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Stabilisation of highly non‐linear continuous‐time hybrid stochastic differential delay equations by discrete‐time feedback control

2019; Institution of Engineering and Technology; Volume: 14; Issue: 2 Linguagem: Inglês

10.1049/iet-cta.2019.0822

1751-8652

Chunhui Mei, Chen Fei, Weiyin Fei, Xuerong Mao,

Stability and Control of Uncertain Systems

IET Control Theory & ApplicationsVolume 14, Issue 2 p. 313-323 Research ArticleFree Access Stabilisation of highly non-linear continuous-time hybrid stochastic differential delay equations by discrete-time feedback control Chunhui Mei, Chunhui Mei School of Science, Nanjing University of Science and Technology, Nanjing, 210094 Jiangsu, People's Republic of China The Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, and School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000 People's Republic of ChinaSearch for more papers by this authorChen Fei, Chen Fei Glorious Sun School of Business and Management, Donghua University, 200051 Shanghai, People's Republic of ChinaSearch for more papers by this authorWeiyin Fei, Corresponding Author Weiyin Fei wyfei@ahpu.edu.cn orcid.org/0000-0001-9864-4258 The Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, and School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000 People's Republic of ChinaSearch for more papers by this authorXuerong Mao, Xuerong Mao Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH UKSearch for more papers by this author Chunhui Mei, Chunhui Mei School of Science, Nanjing University of Science and Technology, Nanjing, 210094 Jiangsu, People's Republic of China The Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, and School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000 People's Republic of ChinaSearch for more papers by this authorChen Fei, Chen Fei Glorious Sun School of Business and Management, Donghua University, 200051 Shanghai, People's Republic of ChinaSearch for more papers by this authorWeiyin Fei, Corresponding Author Weiyin Fei wyfei@ahpu.edu.cn orcid.org/0000-0001-9864-4258 The Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, and School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, 241000 People's Republic of ChinaSearch for more papers by this authorXuerong Mao, Xuerong Mao Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G1 1XH UKSearch for more papers by this author First published: 01 January 2020 https://doi.org/10.1049/iet-cta.2019.0822Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, the authors consider how to use discrete-time state feedback to stabilise hybrid stochastic differential delay equations. The coefficients of these stochastic differential delay equations do not satisfy the conventional linear growth conditions, but are highly non-linear. Using the Lyapunov functional method, they show that a discrete feedback controller, which depends on the states of the discrete-time observations, can be designed to make the solutions of such controlled hybrid stochastic differential delay equations asymptotically stable and exponentially stable. The upper bound of the discrete observation interval is also given in this study. Finally, a numerical example is given to illustrate the proposed theory. 1 Introduction In power system, ecosystem and economic system, since the structure and parameters of these systems may change abruptly, people always use the stochastic differential equations (SDEs) driven by continuous-time Markov chain (also known as hybrid SDEs) to model them. The hybrid SDEs can be described by (1)Here the state takes values in and the mode is a Markov chain taking values in a finite space , is a Brownian motion and f and g are referred to as the drift and diffusion coefficients, respectively. An area of particular importance in the research of hybrid SDEs is the analysis of stability (see, e.g. [1–6]). If the given hybrid SDE (1) is unstable, Mao proposed in [7] that a feedback control based on the discrete-time observations of the state can be designed to make the controlled system (2)becomes exponentially stable in mean square. Here is a constant which stands for the duration between two consecutive state observations, and is the integer part of . Compared with classical continuous-time feedback control, discrete-time feedback control has many advantages, such as lower cost and easier implementation. Subsequently, discrete-time feedback control of stochastic systems has attracted extensive attention (see, e.g. [8–12]). On the other hand, the evolution process of many stochastic systems depends not only on the current state, but also on the past state. Therefore, hybrid stochastic differential delay equations (SDDEs) are widely used in the modelling of such systems (see, e.g. [13–17]). Recently, many papers have studied the stability of hybrid SDDEs whose coefficients do not satisfy the linear growth condition (see, e.g. [18–21]). Such highly non-linear systems exist widely in the real world (see, e.g. [22–24]). For example, the following scalar hybrid SDDE: (3)where the coefficients f and g are defined by (4) is a scalar Brownian motion, is time lag of the system, is a Markov chain on the state space with its generator (5)This paper attempts to design the feedback controls based on the discrete-time state observations in order to stabilise highly non-linear hybrid SDDEs. Compared with the existing articles, the key contributions in this paper are highlighted below: • We observe that there are two time delays in the controlled system (13), one is the delay of the system itself, and the other is the variable delay whose upper bound is . In other words, the controlled system (13) is actually a hybrid multiple-delay stochastic differential equation (18). Due to the influence of two different types of time delays, the mathematical techniques used in this paper are more complex than those in [12]. • Unlike general highly non-linear SDDEs in [18–21], the bounded variable delay in our equation (18) is piecewise differentiable and the derivative is equal to 1 in , thus the previous stability results will no longer apply here. More precisely, we need not only new methods to prove stability, but also new techniques to prove the existence, uniqueness and moment boundedness of solutions. • In this paper, besides the stability results in [12], we also prove the almost surely asymptotic stability of the controlled system (13). 