑖𝑥,𝐶^′_𝑠⊗^𝑖𝑖_𝑚=1𝑙=1𝐷_𝑙^(𝑖,𝑡)𝑃𝐷^{(𝑖,𝑡)′}_𝑚⊗𝐶_𝑡⊗_𝑖𝑥. (25) So we can obtain (23).

Proof of Theorem 4. Applying Lemmas 6–9, we have 𝑉_𝑇(𝑡,𝑥+𝐺(𝑒)𝑥)=^∞_𝑖=1⊗_𝑖𝑥,⊗_𝑖𝐼_𝑛𝑥+𝐺(𝑒)^′𝑃_𝑖(𝑡)×⊗_𝑖𝐼_𝑛𝑥+𝐺(𝑒)⊗_𝑖𝑥,𝜕𝑉_𝑇=𝜕𝑡^∞_𝑖=1⊗_𝑖𝑥,𝑃̇_𝑖⊗_𝑖𝑥,𝜕𝑉_𝑇𝜕𝑥^′𝐴𝑥=^∞_𝑖=1⊗_𝑖𝑥,𝐴^(𝑖)𝑃_𝑖⊗_𝑖𝑥+^∞_𝑖=1⊗_𝑖𝑥,𝑃_𝑖𝐴^(𝑖)′⊗_𝑖𝑥,𝜕𝑉𝜕𝑥^′𝐺(𝑒)𝑥=^∞_𝑖=1⊗_𝑖𝑥,𝐷^(𝑖)(𝑒)𝑃_𝑖⊗_𝑖𝑥+^∞_𝑖=1⊗_𝑖𝑥,𝑃_𝑖𝐷^(𝑖)′⊗_𝑖𝑥,𝜕𝑉_𝑇𝜕𝑥^′𝐵𝑥𝑥^′𝐵^′𝜕𝑉_𝑇𝜕𝑥=4^∞_𝑚=2⊗_𝑚𝑥,_{𝑖+𝑗=𝑚}𝐵^(𝑖)𝑃_𝑖⊗𝑃_𝑗𝐵^(𝑗)′⊗_𝑚𝑥,𝑥^′𝐶^′𝜕²𝑉_𝑇𝜕𝑥²𝐶𝑥=2^∞_𝑖=1⊗_𝑖𝑥,𝑅^(𝑖)𝑃_𝑖⊗_𝑖𝑥. (26)(27)(28)(29)(30)(31) Substituting (26)–(31) and (21) into (8) with terminal conditions 𝑃_𝑖(𝑇)=0(𝑖=1,2,…), we can prove that 𝑉_𝑇(𝑡,𝑥) satisfies HJE (8). By Theorem 2, the 𝐻_∞ control for system (6) can be given as 𝑢^∗_𝑇=−𝑥^′𝐵^′𝜕𝑉_𝑇.𝜕𝑥 (32) Similar to (29), we prove that the 𝐻_∞ control 𝑢^∗_𝑇 can be represented with the form of (18).

By Theorem 4, we can obtain the approximation of 𝐻_∞ control for system (6).

Now we apply the result of tensor power approach to an example.

Example 10. Consider the system (6) with coefficients ,,.𝐴=0.040.020.020.04,𝐵=0.020.020.020.02𝐾=0.020.02,𝐶=−0.020.040.020.02𝐺=0.02−0.020.020.02,𝑀=−0.040.02 (33)𝑁(𝑡) is Poisson measure with parameter 𝜆=2; 𝑊(𝑡) is 1-dimensional Brownian motion and 𝛾=1. Here the approximation of 𝑢^∗_𝑇 is given by 𝑢^∗=−2⁶_𝑖=1⊗_𝑖𝑥,𝐵^(𝑖)𝑃_𝑖⊗_𝑖𝑥, (34) and Figure 1 is the simulation of 𝑚(𝑥)=𝑀𝑥 and 𝐻_∞ control 𝑢^∗, where 𝑢^∗ is the approximation of 𝑢^∗_𝑇 of system (6). For the theories of simulation, we will discuss them in another paper. Here we only give the results of simulation.

Figure 1 

Tracking performance of Example 10.

5. Concluding Remarks

We have discussed the state feedback 𝐻_∞ control for a class of bilinear stochastic system where both Brownian motion and Poisson process are present. In order to solve the HJE given in the paper, we also discuss the method of tensor power series approach.

Conflict of Interests

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

Acknowledgment

This work was supported by the NSF of China, Grant nos. 61174078, 11271007, and 61170054.