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Effect of noise on the solutions of non‐linear delay systems

2018; Institution of Engineering and Technology; Volume: 12; Issue: 13 Linguagem: Inglês

10.1049/iet-cta.2017.0963

1751-8652

Quanxin Zhu, Shiyun Song, Peng Shi,

Advanced Research in Systems and Signal Processing

IET Control Theory & ApplicationsVolume 12, Issue 13 p. 1822-1829 Research ArticleFree Access Effect of noise on the solutions of non-linear delay systems Quanxin Zhu, Corresponding Author zqx22@126.com Key Laboratory of HPC-SIP (MOE), College of Mathematics and Statistics, Hunan Normal University, Changsha, 410081 Hunan, People's Republic of China School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing, 210023 Jiangsu, People's Republic of ChinaSearch for more papers by this authorShiyun Song, School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing, 210023 Jiangsu, People's Republic of ChinaSearch for more papers by this authorPeng Shi, College of Automation, Harbin Engineering University, Harbin, 150001 People's Republic of China School of Engineering and Science, Victoria University, Melbourne, VIC, 8001 AustraliaSearch for more papers by this author Quanxin Zhu, Corresponding Author zqx22@126.com Key Laboratory of HPC-SIP (MOE), College of Mathematics and Statistics, Hunan Normal University, Changsha, 410081 Hunan, People's Republic of China School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing, 210023 Jiangsu, People's Republic of ChinaSearch for more papers by this authorShiyun Song, School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing, 210023 Jiangsu, People's Republic of ChinaSearch for more papers by this authorPeng Shi, College of Automation, Harbin Engineering University, Harbin, 150001 People's Republic of China School of Engineering and Science, Victoria University, Melbourne, VIC, 8001 AustraliaSearch for more papers by this author First published: 01 September 2018 https://doi.org/10.1049/iet-cta.2017.0963Citations: 4AboutSectionsPDF 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 onEmailFacebookTwitterLinked InRedditWechat Abstract In this study the authors are concerned with the effect of noise on the solutions of a class of non-linear delay systems. The systems under investigation are non-linear without linear growth condition requirement, which makes the analysis difficult to ensure the existence and uniqueness of solutions. The authors first introduce a set of new non-linear growth conditions with delays, and then provide the existence and uniqueness of solutions as well as some properties of the solutions. The new condition only requires the knowledge of the coefficient of the highest term, which is much easier to check in practice than the traditional conditions. Finally, an example is given to show the effectiveness and potential of theoretic results obtained. 1 Introduction It is well known that the noise plays an important role in deterministic dynamical systems. There have appeared many results on this topic in the past decades. For example, the stabilisation by the deterministic 'noise' has been extensively studied; see, for example, [1] and the references therein. Furthermore, the stabilisation by a random noise was originally investigated by Hasminskii in [2]. It is worth pointing out that Hasminskii in [2] only studied the stabilisation problem of a linear system by introducing two white noise sources. In fact, there are a large number of non-linear phenomena in real life. Thus, the stabilisation problem of non-linear systems has received a great amount of attention in the past, which overcomes the limitation of application scope. We now introduce some results on the stabilisation of non-linear systems. In [3], it is shown that a non-linear system could be almost surely exponentially stabilised and destabilised when the non-linear function f obeys the local Lipschitz and linear growth conditions. Subsequently, to find some weaker growth conditions for the above issue has been a challenging one. For example, the polynomial growth condition was introduced in [4]. More related works on weaker growth conditions can be found in [5–9]. Note that the condition presented in [9] is the weakest condition since it does not require the fixed form of stochastic term [8]. However, the effect of delays was not considered in [9]. It is well known that delays are always appeared in practical systems, which may depend not only on the present state but also on the past ones. Usually, such systems are called delay systems and they have great importance in physics, mechanics, associative memories, engineering, pattern recognition, aerospace and robotics and control [2, 10–19]. Thus, it is important and useful to study the solution and its properties of non-linear delay systems [20–56]. Motivated by the above discussion, in this paper we are concerned with the effect of noise on the solutions of a class of non-linear delay systems. We first introduce a set of new non-linear growth conditions with delays. In particular, this new condition considers the effect of delays and does not require the fixed forms of both drift and diffusion terms. Furthermore, this new condition only requires the knowledge of the coefficient of the highest term, which is much easier to check in practice than the traditional conditions. Under the new condition, we mainly study the existence and uniqueness of solutions as well as some properties of the solutions. Compared with the previous works, the contributions of our paper can be generalised as follows: (i) We first introduce a set of new non-linear growth conditions with delays, which is weaker than those used in [4–9, 55]. In fact, delays were ignored in [4, 6, 8, 9]. Moreover, the fixed form of stochastic term was required in [3, 5, 7, 8]. (ii) Under these new conditions, we prove the existence and uniqueness of solutions to a class of stochastic high non-linear delay systems. (iii) We obtain some properties of the solutions for a class of stochastic high non-linear delay systems. (iv) The mathematical model presented in this paper can be used to describe some practical systems, such as power systems, manufacturing systems, financial market systems etc. That is, the results developed in this work are not only meaningful in theory, but also have potential in real-world applications. The structure of the paper is organised as follows. In Section 2, the model and new condition are introduced. In Section 3, we present our main results. In Section 4, an example is given to show the superiority of the new condition proposed. Finally, the paper is concluded by Section 5. 