On Exponential Stability of Composite Stochastic Control Systems

and Applied Analysis 3 where Y is the infinitesimal generator for the stochastic process solution of the SCS (7). Theorem 5. Suppose that the origin of the SCS (7) is rESP. Then, there exists a Lyapunov function Φ : Rr → R+ of class C 2 (R r \ {0}) which satisfies all conditions (10)–(13). Note that Definition 3 and Theorem 5 provided in Spiliotis and Tsinias [2] are an extension of Definition 1 and Theorem 2, respectively, established in Khasminskii [1]. Definition 6. Theorigin of the SCS (7) is asESP for some r > 0 if, and only if, there exists a constant c > 0 and a random variable 0 ≤ B υ,u < ∞, such that |V (t)| ≤ B υ,u e −c(t−t 0 ) , (14) for any υ ∈ Rr, t ≥ t 0 , and for almost all β. We now derive the Florchinger’s decomposition [20] of the functions F (υ, y) = f (υ, l (υ, y)) h (υ, y) , G (υ, y) = g (υ, y)√h (υ, y), (15) and we consider the SCS (7) in the form

In recent years, the stablizability of various types of stochastic systems has been studied for different concepts of stochastic stability (see, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]).The global asymptotic stability in probability of stochastic systems by means of strict and output feedback laws has been considered by Krstic and Deng [6] and Deng and Krstic [8], where backstepping design procedure is proposed for stochastic systems.Tsinias [7] extended the input-to-state stability results of Sontag and Wang [18] in deterministic case and obtained exponential input-to-state stability in probability for stochastic system and derived sufficient conditions for global stabilization of this system by means of static and dynamic output feedback.Caraballo and Liu [9] developed some criteria for the almost sure and mean square exponential stability in probability of nonlinear stochastic equations of monotone type.Spiliotis and Tsinias [2] derived control Lyapunov function and established rESP and asESP of stochastic differential systems.The asymptotic and exponential stability of stochastic differential systems (SDSs) has been studied by Mao [10], Liu and Raffool [11], Lan and Dang [12], Luo [13], Abedi et al. [14], Abedi and Leong [15], Khasminiskii [1], and Kushner [16].Michel [17] established asymptotic and exponential stability in probability for some classes 2 Abstract and Applied Analysis of continuous and discrete-parameter stochastic composite systems.Later, Florchinger [3] and Boulanger [4] developed sufficient conditions for asymptotic stability in probability and exponential stability in mean square for a special case of our CSCS (1).The sufficient conditions for exponential stability in probability of nonlinear stochastic systems and special case of our CSCS (1) have been derived by Rusinek [5].
The structure of the paper is as follows.In Section 2, we introduce a class of CSCS and we also recall some basic definitions and results concerning rESP and asESP property.In Section 3, we state and prove the main results of the paper.Finally in Section 4, we give two numerical examples illustrating our results.

Fundamental Definitions and Results
In this section, we introduce the class of stochastic systems and recall some definitions and results concerning exponential stability in probability of these systems.For a complete presentation of exponential stability, we refer the reader to the book of Khasminiskii [1] and the paper of Spiliotis and Tsinias [2].
Let (Ω, , ) be a complete probability space and denote by (  ) ≥0 a standard   -valued Wiener process defined on this space.
We consider the SDS where the following conditions hold: (i)  is given in   , (ii)  :   →   and ℎ :   →  × are Lipschitz functionals in   with (0) = 0, ℎ(0) = 0, and there exists a constant  ≥ 0 such that the following growth condition holds: for any  ∈   .Under restriction on growth (4), for any  ∈   , that guarantees existence and uniqueness of solution () = (,  0 ,  0 ) for (3) starting from  0 at time  0 , we recall the following definition of exponential stability in mean square and a converse Lyapunov theorem given by Khasminskii [1] as follows.
where D is the infinitesimal generator for the stochastic process solution of the SDS (3).
Definition 3. The origin of the SCS ( 7) is rESP for some  > 0 if, and only if, there exist constants  1 ,  2 > 0 such that for any  ∈   , and  ≥  0 .
Definition 4. The SCS ( 7) is said to satisfy the exponential Lyapunov condition if there exists a Lyapunov function Φ : Abstract and Applied Analysis where Y is the infinitesimal generator for the stochastic process solution of the SCS (7).
Theorem 5. Suppose that the origin of the SCS (7) is rESP.
Note that Definition 3 and Theorem 5 provided in Spiliotis and Tsinias [2] are an extension of Definition 1 and Theorem 2, respectively, established in Khasminskii [1].Definition 6.The origin of the SCS ( 7) is asESP for some  > 0 if, and only if, there exists a constant  > 0 and a random variable 0 ≤  , < ∞, such that for any  ∈   ,  ≥  0 , and for almost all .
We now derive the Florchinger's decomposition [20] of the functions  (, ) =  (,  (, )) ℎ (, ) , and we consider the SCS (7) in the form where  is a function mapping   ×   into   and ℎ is a nonnegative function mapping   ×   into .In the following theorem, we use an explicit formula of a feedback law established in [15,20] and exhibit the rESP property for the resulting closed-loop system deduced from (7) or, equivalently, the resulting closed-loop system deduced from (16).
Proof.Suppose that there exists a positive function (()) such that the SCS (7) or, equivalently, the SCS ( 16) satisfies the exponential Lyapunov condition.Then from (13) we have where D 0 is the infinitesimal generator of the resulting closed-loop system deduced from (7) and Y 1 is the infinitesimal generator for the stochastic process solution of the resulting closed-loop system deduced from ( 16) as follows: The desired condition ( 9) is a direct consequence of inequality (10) and (19).Therefore, the resulting closed-loop system deduced from SCS (16) satisfies in rESP property at the origin.
We will now turn the attention to a general composite stochastic system and provide some results related to the rESP of this system.
is rESP.
Suppose that there exists functionals  1 :   →   ,  2 :   →  × ,  1 :   →  × , and  2 :   →  ×× such that for any (, ) ∈   ×  .As mentioned above, we can consider the general SCS (7) into the form of ( 16).This form of ( 16) is the same as (26), which is a special case of (7).By taking into account ( 7), (16) In the next section, we will establish a state feedback law that guarantees the satisfaction of rESP and asESP property for CSCS (27).

