Stability of stochastic impulsive differential equations: integrating the cyber and the physical of stochastic systems
aa r X i v : . [ c s . S Y ] N ov Stability of stochastic impulsivedifferential equations: integrating thecyber and the physical of stochasticsystems
Lirong HuangGuangzhou, ChinaEmail: [email protected]
Abstract: This paper proposes a novel framework for theory of cyber-physical systems (CPS) as well as numerical study of stochastic differentialequations (SDEs), which formulates a hybrid system with stochastic im-pulsive differential equations (SiDEs) to descibe the integrated dynamicsof the physical and the cyber parts of the system that are represented bythe exact and numerical solutions of an SDE, respectively, unlike in theliterature where they are linked by inequalities. This systematic represen-tation provides not only a holistic view but also a seamless, fully syner-gistic integration of the cyber subsystem/the numerical method with thephysical subsystem/the SDE. To establish a foundation of the proposedframework, this paper initiates the study of a new and general class ofSiDEs and develop by Lyapunov methods a stability theory for the gen-eral class of SiDEs. Applying the established theory, this paper studies thetest problem (Q1) how to reproduce the p th moment exponential stabil-ity of the underlying physical system/SDE in the cyber counterpart/thewidely-used Eular-Maruyama method and the converse (Q2) whether the p th moment exponential stability of the underlying physical system/SDEcan be inferred from that of the cyber counterpart/the numerical method.The proposed theory removes the principal barrier to developing CPS andestablishes a new systems science that comprehends cyber and physicalresources in a single unified framework. As an application, this paperpresents the cyber-physical/numerical Lyapunov-Itˆo inequality for linearsystems, which is the necessary and sufficient condition for mean-squareexponential stability of the physical system/the SDE, its cyber counter-part/the Eular-Maruyama method, and the resulting CPS/SiDE. Thiswork initiates the studies of systems numerics and the systems sciencefor CPS, in which there are many open and challenging problems.Key words: cyber-physical systems (CPS); stochastic differential equa-tions; numerical methods; impulsive systems; moment exponential stabil-ity; Lyapunov methods. Introduction
According to Newton’s second law of motion, we describe a mechanical system witha differential equation. Usually, physiccal laws are expressed by means of differentialequations, and so are the models of dynamical systems in many disciplines, rang-ing from biology to finance. The modelling of any practical system is subject touncertainty. Such uncertainty may be due to uncertain model structure and/or pa-rameters, unmeasured disturbances, and measurement noise [1, 6, 7]. Having realizedthe necessity of introducing more realistic models of uncertainty, we are faced withthe problem of finding suitable ways to characterize them. A characteristic featureof practical uncertainty is the impossibility of predicting future values of the systemsprecisely. One successful answer to the problem of modelling uncertainty is to de-scribe dynamical systems as stochastic processes, the theory of which has actuallypartly grown out of attempts to model the fluctuations observed in physical systems,see [1, 15, 26, 40, 57] and the references therein.Over the past decaseds, stochastic systems described by stochastic differentialequations (SDEs) have been intensively studied since stochastic modelling has cometo play an important role in many branches of science and industry (see, e.g., [1, 3, 12,23, 26, 40]). In general, it is difficult to solve an SDE analytically, whose solution is acontinuous-time stochastic process, called the exact solution. For practical purposes,numerical approximations to the exact solution of the SDE, which, called the numer-ical solutions, are discrete-time stochastic processes produced by a numerical scheme,are usually employed. On the other hand, whenever a computer is used in measure-ment, computation, signal processing or control applications, the data, signals andsystems involved are naturally described as discrete-time processes. Prevalent em-ployments of computers and wide use of stochastic modelling have greatly boostednot only popularity of numerical methods for SDEs (see [19, 20, 34, 41, 48, 50]),but also investigations on stochastic systems described by stochastic difference equa-tions, including those many discretizations of SDEs, over the recent years (see, e.g.,[9, 28, 29, 53]), because of their various applications. From the behaviour of thediscrete-time stochastic