Stabilization of the stochastic jump diffusion systems by state-feedback control
Introduction
During the last decades, stochastic jump diffusion systems have become increasingly popular tools to describe the real world [1]. The jump component can capture event-driven uncertainties, such as corporate defaults, operational failures, or insured events. Indeed, such stochastic differential equations are finding a considerable range of applications, including physical sciences [2, Chapter 4], biology, engineering [3], financial economics in nonlinear signal processing [4, Chapter 10], and stochastic resonance [5]. In financial and actuarial modeling and other areas of application, such jump diffusions are often used to describe the dynamics of various state variables. In finance these may represent, for instance, asset prices, credit ratings, stock indices, interest rates, exchange rates or commodity prices.
Each stochastic jump diffusion, consisting of the continuous part and discontinuous jump, can be written as a linear combination of time t, a Brownian motion w(t) and a pure jump process. When the jump part of the system vanishes, one has a stochastic diffusion, known as stochastic differential equation (SDE). When the Brownian motion in the diffusion is also missing, the system reduces to the deterministic system, known as ordinary differential equation (ODE) [6]. This work is concerned with the stochastic jump diffusion systems. The Poisson jump process makes the formulation more versatile with a wider range of applications. In addition, it makes the study more difficult. So our targeted system is much more difficult to deal with than the SDE and ODE.
One of the important issues in the study of stochastic jump diffusion systems is automatic control, with consequent emphasis being placed on the asymptotic analysis of stability. Unlike the deterministic systems, stochastic systems may refer to the different stability concepts, such as asymptotic stability in probability, almost sure (exponential) stability, and mean square (exponential) stability, etc. Stability analysis of stochastic jump diffusion systems has attract much attention. Applebaum and Siakalli [7] discussed the asymptotic stability properties of stochastic differential equations driven by Lévy noise. Yin and Xi [6] investigated the stability of a class of switching jump-diffusion process. Zong et al. [8] discussed stability and stochastic stabilization of numerical solutions of a class of regime-switching jump diffusion systems. In [9], Applebaum and Siakalli showed that the Lévy noise can also be used to stabilize the unstable system almost surely. Bao and Yuan [10] studied the stabilization of partial differential equations by Lévy noise. These studies have not taken the structure of feedback control into account. For the stochastic systems without Poisson jump, there is an intensive literature and we mention, for example, [11], [12], [13], [14], [15], [16], [17] and the references therein. In particular, the two articles [17], [16] are more typical in the feedback control of stochastic diffusion systems. Li and Blankenship [18] studied the stabilization of unstable systems only driven by Poisson jump by feedback controls. There are a few recent studies dealing with the control design problem of jump diffusion systems, for example, Øksendal and Sulem [1] proposed the stochastic control in economics and finance model.
This paper is concerned with the almost sure exponential stabilization and the mean square exponential stabilization of stochastic jump diffusion equations by feedback control. Assume that we are given an unstable linear jump diffusion equationwith , where denotes the state vector, A, Bi and C are matrices in , is an d-dimensional Brownian motion and Nt is a Poisson process (independent of wt) with the jump intensity λ. The two stochastic processes wt and Nt are adapted to a filtration . This can be the natural filtration .
In order to stabilize the unstable diffusion system (1.1), we may need to restrict the control not only in drift term but also in the jump diffusion terms. Because that: There are lots of systems which cannot be stabilized if the control is restricted only in shift or jump diffusion parts, which can be shown similarly with the Example 1 in [17, Chapter 4]. (2) The theory developed in this paper can be applied directly to the case when the control is only in drift, diffusion or jump part (see the Example in Section 5).
We are required to find a state feedback control u(t) so that the corresponding controlled systembecomes stable in the almost sure and mean square sense. Here A Bi, A1, B1i, C, C1 are matrices with appropriate dimension. Because the given stochastic jump diffusion (1.1) is linear, it is natural to use a linear feedback control of the form , with K a constant matrix [17], [19]. Thus the controlled system (1.2) becomes Therefore, the stabilization problem is how to compute a matrix K such that the closed-loop system is (mean square and almost sure) stable.