2 Standing hypotheses and boundedness Throughout this paper, unless otherwise specified, we use the following notations. If both a, b are real numbers, then and . Let . For , denotes its Euclidean norm. If A is a vector or matrix, its transpose is denoted by . For , we let be its trace norm. If A is a symmetric real-valued matrix (), denote by and its largest and smallest eigenvalue, respectively. By and , we mean A is non-positive and negative definite, respectively. Let be a complete probability space with a natural filtration satisfying the usual conditions (i.e. it is increasing and right continuous while contains all -null sets). If A is a subset of , denote by its indicator function; that is, if and 0 otherwise. Let be an m -dimensional Brownian motion defined on the probability space. Let , , be a right-continuous Markov chain on the probability space taking values in a finite state space with generator given by where . Here is the transition rate from i to j if while We always assume that the Markov chain is independent of the Brownian motion . For , denote by the family of continuous functions from with the norm . Let be the family of all -measurable bounded -valued random variables . Consider a non-linear hybrid SDDE (6)on with the initial value , where are Borel measurable functions; is the state vector; positive scalar constant is time lag of the system. In this paper, we maintain the local Lipschitz condition. But as mentioned in the preceding section, we will not confine the coefficients f or g to the linear growth condition, but to a condition similar to the polynomial growth condition. For this reason, we give the following hypotheses. Assumption 1.For each positive number k, there is a constant such that (7)for all with and all . Assumption 2.Assume that there are three constants , and such that (8)for all . Considering highly non-linear hybrid SDDEs, we always assume in condition (8). For the hybrid SDDE (4), it is easy to see that and . We also observe condition (8) implies that and , which are required for the stability purpose of this paper. Obviously, Assumptions 1 and 2 may cause the solution of the hybrid SDDE (6) to explode in finite time. In order to guarantee on the existence of the global unique solution of the SDDE (6), we need to impose another Khasminskii-type condition. Assumption 3.Assume that there are some non-negative constants , such that (9)(where and are the same as in Assumption 2) while (10)for all . It is worth pointing out that in many hybrid SDDEs, and p are different, sometimes p may be any large positive number. For example, consider the hybrid SDDE (4) and let p be arbitrarily large. Then (11)By inequalities Hence (12)That is, the hybrid SDDE (4) satisfies Assumption 3 with any large p and , , , , , . Under Assumptions 1–3, the hybrid SDDE (6) has a unique global solution such that with any initial value (see, e.g. [18], Theorem 3.1 on p. 180). But boundedness does not mean stability. When the given SDDE (6) is unstable, we are required to design a feedback control , based on the discrete-time observations of the state at times , in the drift part so that the controlled system (13)becomes stable, where and the control function is a Borel measurable. In this paper, we will design the control functions to satisfy the following assumption. Assumption 4.Assume that there is a non-negative number such that (14)for all , and . Moreover, assume that for all . Obviously, this assumption implies (15) As pointed out above, the p th moment of the solution of the given SDDE (6) is bounded. The following theorem shows that the controlled SDDE (13) preserves this good property. Theorem 1.Let Assumptions 1–4 hold. The controlled system (13) with any initial value has a unique global solution on . Moreover, the solution obeys (16) Proof.To make the proof more understandable, we divide it into three steps. (1) Let us define a bounded function by (17)Thus, the controlled system (13) can be rewritten as (18)on with the initial value . We observe that is a bounded variable delay, and the controlled system (13) is actually a hybrid stochastic differential multiple-delay equation. Let . Using the Itô formula (19)where the operator is defined by By Assumptions 3 and 4, we derive that Choosing a constant sufficiently small for (20)where From the Young inequality, we get where, here and in the remaining part of this paper, C denotes a positive constant that may change from line to line but its special form is of no use. Hence (21)where (2) Next we will show the existence and uniqueness of the solution of the hybrid SDDE (18) on . Under the local Lipschitz condition (7), there exists a unique maximal local solution to Equation (18) on with any initial value , where is the explosion time (see, e.g. [24], Theorem 2.8 on p. 154). Let be a sufficiently large integer such that . For each integer , define the stopping time where throughout this paper we set (as usual denotes the empty set). Clearly, increases as and a.s. If we can get that a.s., then a.s. That is, the maximal local solution is the unique