The H ∞ Control for Bilinear Systems with Poisson Jumps

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 statespace 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 problemwith randomPoisson jumps, andmany 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 ∞


Introduction
The main purpose of  ∞ control design is to find the law to efficiently eliminate the effect of the disturbance [1,2].Theoretically, study of  ∞ control first starts from the deterministic linear systems, and the derivation of the statespace formulation of the standard  ∞ control leads to a breakthrough, which can be found in the paper [3].In recent years, stochastic  ∞ control systems, such as Markovian jump systems [4][5][6],  ∞ Gaussian control design [7], and Itô differential systems [8][9][10][11][12][13], have received a great deal of attention.However, up to now, most of the work on stochastic  ∞ 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][17][18][19][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  ∞ -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  ∞ 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  under  ∞ control.This paper will follow along the lines of [22] to study the stochastic  ∞ 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  ∞ by solving the HJE which is proved by the completing square method.In Section 4, we discuss the problem of finite horizon  ∞ control with jumps, and using the tensor power series approach, we discuss the approximating  ∞ control given in the paper.For convenience, we adopt the following notation.
S  (R) denotes the set of all real × symmetric matrices;   is the transpose of the corresponding matrix ;  > 0 ( ≥ 0) is the positive definite (semidefinite) matrix ;  is the identity matrix; E is the expectation of random variable ; ‖‖ is the Euclidean norm of vector  ∈ R   and   is the dimension of ;

Preliminaries
For a given complete probability space (Ω, F, P), let   and  be the Brownian motion and the Poisson random measure, respectively, which are mutually independent: (i) a 1-dimensional standard Brownian motion {  } ≥0 ; (ii) a Poisson random measure  on R + ×, where  ⊂ R  is a nonempty open set equipped with its Borel field B(), with the compensator μ(, ) = (), Here  is an arbitrarily given -finite Lévy measure on (, B()), that is, a measure on (, B()) with the property that ∫  (1 ∧ || 2 )() < ∞.We also let where N denotes the totality of -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   be a square integral continuous martingale;   is a continuous adapted process with finite variance.(, ) is locally square integral due to  and ; () satisfies the following Itô type stochastic process: Then for (, ) ∈  1,2 (R + , R   ), we have (see [23] where ⟨⟩ denotes the predictable compensator of martingale . In the paper, for convenience,   is shorten as .Furthermore, for  ∈  1,2 (R + , R   ), if using Itô formula to (,   ) and integrating from  to  (0 ≤  < ), then taking expectation with both sides we can see that E(,   ) is continuous with respect to time .Since we mainly use the results of expectations of some well functions on   and those expectations are continuous with respect to time , so, for briefness, in the rest of this paper the sign  − under integration ∫  is also shortened as .
Taking expectation with both sides and applying (, ) = 0 and (, 0) = 0, we obtain Completing square for  and V, respectively, we have where By HJE (8) and let  =  *  , we have So, the following inequality is true: This proves that  *  is an  ∞ control for system (6).
Remark 3. From the proof of Theorems 2 and (13), we can see that ( *  , V *  ) given by ( 12) is a saddle point for the following stochastic game problem:
In order to prove Theorem 4, we need the following lemmas, and Lemmas 5-8 are given without proofs.

Lemma 5. For any
where  () = ∑   =1    ⊗  (,) , and   is the th row vector of matrix .Lemma 9.For any matrix  ∈ S    (R) and integer , we have where ),   is the th row vector of matrix C, and  () is determined as  () in Lemma 8.