State feedback stabilization problem of stochastic high-order and low-order nonlinear systems with time-delay

: This manuscript considers the state feedback stabilization problem for a class of stochastic high-order and low-order nonlinear systems with time-delay. Compared with the previous results, a distinctive feature to be studied is that the considered systems involve high-order, low-order, intricate stochastic di ff usion terms and time-delay simultaneously. First, the homogeneous domination approach and suitable coordinate transformations are introduced to obtain the updating laws. Then, a state feedback controller is devised to make the closed-loop systems globally asymptotically stable in probability. Finally, a simulation example is shown to prove the proposed approach powerfully


Introduction
Recently, the theory of nonlinear systems with time-delay has been a hot topic, due to its wide application in practical problems, such as physical engineering, biological systems and economic processes. Among these, the Lyapunov-Krasovskii methodology plays a crucial role in dealing with time-delay systems. Based on the above method, Pepe [1] addressed the input state stability of nonlinear systems with time-delay. Zhang [2] designed a stabilized controller for time-delay feed-forward nonlinear systems to achieve system stability. In order to address the stabilization problem of high-order nonlinear systems with time-delay, some researchers try to find new ways to design corresponding controllers. Yang and Sun [3] investigated the state feedback stabilization problem of controlled systems with high-order or/and time-delay via the homogeneous domination idea. With the help of the saturation function technique, homogeneous domination idea and Lyapunov approach, Song [4] studied the stabilization problem of high order feed-forward time-delay nonlinear systems. In addition to the above works, many results in [5][6][7][8][9][10] have established and improved the concept framework of nonlinear systems with time-delay.
Ever since the stochastic stability theory was founded and enriched by Deng and Zhu [11,12], great progress has been made on the global stabilization of stochastic nonlinear systems [13][14][15][16]. Subsequently, Florchinger [17] extended the theory of control with the Lyapunov-Krasovskii functional. With the stochastic stability theory in mind, it is still important and meaningful to address high-order stochastic nonlinear systems with time-delay. Zha [18] investigated the issue of output feedback stabilization. Liu [19] studied the output feedback stabilization problem for time-delay stochastic feed-forward systems. By using a power integrator approach, the work in [20][21][22][23] also considered the state-feedback stabilization problems. However, the state feedback stabilization problem for stochastic high-order and low-order nonlinear systems with time-delay has not been well addressed, which leads us to take the interesting problem into account.
How to deal with the state feedback stabilization problem for high-order and low-order nonlinear systems with time-delay? By using a power integrator approach, Liu & Sun [24] constructed a time-delay independent controller for the aforementioned systems to relax the growth condition and the power order limitations. However, to the best of our knowledge, research on the corresponding stochastic version is limited with scarcely a few convincing results. The main difficulties are explained from two aspects. On one hand, the Itô formula brings the gradient terms and the Hessian terms in the Lyapunov analysis. On the other hand, the particularity of its structure has made many traditional methods inapplicable. Therefore, we need to give a new way to consider stochastic nonlinear systems. Inspired by a large number of results in [25][26][27][28][29], stochastic high-order and low-order nonlinear systems with time-delay will be considered as follows: . , x n (t)] T ∈ R n is state, and u(t) ∈ R is input; the nonnegative real number τ is the time-delay of the states. ω(t) = [ω 1 (t), . . . , ω r (t)] T . The high-order can be revealed by p i ∈ R >1 odd =: { p q |p ≥ q > 0 and p, q are odd integers}. The drift terms f i : R i × R i × R + −→ R and the diffusion terms g i : R i × R i × R + −→ R r , i = 1, . . . , n are considered as locally Lipschitz with f i (0, 0, t) = 0 and g i (0, 0, t) = 0.
The contributions are highlighted in the following: (i) Systems considered are more general. Systems in [24] only solve the control issues for deterministic cases. It is more complex to consider the stochastic disturbance. By using the homogeneous domination idea, one can give a novel perspective to generalize the control strategy for deterministic systems to the corresponding stochastic cases.
(ii) The result extends the works [30][31][32] by relaxing the growth condition and the power order limitations. The low order of the nonlinear terms is successfully relaxed to the high-order and loworder of the nonlinear terms. Based on the above situations, we use a proper Lyapunov-Krasovskii functional to handle the stabilization problem under the weaker assumptions.
Notations: R + ≜ {x|x ≥ 0, x ∈ R}, R n ≜ {x n |x ≥ 0}. For a given vector/matrix D, D T denotes its transpose, T r{D} is the trace when D is square, and the Euclidean norm of a vector |D|. C i is composed of continuous and ith partial derivable functions. K is composed of continuous functions and strictly increasing; K ∞ is composed of functions with K. One sometimes denotes X(t) by X to simplify the procedure.