global solution. By the standard stopping time technique to (19), then using (21), we obtain (22)which shows where But, from (17), we observe that for all . Then We therefore set Noting the sum of the right hand-side terms is increasing in t, we obtain By the Gronwall inequality, we have Consequently Let , we can see , this implies a.s.. With the previous analysis, we can obtain that SDDE (13) with any initial value has a unique global solution on . (3) Finally, we will illustrate the asymptotic boundedness of the p th moment of the solution. Set for . For , by the Itô formula Applying (21), we can compute (23)where In particular This implies (24)where . Furthermore, by , it follows from (23) that Combining this with (24), we deduce that Then (25)Noting from , we have Substituting this into (25), then using (20) and (24), we obtain that Consequently It is straightforward to see that (26)As this holds for any , we hence get the required assertion (16). The proof is therefore complete. □ As we all know, the existence and uniqueness of the solution of the controlled system (13) in Theorem 1 is the precondition of stability. The asymptotic boundedness of the p th moment of the solution will play their fundamental roles when we discuss the stabilisation of the controlled system (13) in subsequent sections. 3 Asymptotic stabilisation In this section, we will discuss the asymptotic stability of the solution of the controlled system. As mentioned earlier, the conditions for asymptotic stability of solutions are stronger than the existence and uniqueness of solutions. So we need to give some new hypotheses. Let us start with the first condition. Assumption 5.For each , design the control function so that we can find constants , and for both (27)and (28)to hold for all (where and have been specified in condition (8) and (9), respectively). In addition, both (29)are non-singular M-matrices. For the theory of M-matrix, readers can refer to (see, e.g. [25], Section 2.6). In fact, many control functions u can meet both Assumptions 4 and 5. For example, if the state of the given SDDE (6) is observable in any mode , we could give the linear control function (obviously satisfies Assumption 4), where A is a symmetric real-valued matrix such that . Then By inequality (30)Recalling (9), we have It then follows from (10) that as well as while which are non-singular M-matrices (see, e.g. [25], Theorem 2.10 on p. 68). That is, the control function meets Assumption 5. In order to lead the second condition, we set (31)As and are non-singular M-matrices, all and are positive. Define a function by (32)while define an operator by (33)By (27), (28) and the Young inequality, we can calculate (34)This observation makes the following assumption possible. Assumption 6.Assume that there exists a function , as well as positive numbers , , , and , such that (35)and (36)for all . Let us go on to show that Assumption 6 can always be met. In fact, by Assumption 2 and (34), we then derive (37)Recalling (9), then using inequality (30) again, we get Substituting this into (37) yields (38)where If , and we can choose positive constants - sufficiently small for Set and It is easy to see that and meet condition (35). Then (39)For the asymptotic stability of the controlled system (13), we will use the Lyapunov functional method. Define two segments and for (where ). For and to be well defined for , we set for and for . The Lyapunov functional used in this paper will be of the form (40)for , where U has been defined by (32) and is a positive constant to be determined later while we set for . By the generalised Itô formula (see, e.g. [25], Lemma 1.9 on p. 49), we get (41)for , where is a continuous local martingale with (see e.g. [22], Theorem 1.45 on p. 48) and is defined by On the other hand, by the basic differential calculation (42)Combining (41) with (42), we get (43)Furthermore, by Assumption 4, we can compute where the function has been defined by (33). It then follows from (43) that (44)where (45) We can now state our first stabilisation result. Theorem 2.Let the conditions of Theorem 1 hold. Assume also is sufficiently small for (46)If Assumptions 5 and 6 hold, then the solution of the controlled system (13) obeys (47)for any and any initial value . Proof.Fix the initial value arbitrarily. Using the generalised Itô formula, we then derive from (44) that (48)for any .Let (Please recall that is a free parameter in the definition of the Lyapunov functional). Using condition (46), it is easy to show that (49)Substituting (49) into (45), then using conditions (15) and (36), we obtain that We note from condition (46) that , consequently It is easy to see that (50)Substituting (50) into (48) yields (51)where By the substitution technique, we deduce that Thus (52)where by condition (46). Similarly, we can show (53)Plugging (52) and (53) into (51), we have (54)where is a constant defined by On the other hand, by the Fubini theorem (55)Using the Hölder inequality and the Itô isometry, we derive that (56)This implies (57)Substituting (57) into (54), we have (58)It is straightforward to show that Using the inequality (30), we can derive that for any . We hence get the required assertion (47). The proof is therefore complete. In general, it does not follow from (47) that . However, using uniformly continuous of and (47), we can show a stronger result that for any . Let us state the second theorem in this section. Theorem 3.Under the same Assumptions of Theorem 2, the solution of the controlled hybrid SDDE (13) satisfies (59)for any and any initial value . Proof. Fix the initial value arbitrarily. Using the Itô formula, we derive that for any . Combining (8), (9) and (15), we further get It then follows from the Theorem 1 that where Thus, is uniformly continuous in t on . Recalling (47), we therefore obtain (60)That is, the assertion (59) holds when . Let us now fix any . By the Hölder inequality, because of , we can calculate (61)It follows for (60) and (61) that implies the required assertion (59).