Globally Asymptotic Stability of Stochastic Nonlinear Systems with Time-Varying Delays via Output Feedback Control

1Department of Mathematics and Computer Science, Tongling University, Tongling 244000, China 2School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, China 3Department of Mathematics, University of Bielefeld, 33615 Bielefeld, Germany 4School of Mathematics and Information Technology, Nanjing Xiaozhuang University, Nanjing, Jiangsu 211171, China


Introduction
In recent years, the stochastic nonlinear system has received much attention and has enjoyed a good development, which has been widely applied in many fields such as engineering and finance [1].Particularly, owing to the reason that a system by the output feedback is more flexible to respond to the information of control systems, the stochastic nonlinear system via the output feedback control has been more widely studied.Marino and Tomei [2] and Battilotti [3] designed linear observer to study the output feedback control for nonlinear systems.As a result, a large number of researchers have been focused on the stability analysis of stochastic nonlinear systems and the design of controller.
Since the backstepping method has been introduced in the nonlinear system, the theory of stochastic nonlinear systems has achieved a remarkable development.According to the different Lyapunov function, there are two ways to solve the problem.One is that Pan and Basar [4][5][6] considered a quadratic Lyapunov function.By using the backstepping method to design the controller, they discussed the risksensitive optimal control problem.The other is that Deng and Krstić [7][8][9][10] used a quartic Lyapunov function to guarantee that the closed-loop system is globally asymptotically stable in probability.Based on their works, Liu et al. [11][12][13] employed the quartic Lyapunov function to design the output feedback control for stochastic nonlinear systems.M.-L.Liu and Y.-G.Liu [14] and Chen et al. [15] used such Lyapunov function to study the state feedback stability for stochastic nonlinear systems with time-varying delays.In [16], Du et al. discussed the global output feedback stability for a class of uncertain upper-triangular systems with the input delay.Finite-time stability for stochastic nonlinear systems in strictfeedback form was considered in [17].More results can be found in [18][19][20].
In this paper, we are concerned with the globally asymptotic stability of stochastic nonlinear system with timevarying delays.We extend the results of the deterministic nonlinear systems in [21] to the stochastic nonlinear systems with time-varying delays by output feedback control and design a Lyapunov-Krasovskii functional to prove the globally asymptotic stability of the closed-system via the linear observer.It is obvious that the form of the linear controller is simpler than that used in [9].And compared with the  1 output feedback controller obtained by the homogeneous observers in [20], the controller designed by the linear observers in this paper is smooth.
The rest of this paper is organized as follows.In Section 2, we present some definitions and establish a new inequality, which plays an important role in the proof of our main results.In Section 3, we construct a Lyapunov-Krasovskii functional and design a linear output feedback control to prove the globally asymptotic stability based on the linear observers.In Section 4, we use an example to illustrate our theoretical results.Finally, we conclude this paper with some general remarks.

Preliminaries
In this section, we will give the following notations, definitions, and some preliminary lemmas.R + denotes the set of all nonnegative real numbers; R  denotes the real dimensional space; Trace{} denotes the trace for square matrix ; || denotes the Euclidean norm of a vector . 1 denotes the family of all functions which are C 2 in the first argument and C 1 in the second argument.K denotes the set of all functions: R + → R + , which are continuous and strictly increasing and vanish at zero; K ∞ denotes the set of all functions which are of class K and unbounded.KL is the set of all functions (, ) : R + × R + → R + , which are of K for each fixed  and decrease to zero as  → ∞ for each fixed .
Lemma 6 (see [1]).Let  :  →  be a continuous positive definite on a domain  =   that contains the origin.Let   ⊂  for some  > 0.Then, there exist class K functions  1 and  2 , defined on [0,r], such that for all  ∈   .If  = R  , the functions  1 and  2 will be defined on [0, ∞) and the foregoing inequality will hold for all  ∈ R  .Moreover, if (()) is radially unbounded, then  1 and  2 can be chosen to belong to class K ∞ .

