State-Feedback Stabilization for Stochastic High-Order Nonlinear Systems with Time-Varying Delays

and Applied Analysis 3 Especially, if W(x(t)) is a positive definite function, then the equilibrium x = 0 is globally asymptotically stable (GAS) in probability. In the remainder of this section, we list several lemmas that serve as the basis for the development of a state-feedback controller for system (1). They are the key tools for adding a power integrator technique. Lemma 3 (see [36]). For x ∈ R, y ∈ R, and p ≥ 1 being a constant, the following inequalities hold: 󵄨󵄨󵄨󵄨x + y 󵄨󵄨󵄨󵄨 p ≤ 2 p−1 󵄨󵄨󵄨󵄨x p + y p󵄨󵄨󵄨󵄨 , (|x| + 󵄨󵄨󵄨󵄨y 󵄨󵄨󵄨󵄨) 1/p ≤ |x| 1/p + 󵄨󵄨󵄨󵄨y 󵄨󵄨󵄨󵄨 1/p ≤ 2 (p−1)/p (|x| + 󵄨󵄨󵄨󵄨y 󵄨󵄨󵄨󵄨) 1/p . (6) If p ≥ 1 is odd, then 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 p ≤ 2 p−1 󵄨󵄨󵄨󵄨x p − y p󵄨󵄨󵄨󵄨 , 󵄨󵄨󵄨󵄨 x 1/p − y 󵄨󵄨󵄨󵄨 ≤ 2 (p−1)/p ( 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨) 1/p . (7) Lemma 4 (see [37]). For x, y ∈ R and positive real number p, the following inequality holds: 󵄨󵄨󵄨󵄨x p − y p󵄨󵄨󵄨󵄨 ≤ p 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 x p−1 + y 󵄨󵄨󵄨󵄨 ≤ c 󵄨󵄨󵄨󵄨x − y 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 (x − y) p−1 + y 󵄨󵄨󵄨󵄨 , (8) where c = p for 1 < p ≤ 2 and c = p2 for p > 2. Lemma5 (see [38]). Let x andy be real variables; then for any positive real numbers a,m, and n, one has a|x| m󵄨󵄨󵄨󵄨y 󵄨󵄨󵄨󵄨 n ≤ b|x| m+n + n m + n ( m + n m ) −m/n a (m+n)/n b −m/n󵄨󵄨󵄨󵄨y 󵄨󵄨󵄨󵄨 m+n , (9) where b > 0 is any real number. 3. Controller Design and Main Result The objective of this paper is to design a state-feedback controller of the form u = u (x) , (10) such that system (1) is GAS in probability at the origin. To this end, the following assumptions regarding system (1) are imposed. Assumption 6. For i = 1, . . . , n, there are positive constants c i1 and c i2 such that c i1 ≤ h i (t) ≤ ci2. (11) Assumption 7. For i = 1, . . . , n, there are nonnegative smooth functions a i1 (x i ), a i2 (x i ) and constants τ 1 ≥ τ 2 ≥ ⋅ ⋅ ⋅ ≥ τ n ≥ 0 such that 󵄨󵄨󵄨󵄨fi (t, xi (t) , xi (t − d (t))) 󵄨󵄨󵄨󵄨