Problem statement
Now, the time-delay stochastic nonlinear systems are addressed as follows: is an initial data, and ω(t) denotes a Brownian motion with dimension r defined on a complete probability space (Ω, F , {F t } t≥0 , P).
The following assumptions are needed: Assumption 1. For i = 1, . . . , n, there exist two constants a 1 > 0 and a 2 > 0 such that in which θ= m n ≥ 0, n is an odd integer, m is an even integer, and r ′ i s have the following definitions: Remark 1. Assumption 1 encompasses and extends high-order and/or low-order results. We discuss this point from two cases. Case I: Condition (2.2), when τ = 0 it reduces to high-order growth condition with θ ≥ 0, and low-order growth condition with θ = 0, We further discuss its significance from value ranges of both low-order and high-order. From θ ∈ (− 1 p j ...p i−1 , 0], it is easy to see that 0 < r i +θ r j ≤ 1 p j ...p i−1 , which implies that both low-order and high-order in Assumption 1 can take any value in (0, 1 p j ...p i−1 ], [ 1 p j ...p i−1 , +∞), respectively. Case II: When τ 0, several new results [18][19][20][21][22] have been achieved on feedback stabilization of high-order nonlinear time-delay systems. The nonlinearities in [18][19][20][21][22] only have high-order terms. The nonlinearities in [24] include linear and nonlinear parts, and their nonlinear parts only allow low-order 1 p j ...p i−1 and high-order r i +θ r j with θ ≥ 0. While in this paper, (2.2) not only includes time-delays but relaxes the intervals of low-order and high-order.

Remark 2.
When p i = 1, i = 1, 2, . . . , n − 1, and τ = 0, equation (1) reduces to the well-known form, for which the feedback control problem has been well developed in recent years [16,24,26]. Proposition 1. For r 1 , . . . , r n and σ = p 1 . . . p n r n+1 having the following properties: Remark 3. It is not difficult to see that system (1.1) is a class of high-order and low-order stochastic nonlinear systems with time-delay satisfying Assumption 1. Compared with [30], it is significant to point out that system (1.1) addressed here is more general. The systems can be composed by timedelay and the coupling of the high-order and low-order terms. Moreover, if g = 0, Assumption 1 will generate the same assumption as in [24]. When p i > 3, the state feedback stabilization problem under constraint p i = p can give similar results as [19]. Under Assumption 1 with τ = 0, we can obtain the same results with [30], if there are no low-order nonlinearities.
Remark 4. For the case of τ = 0 in system (1.1), with the help of adding a power integrator, fruitful results have been achieved over the past years. However, for the case of τ 0, some essential difficulties will inevitably be encountered in constructing the desired controller. For instance, the time-delay effect will make the common assumption on the high-order system nonlinearities infeasible, and what conditions should be placed to the nonlinearities remains unanswered. Second, due to the higher power, time-delay and assumptions on the nonlinearities, it is more complicated to find a Lyapunov-Krasovskii functional which can be behaved well in theoretical analysis.

Useful definitions and lemmas
For ease of the controller design, some helpful definitions are presented.
For any given C 2 function V(x, t), the differential operator L is defined as follows: . . , ε h n x n ), and h i is referred to as the weights. And one defines dilation weight as △ = (h 1 , . . . , h n ). • γ , for any x ∈ R n , where γ ≥ 1. We use ∥x∥ △ or ∥x∥ △,2 to a exhibit 2-norm.
With the above definitions, we give some lemmas which will be crucial for controller design.
odd , ∀a ∈ R and ∀b ∈ R, there hold Lemma 2.
[13] For given a, b ≥ 0 and a given positive function f (x, y), there exists a positive function g(x, y), such that Lemma 3.

Control design procedures
Consider the stochastic high-order and low-order nonlinear systems with time-delay as follows: (3.1) Step 0: To begin with, introducing the complete form of the controller, The purpose of this work is to construct a state controller to render system (1.1) globally asymptotically stable in probability. To achieve this goal, propositions are presented as follows.

Stability analysis
The main result of this manuscript will be stated as follows. (ii) The equilibrium at the origin is globally asymptotically stable in probability.
Proof. Four steps are used to verify Theorem 1.
Step 1: By the definition of ϱ > 0, we know that p 1 . . . p j−1 − 1 > 1, which implies that 4 − 1 p 1 ...p k−1 −1 > 2, 4σ−r k+1 p k r k p k−1 ...p 1 > 2. Therefore, is continuous, and u p n = L k n+1 ν p n is C. As is known to all, the function is C. The closed-loop system satisfies the locally Lipschitz condition based on f i and g i being locally Lipschitz.
Step 4: Because (3.20) is an equivalent transformation, the system composed by (1) and u p = L k n+1 v p is similar to the systems (3.20) and (3.22). □ Remark 5. Compared with [24], we construct a state-feedback controller independent of time delays for the stochastic nonlinear system. Compared with [30], we use the methods of adding a power integrator to relax the nonlinear growth condition to cover both high-order and low-order nonlinearities. Not only does it not need to know anything information about the unknown function, but also it can reduce burdensome computations.
Remark 6. The homogeneous domination method is used for the first time to solve the stabilization problem of stochastic high-order and low-order nonlinear system (1.1) with time-delay.
Remark 7. In this paper, it is hard to adopt a Lyapunov-Krasovskii functional. In order to solve the above the problem, a suitable Lyapunov-candidate-function is designed to guarantee good system performance, and stabilization analysis is proposed to save better resources Remark 8. The construction of the controller effectively keeps away from the zero-division problem of ∂ 2 ξ * µ/r i i ∂ξ 2 j . It need be noted that the non-zero-division problem and the locally Lipschitz condition (see Step 1 in the proof of Theorem 1) should to be guaranteed simultaneously, which will increase more difficulties.

Conclusions
In this technical note, we investigate the state feedback stabilization problem of stochastic highorder and low-order nonlinear systems with time-delay successfully. According to the homogeneous domination method and the design of integral Lyapunov functions, the control strategy is achieved with the controller design. The above results indicate that the closed-loop system is globally asymptotically stable in probability. There still remain problems to be investigated, such as how to take into account output feedback control and how to extend our results under weaker conditions.