Exponential input-to-state stability of composite stochastic systems
© Abedi et al.; licensee Springer 2013
Received: 13 March 2013
Accepted: 26 June 2013
Published: 11 July 2013
Sufficient conditions for the exponential input-to-state stability in probability in r th mean and for the almost sure exponential input-to-state stability in probability of a composite stochastic system are established. Illustrating example is provided to validate our results.
MSC:60H10, 93C10, 93D05, 93D15, 93D21, 93E15.
The stability of deterministic control systems has been widely developed in the past few decades (see, for instance, Sontag and Wang , Grune , Karafyllis and Tsinias , Ning et al. ). In these papers, necessary and sufficient conditions for input-to-state stability at the equilibrium state of deterministic control systems were provided. On the other hand, the asymptotic stability, exponential stability and input-to-state stability in probability at the equilibrium state of stochastic control systems (SCS) were established by Mao , Liu and Raffool , Lan and Dang , Abedi et al. [8, 9], Abedi and Leong , Khasminiskii , Kushner , Tsinias , Ito , Liu et al. , Duan et al.  and Yu and Xie . Particularly, Ito  developed the properties of input-to-state stability and gave the design of controller by using nonlinear small-gain conditions. While Liu et al.  discussed the property of practical stochastic input-to-state stability and its application to stability of cascaded nonlinear systems. The concept of integral input-to-state stability to the stochastic nonlinear systems was generalized by Yu and Xie  and Duan et al.  and state feedback controller that guarantees that all the signals of the resulting closed-loop system are bounded almost surely was also given.
Michel  developed exponential stability in probability of a continuous and discrete parameters composite stochastic system. Later, Boulanger  derived sufficient conditions for asymptotic stability in probability and exponential stability in mean square for a special case of our composite stochastic control system (CSCS). The sufficient conditions for exponential stability in probability of nonlinear stochastic systems, which are the special case of our CSCS, were established by Rusinek .
The purpose of the paper is to fill the gaps in the literature by studying the exponential input-to-state stability in probability in r th mean (REISSP) and the almost sure exponential input-to-state stability in probability (AEISSP) of a CSCS. We take this fact into account to extend the REISSP and AEISSP results established by Spiliotis and Tsinias  for SCS to the broader class of CSCS driven by two independent Wiener processes that were considered in Boulanger  and Rusinek . The main results of our work enable us to derive sufficient conditions for REISSP and AEISSP of the CSCS. We also establish the existence of an explicit formula for a feedback law exhibiting the REISSP and AEISSP property.
The rest of the paper is organized as follows. In Section 2, we introduce a wider class of CSCS and we also recall some basic definitions and results relating the REISSP and AEISSP property. Finally in Section 3, we use the results of Section 2 and prove that the CSCS satisfies the REISSP and AEISSP property. We also provide a numerical example to illustrate our results.
2 Assumptions and preliminaries
The aim of this section is to introduce a class of stochastic differential systems (SDS) and recall some basic definitions and theorems concerning an exponential control Lyapunov function for the exponential stability in probability of this system that we are dealing with in the rest of the paper. We also focus on the properties of the exponential control Lyapunov function, which plays an important role in the exponential stability, in Section 3.
Let be a complete probability space, and denote by a standard -valued Wiener process defined on this space.
is the infinitesimal generator for the stochastic process solution of SDS (1). Then we can recall the definition of exponential stability in probability in mean square and the stochastic version of the converse Lyapunov theorem established by Khasminskii  as follows.
where ∇ is the gradient operator and D is given in (3).
u is an -valued measurable control law,
- (2)F and G are Lipschitz functionals mapping from into , and , respectively, such that they are vanishing at the origin and there exists constant such that for any and , the following growth condition holds:(8)
Under restriction on growth (8) (see, for instance, Arnold ), for every input , and , there exists a unique solution of (7) starting from at time which is defined for all and almost all . In the following we assume and we recall the definition of RESP, AESP and the stochastic version of the converse Lyapunov theorem proved by Spiliotis and Tsinias  as follows.
where Y is the infinitesimal generator for the stochastic process solution of SCS (7).
Theorem 2.5 Suppose that the origin is RESP with respect to SCS (7). Then there exists a Lyapunov function of class which satisfies all conditions (10)-(12).
Note that Definition 2.3 and Theorem 2.5 in Spiliotis and Tsinias  are an extension of Definition 2.1 and Theorem 2.2, respectively, established in Khasminskii . Now, we recall the definition of REISSP and AEISSP exposed in  as follows.