2 Model and new conditions Let us consider the following deterministic non-linear delay system: (1)where is Borel measurable, and and is a time-varying delay such that , where is a positive constant. We denote is a continuous function from to with the uniform norm . Then, we can give the corresponding stochastic non-linear delay differential system as follows: (2)where is an r -dimensional Brownian motion defined a complete probability space , is a measurable valued random variable such that , is Borel measurable. To guarantee the existence and uniqueness of solutions to system (2), we first need to introduce the following two conditions imposed on the drift function f and the diffuse function g. Assumption 1.The functions f and g satisfy the traditionally local Lipschitz condition, i.e. for every integer , there are and satisfying for every with and (3) (4) Assumption 2.There are constants , , , , , , , such that , (5) (6) (7)where and The notation means some functions such that . Remark 1.Assumption 1 is the local Lipschitz condition and it is easy to check in real systems. Moreover, under this condition, it follows from [3] that for any initial value , system (2) has a unique maximal local solution on , where is the explosion time. Remark 2.Assumption 2 is a new non-linear growth condition with delays. In particular, it considers the effect of delays and does not require the fixed form of diffuse term. Thus, our condition is very general and weaker than those given in [4–9, 55]. In fact, the effect of delays was ignored in [4, 6, 8, 9, 55], and the fixed form of diffuse term was actually required in [3, 5, 7, 8, 55]. Remark 3.Compared with the traditional conditions, our new condition only requires the knowledge of the coefficient of the highest term, which is much easier to check in practice. 3 Main results and their proofs In this section, we present our main results and their proofs. Theorem 1.Suppose that Assumptions 1 and 2 hold. If one of the following conditions holds: (i) , ; (ii) , ; (iii) ; (iv) , then (2) has a unique global solution . Proof.Obviously, it follows from Assumption 1 and Remark 1 that for any given initial value , system (2) has a unique maximum local solution . Our aim is to prove that is global. Thus, we have to prove . Let us define a stopping time for all . It is easy to prove that is increasing on k. Letting , then it is obvious that . Hence, in order to show , we only need to prove , i.e. we need to prove that holds for all . When , the conclusion is trivial. So we only need to discuss the case of . For any and , it follows from the Dynkin formula that (8)where Define Then, we have Next, we will discuss the dominated term coefficient of under four cases. Case (i): , . Under this case, is dominated by the term and is dominated by the term . Noting that and , we can easily find an appropriate constant such that and . Case (ii) , . is the same as in Case (i), and is dominated by the term . Noting that and , we can easily find an appropriate constant such that and . Case (iii): Under this case, is dominated by the term and is the same as Case (i). Noting that and , we can easily find an appropriate constant such that and . Case (iv): . is the same as in Case (iii), and is the same as in Case (ii). It follows from Cases (ii) and (iii) that and holds for all . From the above discussion, if we choose an appropriate constant , then the coefficient of dominated term of is always less than zero. Hence, there exists satisfying . Then, it follows (8) that (9)Hence, according to the definition of and (9), we have So holds. Finally, we obtain , which imply that when t is arbitrary. □ Remark 4.Theorem 1 reveals that there always exists a non-linear diffuse term that satisfies Assumption 2, to ensure that the stochastic system (2) exists a unique global solution, even the solution of the deterministic system (2) may go to infinite in a finite time. Remark 5.Owing to the flexible form of , some published results can be regarded as special cases of Theorem 1. For instance, the stochastic feedback in [55] was bounded in quadratic polynomial like the following: which is a special case of (6) when . In [5], the stochastic feedback is required to be linear, i.e. , where is a matrix. In [8], , where is a constant, and is a matrix. In [6, 7], , where and are two constants. Obviously, all the stochastic feedback in [5–8, 55] is only a special case of ours. Furthermore, the stochastic feedback in [5–8, 55] did not consider the delay term , and the systems in [6, 8] ignored the effect of delays. Remark 6.If we ignore the effect of delays in Theorem 1, then our result can be reduced to that in [9]. Therefore, our result generalises and improves that given in [9]. Next, we will discuss several properties of the global solution. The first result is that the moment of the solution is bounded. Theorem 2.Let Assumptions 1 and 2 hold. Then, for any suitable , there is a constant such that is bounded in the sense of moment: where satisfies one of the following conditions: (i) If , , we choose . (ii) If , , we choose . (iii) If , we choose . (iv) If , we choose . Proof.For any sufficiently small positive constant , we can construct the following