□ At the end of this section, we will give the almost surely stability of the controlled system (13). i.e. a.s. Theorem 4.Under the same assumptions of Theorem 2, then the solution of the controlled system (13) satisfies (62)for any initial value . Proof.It follows from (58) in Theorem 2 and the well-known Fubini theorem that which implies We now claim that (62). If this is false, we can find a positive number sufficiently small, such that (63)We can add the same stopping time to the proof of Theorem 3.3 as in Theorem 2.5. Obviously, (54) can be rewritten as where Then, we have This, together with (57), implies There is a positive integer such that Letting , we obtain that Choosing a sufficiently large integer , for This means (64)Then similar to the discussion in [8], the required assertion (62) must hold. The proof is therefore complete. □ 4 Exponential stabilisation In the previous part, we have given the upper bound of when the controlled system (13) satisfies asymptotic stability. Next, we will illustrate how to design a feedback control based on the discrete-time state observations to make the controlled system (13) become exponentially stable either in () or almost surely. As the exponential stability condition is stronger than the corresponding asymptotic stability, the upper bound of the discrete observation time interval given by us is smaller than that in the previous section. Theorem 5.Let the conditions of Theorem 1 hold. Assume also is sufficiently small for (65)If Assumptions 5 and 6 hold, then the solution of the controlled system (13) satisfies (66)and (67)for any and any initial value . Proof.Fix the initial value arbitrarily. is a sufficiently small positive number to be determined later. Similar to the proof of Theorem 2, we can compute (68)for all . Recalling the structure of V, we then have (69)where and Noting , we can rewrite (50) as (70)By (56), we then derive (71)Recalling (30), we get Substituting this and (71) into (69), by similar techniques in (52) yields (72)where On the other hand, it is straightforward to show that We further make sure to be sufficiently small for It then follows from (72) immediately that (73)Finally, by (61) and (73), and applying the Hölder inequality, we obtain (74)for any . This implies the required assertion (66).By slightly modifying the proof of Theorem 5.4 in [12], we can obtain another assertion, (67), from (74). The proof is therefore complete. □ 5 Example We will illustrate our results with an example. In order to maintain the coherence of the paper, we will take the hybrid SDDE (3) as an example. Let us recall that the coefficients f and g in SDDE (3) are defined by (4), where is a scalar Brownian motion and is a Markov chain on with the generator defined by (5). Through computer numerical simulation (we set and the initial value on and ), we can find that hybrid SDDE (3) is unstable. This result can be referred to in Fig. 1. Fig. 1Open in figure viewerPowerPoint Computer simulation of the sample paths of the Markov chain and the SDDE (3) with using the Euler–Maruyama method with step size We will choose the control function defined by (75)We see easily that Assumption 4 hold with . It follows from Theorem 1 immediately that the controlled system (76)has a unique global solution on for any initial value and the solution satisfies that (77)By a simple calculation, for , we get and It is easy to see that Hence are both M -matrices. By (31), we then have and while Assumption 5 is satisfied. The function U defined by (32) becomes Recalling (34), we can compute that To verify Assumption 6, we let , and . Noting (78)where and . That is, condition (36) is also met. After calculation, condition (65) becomes . By Theorems 5, we can therefore conclude that the controlled system (76) with the control function (75) is not only exponentially stable in for any but also almost surely provided . To perform a computer simulation, we set and the initial value on and . The sample paths of the Markov chain and the solution of the SDDE (76) are plotted in Fig. 2. The simulation supports our theoretical results clearly. Fig. 2Open in figure viewerPowerPoint Computer simulation of the sample paths of the Markov chain and the SDDE (76) with the control function (75) and using the Euler–Maruyama method with step size 6 Conclusion In this paper, we have discussed the stabilisation of highly non-linear hybrid SDDEs by the feedback controls based on the discrete-time observations of the states. It should be noted that the results of stabilisation of existing non-linear stochastic systems mainly depend on linear growth conditions, and do not take into account the existence of delay in the system itself. Hence, developing a new theory on the stabilisation for the highly non-linear SDDE models is needed. In this paper, we consider a class of hybrid SDDEs which are not stable but their solutions are bounded in p th moment. We use a new technique to show that the controlled SDDE can maintain moment boundedness as long as the control function satisfies Lipschitz condition. We then show how to design the control functions more wisely so that the controlled SDDEs become stable. The stability discussed in this paper includes the -stable in , asymptotic stability in th moment, almost surely stability, p th moment exponential stability and almost surely exponential stability. The key technique used in this paper is the method of Lyapunov functionals. 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