Introduction
It is well known that stochastic modeling has come to play an important role in the field of engineering where stochastic differential equations have been applied for the analysis and control of stochastic systems.As a consequence, the study of stochastic nonlinear systems has drawn an increasing attention in the past decades [1][2][3][4][5][6][7][8][9].Especially, by applying the so-called method of adding a power integrator [10], which can be viewed as the latest achievement of the traditional backstepping method, a series of research results for stochastic high-order nonlinear systems have been achieved [11][12][13][14][15][16][17][18].
On the other hand, time delay extensively exists in various engineering systems, such as electrical networks, microwave oscillator, and hydraulic systems.It is well known that the existence of time delay often deteriorates the control performance of systems and even causes the instability of closedloop systems [19].Therefore, the control design and stability analysis of time-delay systems has been an active research area within the automation and control community.In recent years, by employing the Lyapunov-Krasovskii method or Lyapunov-Razumikhin method to deal with the time delay, control theory and techniques for time-delay linear systems were developed and many advanced methods such as timedelay partitioning method and input-output method have been established; see, for instance, [20][21][22][23][24][25] and reference therein.However, for the time-delay nonlinear systems, there exist many open problems which are so important and interesting at least from the theoretical point of view and have been paid careful attention; see, for example, the lastly published papers [26][27][28][29].
It is worth pointing out that, when   = 1 and ℎ  () = 1, system (1) degenerates to the stochastic time-delay system in strict-feedback form, whose study on feedback control problem has achieved great development in recent years; see, for example, [30][31][32][33] and reference therein.However, when   > 1, the control design for system (1) is challenging because on one hand, the presence of the fractional power makes system (1) not only nonsmooth but also having the uncontrollable unstable linearization, and on the other hand, the existence of time-delay effect makes the common assumption on the high-order systems nonlinearities infeasible and which conditions should be placed to the nonlinearities remains unanswered.To the best of the authors' knowledge, up to now only [34,35] two papers considered the state-feedback stabilization problem of system (1) under the strict power order restriction ( 1 =  2 = ⋅ ⋅ ⋅ =   = ) and some rigorous growth condition.However, from both practical and theoretical points of view, it is somewhat restrictive to require system (1) to satisfy such restrictions.Immediately, the following interesting questions are proposed.Is it possible to relax the power order restriction and the nonlinear growth condition?Under these weaker conditions, how to design a state-feedback controller such that the closed-loop system is globally asymptotically stable in probability?
Inspired by the recent works [15,29], we will solve the aforementioned problems here.The contribution of this paper is highlighted as follows.
(i) By comparison with the existing results in [34,35], the power order restriction is completely removed, the nonlinear growth condition is largely relaxed, and a much weaker sufficient condition is given.
(ii) Different from the result in the deterministic case [29], in which the controller need not satisfy the locally Lipschitz condition, for stochastic system, to guarantee the existence and uniqueness of the solution, the designed controller must guarantee that the closed-loop system satisfies the locally Lipschitz condition.Hence, how to construct the controller constitutes one of the main contributions.
The remainder of this paper is organized as follows.Section 2 offers notations and some preliminary results.Section 3 presents the control design procedure and the main result, while Section 4 gives a simulation example to illustrate the theoretical finding of this paper.Finally, concluding remarks are proposed in Section 5.

Notations and Preliminary Results
Throughout this paper, the following notations are adopted. + denotes the set of all nonnegative real numbers and   denotes the real -dimensional space.
In the remainder of this section, we list several lemmas that serve as the basis for the development of a state-feedback controller for system (1).They are the key tools for adding a power integrator technique.
Lemma 5 (see [38]).Let  and  be real variables; then for any positive real numbers , , and , one has where  > 0 is any real number.

Controller Design and Main Result
The objective of this paper is to design a state-feedback controller of the form such that system (1) is GAS in probability at the origin.
The assumption plays an essential role in ensuring the locally Lipschitz condition of the closed-loop system, which guarantees the existence and uniqueness of the solution.Moreover, it is worth pointing out that Assumption 7 encompasses the assumption in existing results [34,35].Specifically, when   =  ∈  >2 odd ,   = , and  1 (  ) =  2 (  ) = , Assumption 7 can be seen to reduce to the condition used in [35].Moreover, when   =  and  1 (  ) =  2 (  ) =  by choosing   =  − 1, Assumption 7 reduces to the following condition: which is equivalent to that in [34].
Remark 9.It should be emphasized that the available statefeedback results for high-order stochastic time-delay system (1) in [34,35] are all based on the strict power order restriction ( 1 =  2 = ⋅ ⋅ ⋅ =   = ).In this paper, for   , we only need a much more general condition:   ∈  ≥1 odd ,  = 1, . . ., , which means that the power order restriction is completely removed in this paper.
Without loss of generality, we assume that   =   /  with   being any even integer and   being any odd integer, under which and with the definition of   in Assumption 7, we know that   ∈  + odd .For general   , a similar technique in [39] can be used.
(a) Choose  =  0 if condition (i) of Assumption 7 is satisfied.
(b)  can be chosen as any  ∈  + odd satisfying   +   ≥  ≥ max 1≤≤ {(  +   )/, 2  } if condition (ii) of Assumption 7 holds.Now, we are in the position to give the controller design procedure by applying the method of adding a power integrator.To simplify the deduction procedure, we sometimes denote () by .
Step 1. Define  1 =   1 and choose the Lyapunov functional . With the help of Assumptions 6 and 7 and Lemma 5, we have where Obviously, the first virtual controller with  1 > 0 being smooth results in Remark 10.It is necessary to mention that in the first design step, the functions  11 ( 1 ) and  12 ( 1 ) have been provided with explicit expression in order to deduce the completely explicit virtual controller.However, in the later design steps, sometimes for the sake of briefness, we will not explicitly write out the functions which are easily defined.

