Global stabilisation for a class of stochastic continuous non-linear systems with time-varying delay

This paper studies the problem of global stabilisation for a class of continuous but nonsmooth stochastic non-linear systems with time-varying delays. Other than the traditional backstepping recursive control, a concise controller of the nominal system is ﬁrstly designed by employing the homogenous domination approach, which overcomes the problem of calculating explosion produced by the diffusion and Hessian terms. Then, by means of an appropriate Lyapunov–Krasoviskii functional and a design parameter, the negative effects of the time delays and non-linear terms generated in the controller design process are dominated; the global asymptotic stability of the closed-loop system can be ensured by the simple but effective controller, which provides signiﬁcant cost savings. Finally, a simulation example is given to demonstrate the effectiveness of the presented scheme.

tems was studied in [12,13]. Moreover, the output-feedback controller was designed in [14] for a class of stochastic timedelay non-linear systems. Thereupon, [15] promoted the stabilisation result to a more general class of systems. Furthermore, [16,17] investigated the fuzzy control and neural control of the stochastic time-delay systems. The adaptive control and outputfeedback control were discussed in [18] and [19] for a class of stochastic time-delay systems with perturbations or input saturation. Additionally, some excellent results for robust stability or robust H ∞ control of stochastic non-linear systems were presented in [20][21][22][23][24].
It is worth emphasising that the systems discussed above are strict-feedback stochastic time-delay systems; that is to say, the powers of systems are equal to one. On the other hand, stochastic high-order non-linear system is also an important class of stochastic systems. Due to the Jacobian linearisation of high-order non-linear systems are neither controllable nor feedback linearisable, with the effect of stochastic noise, the control and analysis is nontrivial. In fact, many excellent works have been done on the stabilisation of stochastic high-order IET Control Theory Appl. 2021;15:297-306.
wileyonlinelibrary.com/iet-cth 297 non-linear systems [31][32][33][34]. For example, by employing homogeneous domination manner, the global stabilisation of stochastic time-delay non-linear systems with high-order power was studied in [31] in feed-forward form, and then, [32] discussed the finite-time stabilisation of stochastic high-order non-linear systems in strict-feedback form. Moreover, based on the adding a power integrator and the L-K functional, the global tracking control was solved for stochastic high-order non-linear systems with time-varying delay by output-feedback [33] and adaptive design [34], respectively. However, when a stochastic non-linear system is continuous but not smooth, in which the power of the considered system is less than one, those methods developed in the literature above are inapplicable because they still require some smoothness of the system. Therefore, the stabilisation problem of low-order stochastic time-delay non-linear systems has been rarely studied. In particular, for the deterministic systems in which the effects of stochastic factors are not considered, the global finite-time stabilisation was settled in [35] for a class of continuous nonlinear systems; [36,37] solved the state-feedback stabilisation for a class of stochastic non-linear systems with input delay or state delay. Moreover, for stochastic delay-free systems, the finitetime control for a class of low-order stochastic non-linear systems was addressed in [38][39][40] by output feedback or state feedback. However, so far, by taking the effects of both time-delay and the completely undifferentiable produced by low-order into account, the stabilisation problem is still open.
In this paper, the global stabilisation for a class of stochastic time-delay non-linear systems with low-order power will be discussed. The main contributions and obstacles are characterised by threefold. (i) The paper shows a first viewpoint to investigate the global stabilisation of stochastic non-linear systems with time-varying delay and low-order, which is only continuous but non-smooth; thereby the existing control methods applied for those smooth non-linear systems are no longer applicable. (ii) By combining the homogenous domination control with the backstepping method, a concise but effective controller is constructed to reduce the large control costs and counteract the disadvantage of the computation explosion caused by the only backstepping recursive design. (iii) Without involving any approximation algorithms, which can only aim at the stability of the closed-loop system rather than global asymptotic stability, the control parameter is presented to dominate the nonlinearities produced during control design. Meanwhile, a sophisticated L-K functional is selected to offset the negative effects of time delays.
Notations This note adopts the following notations. is negative definite: − (x) is positive definite. is positive definite, decrescent and radially unbounded:

Preliminaries
Consider stochastic time-delay non-linear system ] is a Borel measurable function; (t ) is an r-dimensional standard Wiener process defined on a complete probability space {Ω,  , P}, where Ω is a sample space,  is a -field, P is the probability measure; f (⋅): ℝ + × ℝ n × ℝ n → ℝ n and g T : ℝ + × ℝ n × ℝ n → ℝ n×r are unknown continuous functions.
Two definitions and four key lemmas are provided in advance.