Exponential Stability of Composite Systems
Our aims of this section are twofold.On one hand, we study the problem of finding state feedback law that guarantees that the CSCS (27) satisfies rESP property.We derive sufficient conditions for rESP of this system.On the other hand, we establish sufficient conditions for asESP of the CSCS (27).
In the following theorem, we suppose that the function  and  are bounded on   ×   and  is the set of admissible control and establish a sufficient conditions for rESP property of CSCS (27).Theorem 9 is a stochastic extension of Proposition 3.1, Theorem 5.1, and Theorem 4.1 stated in Spiliotis and Tsinias [2], Florchinger [3], and Boulanger [4], respectively, to a general composite stochastic system.Both the results and the proof used in this theorem, however, are different from those in [2][3][4].Furthermore, we can consider the exponential stability in mean square results of Florchinger [3] and Boulanger [4] as a special case of our rESP results (Theorem 9) where  = 2.
where (()) = Φ 2 () + (, ), that guarantees that the resulting closed-loop system satisfies the rESP property.Furthermore, assume that Then, the state control law where Φ 1 is a smooth Lyapunov function corresponding to the SCS (28), guarantees that the CSCS (27) satisfies rESP.
Proof.Suppose that the origin of the stochastic system (28) satisfies the rESP property.Then by the converse Lyapunov Theorem 5, there exists a Lyapunov function Φ 1 () and positive constants   , 1 ≤  ≤ 5, such that (10)-( 12) hold and On the other hand, since the origin is a rESP for the closedloop system (30), then by converse Lyapunov Theorem 5, there exists a Lyapunov function Φ 2 () and positive constants    , 1 ≤  ≤ 5, such that the following conditions hold: where Y 2 is the infinitesimal generator for the stochastic process solution of the resulting closed-loop system (30).Consider the composite function for any (, ) ∈   ×   .Obviously, Φ(()) is a Lyapunov function.We will show that Φ(()) satisfies in the exponential Lyapunov condition.A simple calculation shows that conditions (10)-( 12) are satisfied.It remains to establish condition (13).Substituting (, ) into the closedloop system deduced from CSCS (27), we have Denoting D  as the infinitesimal generator of the stochastic process solution of the resulting closed-loop system (39) yields for any (, ) ∈   ×   .Substituting (31) into (40), we get Thus, we obtain by taking into account (33) and (37) that From ( 42 where Φ 1 and Φ 2 are Lyapunov functions corresponding to the SDS (28) and SCS (30), respectively, and   is the infinitesimal generator of the stochastic process solution of the resulting closed-loop system (39).The desired condition ( 9) is a direct consequence of inequality (10) and (43).Therefore, CSCS (27) satisfies in rESP property at the origin.
(i) Theorem 9 is a stochastic extension of Proposition 3.1, Theorem 5.1, and Theorem 4.1 stated in Spiliotis and Tsinias [2], Florchinger [3], and Boulanger [4], respectively, to a general composite stochastic system.Both the results and the proof used in this theorem, however, are different from those in [2][3][4].Furthermore, we can consider the exponential stability in mean square results of Florchinger [3] and Boulanger [4] as a special case of our rESP results (Theorem 9) where  = 2.
(ii) The rESP and asESP results for stochastic system proved in Spiliotis and Tsinias [2] are a special case of our rESP and asESP results for CSCS (21).In addition, the existing exponential stability results established in [2][3][4][5][6][7][8] do not permit us to make a conclusion about rESP and asESP for the broader class of CSCS (21) at the origin, whereas the results of this paper are still valid.

Applications
This section illustrates applicability of our results by designing the following two numerical examples.
Example 12. Consider the SDS where (  ) ≥0 is a standard real-valued Wiener process and  ∈ .Assume that Φ() =  2 is a smooth Lyapunov function corresponding to the SDS (47).Then, a simple calculation shows that the SDS (47) satisfies rESP.
Next, consider the SCS where (  ) ≥0 is a standard real-valued Wiener process,  ∈ , and  is a real-valued measurable control law.Obviously,

Conclusions
In this paper, we have provided the new results for rESP and asESP of the CSCS