processes generated by some numerical scheme that are theapproximate realizations of the exact solution, one would learn and/or infer somedynamical properties of the underlying SDE (see, e.g., [2, 52]). It should be stressedin this situation that the SDE is the physical model of a dynamical system, which isestablished in a human brain with some physical laws, while the stochastic differenceequation derived from the SDE by some numerical method is the cyber model of thedynamical system in a computing machine, which, conversely, generates numericalsolutions to the SDE. This is typically teamwork of a human and a computing ma-chine, in which the physical model of the dynamical system established in the brain,as a natural logic machine, is transformed, by some numerical method, into its cybercouterpart in the computer, an artificial logic machine, and the machine, as an ana-logue of the brain, executes the computations, part of the task it excels at. NorbertWiener [58] was a pioneer who considered the mechanism and cooperation of such2atural and artificial machines, see also [6, 37, 46]. The situation we now face is onein which pervasive computing, sensing, and communications are common and the waythat we interact with machines and they interact with each other is changing rapidly[6]. It is an urgent need in such an information rich world to understand the mech-anism of interaction of the physical world with its cyber counterpart in machines,see [6, 7, 37, 46]. The physical model refers to the dynamical system itself (particu-larly, for a human-designed system) in the physical world while its cyber couterpartsymbolizes it in the cyber world, and hence they are also referred to as the physicalsystem and the cyber system, respectively. It is natural and necessary (i) to figureout the relationship between the physical system and its cyber counterpart, which arerepresented by the SDE and the numerical method, respectively; and (ii) to ensurethat they both share some dynamical properties, such as stability in this work, whichare concerned in a study.The principal aim of this paper is to address the above problems. As a matterof fact, these problems are of fundamental importance and have been studied in avast literature. Results that address the problems can be found in those many onconvergence and stability of numerical methods for SDEs, where the physical system/the SDE and its cyber counterpart/the numerical method are linked by inequalitiesin some moment sense on any finite time interval [20, 22, 30, 34, 43, 47], and theability of a cyber model/a numerical method to reproduce the stability of the physicalmodel/the SDE is shown in a number of works. Among these results, the problem howto reproduce the stability of the physical system/the SDE in its cyber counterpart/anumerical method, which is called the test problem [18], has been studied for SDEs[17, 21, 42, 48]. The key question in a test problem is [17](Q1) for what stepsizes ∆ t deos the cyber system/the numerical method share thestability property of the physical system/the SDE?This naturally provokes the converse key question [21, 42](Q2) does the stability of the cyber system /the numerical method for some smallstepsizes ∆ t imply that of the physical system/the SDE?These questions deal with asymptotic ( t → ∞ ) properties and hence cannot be an-swered directly by applying traditional finite-time convergence results, see [18, 21, 42].Results that answer questions (Q1) and (Q2) can be found in the literature. Forexample, results for scalar linear systems were given in [17, 48]. Particularly, formulti-dimensional systems with global Lipschitz condition, Higham, Mao and Stuart[21] introduced a natural finite-time strong convergence condition [21, Condition 2.3],which links a cyber system/a numerical method with the physical system/the SDEin some moment sense by an inequality over any finite time interval, and proved thatthere is a sufficiently small ∆ t ∗ > t ∈ (0 , ∆ t ∗ ], the mean-square exponential stability of the physical system /the SDE is equivalent to that ofits cyber counterpart /the numerical method [21, Lemmas 2.4-2.5 and Theorem 2.6].3ecently, Mao [42] developed new techniques to handle the small p th moment (with p ∈ (0 , p th moment condition [42, Assumption2.4] and a natural finite-time convergence condition [42, Assumption 2.5], that, forevery ∆ t ∈ (0 , ∆ t ∗ ] with sufficiently small ∆ t ∗ >