Definition 1 ([3]
). For any given function V (x, t ) ∈  2,1 of system (1), the differential operator  is defined as Tr{g T 2 V x 2 g} is known as the Hessian term of .

then (i) there exists a unique strong solution on [− , ∞);
(ii) the equilibrium x(t ) = 0 is globally asymptotically stable in probability and P{ lim

Problem formulation
Consider the stochastic time-delay non-linear system where z = [z 1 , … , z n ] T ∈ ℝ n is measurable state, u ∈ ℝ and y ∈ ℝ are control input and system output.
The objective of this paper is to construct a state-feedback controller such that the solution of the closed-loop system is globally asymptotically stable in probability. To this end, the following assumption is needed: and where ρ > 0, r is the power of system (2), andr = 1+r 2 .

Remark 1. Assumption 1 indicates that both state delay and input delay are considered in diffusion and
Hessian terms simultaneously, which destroys the original form of lower-triangular where the traditional backstepping method can usually be directly used to control design. Furthermore, the appearance of z r j (where 0 < r < 1) leads to the obstacle of completely non-differentiable, which results in the fact that all the control methods that require smoothness in the existing literature are no longer applicable. Therefore, the results on the global stabilisation for the only continuous but nonsmooth stochastic non-linear systems are quite scarce. Hence, it is significant and challenging to study the global stabilisation of only continuous but nonsmooth stochastic non-linear systems. On the other hand, "smooth" means "sufficient often differentiable," most often  ∞ (infinitely often differentiable). This indicates that the requirement of "smooth" system is much stronger than "continuous" one. That is to say, a stochastic non-linear system may be continuous but non-smooth, but the smooth stochastic non-linear system must be continuous. Hence, the results proposed in this paper can be applied to a wider range of systems and is more general in theory.

Remark 2.
Compared with the exiting literature, both the effects of timevarying delays and the problem of completely non-differentiable produced by low-order powers have to be faced in this paper. Actually, the globally adaptive control and output-feedback control were concerned in [18,19] for a class of stochastic non-linear time-delay systems with the power r = 1.
In [13,[31][32][33][34], the global stabilisation and tracking control were investigated for the stochastic high-order non-linear time-delay systems with various structures and conditions. In addition, without involving the effects of time delays, [38][39][40] designed the finite-time controller for a class of stochastic low-order non-linear systems in a lower-triangular form or upper-triangular form. In view of these facts, it is meaningful and challenging to study the global stabilisation for a more general class of stochastic non-linear systems with both time-varying delays and low-order power.

MAIN RESULTS
The main results are stated in the following: Theorem 1. If system (2) satisfies Assumption 1, under the state feedback controller of the form with the parameters l , l = 1, … , n determined in (9), (11) and (15), then the following hold: (i) the closed-loop system has a unique strong solution on [− , ∞); (ii) the closed-loop system is globally asymptotically stable in probability.
Proof. The proof is divided into three parts. On the basis of homogeneous domain technique, the controller of the the nominal system (6) is constructed in the first part, and the second part designs the controller of the original system (2). Additionally, the third part aims to present the theoretical analysis of the closed-loop system composed by (2) and (5). □ Part I: Design the controller for nominal system (6). First of all, the nominal system of (2) can be represented as Then, we introduce where 2 , … , n+1 are virtual control laws in which 1 , … , n are positive constants. Now, we focus on revealing the backstepping recursive procedure.
Step 1. Considering the Lyapunov function V 1 = 2 2 1 , it can be verified from Definition 1 that where is a positive parameter. Constructing and substituting (9) into (8), it yields that Step k (2 ≤ k ≤ n − 1). Assume at step k − 1, there exist a series of virtual control laws that can assure the Lyapunov function candidate Letting k = 2, it is clear to see that (12) becomes (10). Choosing (7) and (12) imply that According to Lemmas 1 and 2, one can get with the constant . Therefore, the kth virtual control law is chosen as Substituting (14) and (15) into (13), one obtains Step n. With the recursive manner in mind, positive constants 1 , … , n can be determined one by one.
It can be deduced that Part II: Construct the actual controller u for original system (2). Take the same Lyapunov function and Definition 1 into account, it follows from (2) and (19) that Then, analyse each term of the last inequality (20) one by one. Firstly, by using the same controller as in the Part I, that is to say, by choosing one obtains To proceed further, there exist positive constants 11 , 12 , 21 , 22 such that Please see Appendix for the specified proof of (23). Thus, by substituting (22) and (23) into (20), one has 1+r j 1+r j 1+r j Define with c 0 = 12 + 22 is a positive parameter. It follows from (24) and (25) that V ∈  2,1 and Moreover, if one selects appropriate parameter to satisfy there holds Remark 3. The condition 0 <̇(t ) ≤ < 1 is necessary, which ensures that the positive time-delay terms in (26) can be canceled out by selecting the appropriate L-K functional in (25), so that (28) can be obtained. Furthermore, the condition is general, which also can be found in a lot of literature with stochastic time-delays, such as [16,19,33,34].
Part III: Theoretical analysis of the closed-loop system composed by (2) and (5).
On the one hand, it can be inferred from (25) and (28) that V is  2 , non-negative, radially unbounded and V ≤ 0. Thus, for any initial data, Lemma 4 implies that the closed-loop system (2) and (5) has a solution.
On the other hand, defining = [ 1 , … , n ] T , and ( ) ≜ ∑ n j =1 1+r j , and considering the fact that 1 + r is an even num-ber, it is clear to see that ( ) is continuous, positive definite, and radially unbounded. With the aid of lemma 4.3 in [43], there exist two class  ∞ functions 1 (⋅), 2 (⋅) that satisfy It follows from (28) and (29) that Hence, there exists a function 1 (‖ ‖) ∈  ∞ such that In addition, letting ( ) = 0, ∈ [− , 0), with the mean value theorem and Lemma 3 in mind, one has with ∈ [t − (t ), t ] and 2 ∈  ∞ . Consequently, with the aid of Lemma 4, from (28), (31), (32), one can obtain that the closed-loop system (2) and (5) has a unique strong solution on [− , ∞). Moreover, the equilibrium = 0 is globally asymptotically stable in probability. From (7) and Definition 2, it can be directly verified that the origin z = 0 of system (2) is also globally asymptotically stability in probability. This completes the whole proof.