Inductive
Step.Suppose at step −1 that there are a  2 , proper and positive definite Lyapunov function  −1 , and a set of virtual controllers  * 1 , . . .,  * −1 defined by with  1 > 0, . . .,  −1 > 0 being smooth, such that To complete the induction, at the th step, we choose the following Lyapunov functional candidate where Noting that /  ≥ 2, using a similar method as in [40],   can be shown to be  ) where ,  = 1, . . .,  − 1,  ̸ = .Using (19), (20), and (21), it follows that In order to proceed further, an appropriate bounding estimate should be given for each term on the right-hand side of (22).This is accomplished in the following propositions whose technical proofs are given in the Appendix.Proposition 11.There exists a positive constant  1 such that Substituting ( 23)-( 29) into ( 22) yields Now, it easy to see that the virtual controller renders Finally, when  = ,  +1 =  * +1 =  is the actual control.By choosing the actual control law, we get where We are now ready to state the main theorem of this paper.
Theorem 18.If Assumptions 6 and 7 hold for stochastic highorder nonlinear time-delay system (1), under the state-feedback controller (33), then the closed-loop system has a unique solution on [−, ∞) and the equilibrium at the origin of the closed-loop system is GAS in probability.
Proof .We prove Theorem 18 in two steps.
Step 2. Then we prove that the origin of closed-loop system is GAS in probability.
From the definitions of   's, we easily see that ∑  =1   , as the function of , is positive, definite, and radially unbounded.Then, by (34), (35) With the help of Lemma 2, we conclude that the closed-loop system has a unique solution on [−, ∞), and the equilibrium  = 0 is GAS in probability.This together with the definitions of  *  's directly concludes that the origin  = 0 of system (1) is also GAS in probability.Thus, the proof is completed.Remark 19.In the deterministic work [29], strong stability theory [42] guarantees the existence of a solution; hence, the locally Lipschitz condition need not to be satisfied.While for stochastic system, according to the existing stochastic stability theory, to guarantee the existence and uniqueness of the solution, the designed controller must guarantee that the closed-loop system satisfies the locally Lipschitz condition.
What should be emphasized is that how to guarantee the constructed Lyapunov functional being  2 and the closedloop systems satisfying the locally Lipschitz condition simultaneously is one of the difficulties in this paper.
Remark 20.Since Itô stochastic differentiation involves not only the gradient but also the higher order Hessian term in the Lyapunov design procedure, many tedious nonlinear terms will arise in the design process, especially in the case of the appearance of high order, time-varying delay and nonlinear assumption.How to deal with these terms is another difficulty of this paper.
Remark 21.From the above design procedure, we can see that the upper bound of the change rate of time delays has important impact on the control effort.To keep the control effort within the certain range, the upper bound of the change rate of time delays cannot be arbitrarily close to 1, which should be considered in practical engineering design.

Simulation Example
To illustrate the effectiveness of the proposed controller, we consider the following low-dimensional system where () = (1/3)(1 + sin()).

Conclusion
This paper deals with the state-feedback stabilization problem for a class of stochastic high-order nonlinear time-delay systems under weaker conditions.The designed state-feedback controller ensures that the origin of the closed-loop system is GAS in probability.It should be noted that the proposed controller can only work well when the whole state vector is measurable.Therefore, a natural and more interesting problem is how to design output feedback stabilization controller for the systems studied in the paper if only partial state vector is being measurable, which is now under our further investigation.

Appendix
Proof of Proposition 11.There are two different cases for the proof.
If where  4 is a nonnegative smooth function.
In a similar way, Propositions 15-17 can be proved and they are omitted here.