To see the stability impact from Brownian motion and Poisson noise clearly, we take into account a simple one-dimensional linear stochastic systemwith the initial data , where are scalar constants. It is known that the explicit solution of the linear scalar system (1.3) is [20] whose second moment is It is evident that the sample paths tend to the origin almost surely if and only if (see [20]), and the second moment will tend to zero exponentially if and only if . That is to say, if we choose the constant c such that , then the Poisson process will work positively for stability (in the sense of mean square or almost sure). For the impact of Brownian motion, it is easy to see that Brownian motion works as destabilizing effect for mean square stability but plays positive impact for almost sure stability. In addition, the mean square stabilization impact of Levy noise has limitation, due to that in order to guarantee the condition hold, is the maximum stabilization contribution of Poisson noise for the fixed Poisson process N(t).
The rest of the paper is organized as follows: Section 2 begins with the related definitions and theorems on stochastic stability and gives the detailed analysis about the positive effect of Brownian motion and Poisson process in the mean square and almost sure stability. Section 3 proceeds with the study of mean square stabilization and almost sure stabilization of the stochastic linear time invariant (SLTI) systems. Section 4 furthers our investigation by examining the stabilization of uncertain system. An example is given in Section 5. Conclusions will be presented in Section 6.
Notations: Throughout this paper, denotes the n-dimensional Euclidean space and is the set of all n×m real matrices. Let be the Euclidean norm in . The superscript ‘T’ stands for matrix transposition and the star symbol ⁎ in a symmetric matrix is used to denote the transposed element at the symmetric position. If A is a matrix, its trace norm is denoted by . If A is a symmetric matrix, its largest and smallest eigenvalues are denoted by and , respectively. Let be a complete probability space with a filtration satisfying the usual conditions, that is, it is right continuous and increasing while contains all P-null sets. For symmetric matrices X and Y, the notation (respectively, ) means that the matrix is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.
Section snippets
Stability of stochastic jump diffusion systems
Let us consider the following stochastic jump diffusion system:with , where denotes the state vector and is an d-dimensional Brownian motion and Nt is a -adapted Poisson process (independent of wt) with the jump intensity λ. The compensated process of Poisson process Nt is . The vector or matrix-valued function f and g are assumed to be of appropriate dimensions. The solution of Eq. (2.1) is denoted by
Stabilization of the stochastic linear jump diffusion systems
In this section, two stabilization theorems are proposed by designing the state-feedback control law such that the closed-loop stochastic system (1.2) is mean square exponentially stable and almost sure exponentially stable. Theorem 3 The equilibrium of the stochastic system (1.2) is mean square exponentially stable with respect to state-feedback gain , if there exist a positive definite matrix X, a matrix Y and real number such that following LMIs hold:where
Stabilization of uncertain jump diffusion systems
It is very known that the environmental uncertainty exists in many dynamical systems and cannot be neglected. Thus, we develop our theory to cope with the uncertainty of system parameters in this section. More precisely, let us consider the uncertain stochastic controlled system of the formwith the state-feedback control . The closed-loop system is
Example
Let us consider the two dimensional stochastic jump diffusion systemwith the Poisson intensity , the initial value , and
It is easy to see that this system is mean square unstable (see Fig. 1).
Assume that we are required to define a state-feedback control in the noise terms, then we consider the following stochastic controlled system:where
Conclusions
In this paper we devise a linear matrix inequality approach to the state-feedback control of the stochastic systems driven by Brownian motion and Poisson process. The control structure of this paper appears not only in drift part but also in jump diffusion part of the stochastic system, and both theoretical results and illustrative examples are provided. We also give the proof of two types of exponential stability of stochastic jump diffusion systems, namely mean square exponential stability
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