Lyapunov function: where satisfies one of the conditions (i)–(iv). By using the Itô formula, one gets Obviously, the above equality can be rewritten as follows: (10)where It follows from Assumption 2 that Noting that , then we have Now, let where and Then, we can rewrite as follows: Similar to the proof of Theorem 1, it is easy to prove that the coefficient of dominated term of is always less than zero. Therefore, there is satisfying which implies (11)Define the stopping time . Then, by (10) and (11), one has Hence, letting , which yields that , we can easily get which gives Thus, letting on both sides of the above inequality, we obtain which verifies that Theorem 2 holds for . □ Remark 7.It is worth pointing out that we obtain some less conservative results in the area of stochastic non-linear systems by computing the operator based on our new condition, i.e. Assumption 2. In what follows, we will give another property of the solution . Theorem 3.Let the conditions in Theorem 1 hold. Then, we have where is taken different values according to the following four cases: (i) If , , we choose . (ii) If , , we choose . (iii) If , we choose . (iv) If , we choose . Proof.Take . Then, by the Itô formula and Assumption 2, we have where Thus, we get (12)where is a local martingale and satisfies Then, we have that for every positive integer k and : Noting that , it easily derives that for almost every holds for all . For convenience, letting then we see that (12) yields (13)where and Similar to the proof of Theorem 2, we easily prove that the coefficient of dominated term of is always less than zero. Hence, there is satisfying (14)It follows from (13) and (14) that Hence, we have where . Letting and in both sides of the above inequality, we obtain where is taken different values according to the following four cases: (i) If , , we choose . (ii) If , , we choose . (iii) If , we choose . (iv) If , we choose . Remark 8.From Theorem 3, we know that the relation of the growth speed between the solution and time t. Moreover, Theorems 1–3 tell us that the diffusion term can be designed to ensure that there is a unique global solution of stochastic system (2), and the solution is both limited in the sense of moment and growing at most polynomially, even the corresponding deterministic system may go to infinite in a finite time. Remark 9.In Theorem 3, we consider the effect of delays and do not require the fixed form of diffusion term. As discussed in Remark 2, our result is more general than those given in [4–9, 55]. Remark 10.The mathematical model presented in this paper can be used to describe some practical systems, such as power systems, manufacturing systems, Duffing systems with delays and financial market systems etc. That is, the results developed in this work are not only meaningful in theory, but also has potential in real-world applications. 4 Example A simple example is given to check the theoretical analysis in this part. Example 1.A stochastic non-linear delay system is as follows: (15)where , and are two independent one-dimensional Brownian motions.For the coefficient of drift term, it is easy to compute where c is a constant.For the coefficient of diffuse term, it follows from a direct computation that and where and .Obviously, , , , , , , . It is easy to check that Assumptions 2.1 and 2.2 are satisfied. Moreover, we obtain . Thus, according to Theorems 1–3, the corresponding stochastic non-linear delay system (15) has a unique global solution , which obeys and where . Especially, by taking , we obtain which verifies that the growth speed of is as fast as .By using the Euler–Maruyama numerical scheme, the simulation results are as follows: and step size , the initial condition for . Fig. 1 shows that the solution of deterministic delay system [i.e. system (15] without the stochastic feedback term) explodes, while Fig. 2 verifies that the solution of the stochastic system (15) has degressive trend. Remark 11.It should be mentioned that in Example 1, and . Thus, all the results in [5–9, 55] cannot be applied to our example. Fig. 1Open in figure viewerPowerPoint State response of deterministic delay system Fig. 2Open in figure viewerPowerPoint State response of stochastic case 5 Conclusion In this paper we are interested in the effect of a noise on the solutions of a class of non-linear delay systems. We have introduced a set of new non-linear growth conditions with delays, and under it we have proved the existence and uniqueness of solutions as well as some properties of solutions. It is worth pointing out that our new condition only needs to know the coefficient of the highest term, which is much easier to check in practice than the traditional conditions. Clearly, our new condition overcomes many limitations and have a wider application, e.g. our conditions can allow some more complex Duffing systems with delays than those in [8] since delays were ignored in [8]. In our future works, we will apply our new condition and method to study some real non-linear systems. 6 Acknowledgments This work was partially supported by the National Natural Science Foundation of China (61773217, 61374080,11531006, 61773131, U1509217), the Natural Science Foundation of Jiangsu Province (BK20161552), the Alexander von Humboldt Foundation of Germany (Fellowship CHN/1163390), the Australian Research Council (DP170102644), Qing Lan Project of Jiangsu Province, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the 111 Project (B17048, B17017). 7 References 1Meerkov S.: ' Condition of virbrational stablizability for a class of nonlinear systems', IEEE Trans. Autom. Control, 1982, 27, pp. 485– 487CrossrefWeb of Science®Google Scholar 2Hasminskii R.Z.: ' Stochastic stability of differential equations' ( Sijthoff and Noordhoff, Dordrecht, Netherlands 1981) Google Scholar 3Mao X.: ' Stochastic stabilisation and destabilization', Syst. 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