Remark 4.
On the basis of the homogenous domination approach, a concise but effective controller is constructed for the nominal system, which avoids the explosion of computation caused by the backstepping recursive control to the diffusion and Hessian terms directly. Then, the control parameter is presented deliberately to regulate those non-linearities produced in the controller design for the original complete system. Afterward, an appropriate Lyapunov-Krasoviskii functional is chosen so that the globally asymptotic stability of the closed-loop system can be assured with less control cost. Furthermore, the saving of control cost is reflected in three aspects: (i) The Equation (5) shows that the controller is designed to be more concise in form; (ii) Comparing with [34,38], the value of control u is reduced to ensure the stability of the closed-loop system in about the same time; (iii) Under the same control value, more negative effects of time delays are dealt with in this paper than [39,40].

SIMULATION EXAMPLE
Consider the stochastic continuous time-delay non-linear system For demonstration, we choose the initial condition of the system as  Figures 2 and 3 show that the controller (34) stabilises the closed-loop system and reveal the responses of (33) and (34) where the effectiveness of the provided control strategy is clarified.  Figures 4-6 show that the convergent time of this paper is much less than that of existing result [40], and although the convergent time of this paper is roughly the same as in [38], the value of control required in this paper is much less than [38]. This paper also deals with the undesired negative effects of time-varying delay in addition. Therefore, the controller pro- The trajectories of x 2 posed in this paper is more efficient than those in existing results [38] and [40].
Besides, the relation between the convergent time and control u and the values of 1 , 2 is verified through Figures 7  and 8. Specifically, Figure 7 shows that the convergence rate of the states becomes faster if 1 , 2 become larger, but it can be seen from Figure 8 that the cost is the huge increase of con- The trajectories of u trol value. Therefore, in order to save control cost, we choose smaller 1 , 2 in (34) to achieve satisfactory results.

CONCLUSION
This note has studied the global stabilisation problem for a class of stochastic low-order non-linear systems corrupted by both state delay and input delays. Based on the homogeneous domination approach, a delay-independent controller has been constructed for the nominal system, which avoided the disadvantage of the explosion of computation caused by backstepping recursive method to drift and diffusion terms. Then, by introducing the appropriate Lyapunov-Krasoviskii functional and the design parameter, the globally asymptotic stability of the closed-loop system has been guaranteed. A natural problem under investigation is how to achieve the finite-time stabilisation of the stochastic time-delay non-linear systems. In addition, how to design the output-feedback controller for stochastic low-order non-linear systems is also a difficult but valuable topic. Besides, another open problem is whether the scheme could be used to control the physical systems.