Definition 2.8 We say that the origin is AEISSP with respect to SCS (7) if, there exist a positive definite function , constant , and a random variable such that (13) and (14) hold for any , and .
We shall now turn the attention to a broader class of CSCS and provide some results related to the REISSP and AEISSP of this system.
Let us first introduce a general composite stochastic system.
where the following conditions hold:
(H1) , , and D is a matrix function with value in ,
(H2) f and h are functionals in and , respectively, such that ,
(H4) u is an -valued measurable control law,
By considering the results on EISSP exposed in , we can establish the following outcome.
Proof The proof of this lemma is a direct consequence of Definitions 2.3, 2.6-2.8 and the fact that each solution of (18) that corresponds to some coincides with the solution of (17) with the same initial , and and corresponding to , namely, satisfying (14). Conversely, each solution of (17) under restriction (14) is a solution of (18) with input and the same initial value. □
In the next section we establish the state feedback law that guarantees the satisfaction of the REISSP and AEISSP property for CSCS (17).
3 Main stability results
The REISSP and AEISSP results proved in this paper employ the technique which is a combination of those by Sontag and Wang [1, 23], i.e., decomposition for a deterministic control system, and Spiliotis and Tsinias , i.e., decomposition in stochastic case for the SCS. We use these decompositions to obtain the existing exponential stability results for the CSCS. In the following theorem, we assume that the functions g and q are bounded on and U is the set of admissible control. To establish sufficient conditions for REISSP and AEISSP of CSCS (17), we use Lemma 2.9 and show that the origin is RESP and AESP for CSCS (18). Theorem 3.1 can be viewed as a stochastic extension of Proposition 4.1 and Theorem 4.1 stated in Spiliotis and Tsinias  and Boulanger , respectively, to a general composite stochastic system. Both the results and the tools used in this theorem, however, are different from those in [19, 21]. Furthermore, we can consider the exponential stability results of Boulanger  as a special case of our REISSP results (Theorem 3.1) where .
where is a smooth Lyapunov function corresponding to SDS (1) and is a state control law corresponding to the resulting closed-loop system deduced from (16), renders CSCS (17) satisfying the REISSP and AEISSP property.
Hence, the desired condition (9) is a direct consequence of inequalities (10) and (25). Therefore, CSCS (18) satisfies the RESP property at the origin. It turns out by Lemma 2.9 that CSCS (17) satisfies the REISSP property.
From (28) we have for all , almost surely, for some random . The latter in conjunction with (10), (25) and (26) implies (13). Thus, CSCS (18) satisfies the AESP property at the origin. It turns out by Lemma 2.9 that CSCS (17) will also satisfy the AEISSP property, which completes the proof. □
The following proposition (Proposition 3.2) is a stochastic extension of Proposition 4.2 stated in Spiliotis and Tsinias  to a general composite stochastic system.
for some positive constant k and certain , . The rest of the proof is a consequence of Theorem 3.1 and is, therefore, omitted. □
Theorem 3.1 is the stochastic extension of Proposition 4.1 and Theorem 4.1 stated in Spiliotis and Tsinias  and Boulanger , respectively, to a general composite stochastic system. Both the results and the tools used in this theorem, however, are different from those in [19, 21]. Furthermore, we can consider the exponential stability results of Boulanger  as a special case of our REISSP results (Theorem 3.1) where .
Proposition 3.2 is the stochastic extension of Proposition 4.2 stated in Spiliotis and Tsinias  to a general composite stochastic system. In addition, the existing exponential stability results established in [19, 20] and  do not permit us to make a conclusion about REISSP and AEISSP for the broader class of CSCS (15) at the origin, whereas the results of this paper are still valid.
Finally, we conclude the paper by designing a numerical example to validate our results.
guarantees that CSCS (30) satisfies the REISSP and AEISSP property.
We have developed the exponential input-to-state stability in probability of a larger class of multi-input composite system (15). We have used the stochastic version of converse Lyapunov theorems derived by Khasminiskii  and Spiliotis and Tsinias  to the concept of stochastic control Lyapunov function and extended the REISSP and the AEISSP results provided by Spiliotis and Tsinias  for a stochastic system to the larger class of composite stochastic systems driven by two independent Wiener processes that were considered in Boulanger  and Rusinek . The main results of our work enable us to derive the sufficient conditions for REISSP and AEISSP of a composite stochastic system. We have also established the existence of an explicit formula of a feedback law exhibiting the REISSP and AEISSP property of a composite stochastic system.
The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.
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