The Output Feedback Model and Control Design
In this section, we will design a linear observer system for a class of stochastic nonlinear systems with time-varying delays.Using the backstepping method, a simple linear controller will be constructed to guarantee that the closedloop stochastic system is globally asymptotically stable in probability.
Remark 11.It should be pointed out that Zhai and Zha [23] considered the global adaptive output feedback control for system (18) without the diffusion terms.And Duan and Xie [24] discussed the globally asymptotic stability for system (18) The linear observer system is designed as where  ≥ 1 is a constant to be determined and   > 0,  = 1, . . ., , are coefficients of the Hurwitz polynomial The observation error   = (  − x )/ −1 satisfies where  is a Hurwitz matrix.Therefore, there is a positivedefinite matrix  such that Theorem 13.Suppose Assumption 12 holds.Then, the equilibrium at origin of closed-loop stochastic nonlinear system (18) and (20) with the linear controller (50) below is globally asymptotically stable in probability and {lim →∞ |()| = 0} = 1.Furthermore, the closed-loop system has a unique solution on [−, ∞) for each  ∈ R  .
By Lemmas 8 and 9 and Assumption 12, one gets Substituting ( 25) into ( 24), we have where then Substituting   = x +  −1   ,  = 1, . . ., , into (29), one gets where Next, we will combine the backstepping design method with the mathematical induction to design the linear output feedback control.
Define  2 = x2 − x * 2 with x * 2 being a virtual control.Then, we have Choose the virtual controller where  1 > 0 is independent of .Hence, Step .Suppose that, at Step , there exists a smooth Lyapunov function   which is positive definite, radially unbounded, and twice continuously differentiable, satisfying with a set of virtual controllers x * where Obviously,  0 ,  1 , . . .,  +1 are suitable real numbers and they are independent of the gain constant .
Let us consider the following Lyapunov function: Then, combining (36) and (38), we obtain It follows from Lemma 7 that In particular, the last term of (41) yields ) −3 Hence, where ) , Now, we choose the following linear controller: with  +1 > 0 being independent of .
So it follows from (44) that Step .Using the inductive argument step by step, at the ( − 1)th step, we get Noting the function   =  −1 + (1/4 4−4 ) 4  , we have We now design the linear controller where   = 1 +   0 +   1 + ⋅ ⋅ ⋅ +   −1 + 8 1 +   > 0 is a real number independent of the gain parameter .Thus, where  3 (, where  1 and   are positive parameters and It is easy to verify that  is C 2 on .Suppose that () is continuous, positive definite, and radially unbounded.Then by Lemmas 5 and 6, there exist positive constants  and  and two class K ∞ functions  1 and  11 such that one gets Therefore, by Lemma 4, the equilibrium of closed-loop nonlinear stochastic system ( 18), (20), and (50) is globally asymptotically stable and the closed-loop system has a unique solution on [−, ∞) for each  ∈ R  .
Remark 15.It should be pointed out that, compared with the  1 output feedback controller in [20], we see that the linear controller (50) based on the observable linearization is a smooth one.
Remark 16.In this paper, combining the backstepping design with mathematical induction, we extend the method in [21] to the stochastic nonlinear system with time-varying delay.Although the globally asymptotic stability of stochastic nonlinear system has been discussed by Liu and Zhang in [12], they did not consider the term of time-varying delay.Therefore, our conclusion is proposed for the first time.
Remark 17.In [22], Liu and Xie considered the state feedback stability of stochastic feedforward nonlinear systems with time-varying delays.They introduced the homogeneous domination approach to construct a state feedback controller.
It is generally known that in real lives the output feedback control is easier to achieve than the state feedback control and has great theory significance and utility value.Therefore, in this paper we combine the backstepping method with the mathematical induction to design a Lyapunov-Krasovskii functional and construct a linear output feedback controller for stochastic nonlinear systems with time-varying delay.Moreover, we impose different restricted conditions on functions   and   ( = 1, . . ., ) to ensure the global stability of stochastic nonlinear systems with time-varying delays.

An Example
In this section, we will use a simulation example to illustrate our main results.
Consider the following stochastic nonlinear system:

Conclusion
In this paper, we combine the backstepping method with the mathematical induction to prove that a class of stochastic nonlinear systems with time-varying delays by the output feedback control is globally asymptotically stable in probability.It should be pointed out that we not only extend the results in deterministic nonlinear systems to more complex stochastic case but also add time-varying delays to stochastic nonlinear systems.For the term of time-varying delays, we design a Lyapunov-Krasovskii functional and a linear output feedback controller to render the closed-loop system globally asymptotically stable.