0, the p th moment exponentialstability of of the physical system /the SDE is equivalent to that of its cyber coun-terpart/the numerical method [42, Lemmas 2.6-2.7 and Theorem 2.8]. As is pointedout in [42], there are many open problems in this research. For example, althoughthe existence of the (sufficiently small) upper bound ∆ t ∗ > t ∗ ofstepsizes and facilitate its computation, see Sections 4 and 5 below.In the literature, the above problems of fundamental importance are addressedby employing dynamical union of the physical system/the differential equation andits cyber counterpart/a numerical method such as linking the cyber system /thenumerical method with the physical system/the differential equation by inequali-ties [20, 22, 30, 34, 43, 47, 52], in which, however, they have remained as two sys-tems, largely seperate. During the past a few years, a trend now known as cyber-physical systems (CPS, the term coined at the National Science Foundation in theUnited States) for integrating the cyber system with the physical system has beenan emerging area that refers to the next generation engineered systems, which hasbeen placed as one of the top for priorities for research investment and recognized asa paramount and prospective shift towards future networking and information tech-nology [6, 7, 14, 35]. From an engineering viewpoint, ‘as an intellectual challenge,CPS is about the intersection, not the union, of the physical and the cyber. It is notsufficient to separately understand the physical components and the computationalcomponents. We must instead understand their interaction’ [37]. ‘The principal bar-rier to developing CPS is the lack of a theory that comprehends cyber and physicalresources in a single unified framework’ [7]. Cyberphysicality spans the gamut ofengineering domains. The tight integration of physical and information processes inCPS necessitates the development of a new systems science [35].This paper proposes a novel framework for numerical study of SDE and theory ofCPS, in which we formulate a new and general class of stochastic impulsive differentialequations (SiDEs) that can be used to describe the dynamics of a CPS, a seamless,fully synergistic integration of a physical system /an SDE and its cyber counterpart (anumerical method). Impulsive systems described by impulsive differential equationshave been studied in systems and control theory for several decades with the emphasis4eing on stability analysis and its applications [31, 32, 36, 49, 60–62]. Stochasticimpulsive systems described by SiDEs, which comprise an extension of impulsivesystems that combines continuous change and instantaneous change and that includeseffects of stochastic perturbation [54], have been studied in many works, e.g., [45] and[63] gave the stochastic extensions of some results in [8] and [16], respectively, while[5, 44] developed Razumikhin-type theorems for SiDEs with delays. However, theimpulsive systems including the SiDEs considered in the literature [31, 32, 36, 38, 45,49, 51, 54, 59, 63] do not apply to our motivation but can be regarded as some specificcases of our proposed SiDEs, see Section 2. Unlike the results in the literature wherethe cyber system /the numerical method is linked with the physical system /the SDEby (more or less restrictive) inequalities such as the natural finite-time convergencecondition in [21, 42], we formulate a hybrid system with SiDEs to represent the CPSthat is a seamless integration of the physical /the SDE and the cyber/the numericalmethod, in which they are interactive subsystems. In this systematic framework,one could not only exploit some useful ideas/methods in systems and control theorybut also develop novel approaches, methods, techniques and theories for numericalsimulations of SDEs and design science of CPS.This systematic presentation provides not only a seamless, fully synergistic inte-gration but also a holistic view of a physical system/an SDE and its cyber counter-part/a numerical method. It reveals the intrinsic relationship between the physicalmodel/the SDE and its cyber counterpart /the numerical method by representing thephysical and the cyber of the CPS as interactive components in a hybrid model ofSiDE. From the viewpoint of cybernetics, an essential problem to study is whetherand how the CPS (the hybrid system of SiDE) reproduces some dynamical propertiessuch as stability of either the physical subsystem/the SDE or its cyber counterpart/the numerical method, since ‘the primary concern of cybernetics is on the qualita-tive aspects of the interrelations among the various components of a system and thesynthetic behavior of the complete mechanism’ [55]. Using the semantics of CPS, werephrase questions (Q1) and (Q2) as follows.(Q1) For what stepsizes ∆ t of the cyber subsystem/the numerical method deos theCPS/the SiDE reproduce the stability property of the physical subsystem /theSDE?(Q2) Does the stability of the cyber subsystem/the numerical method for some smallstepsizes ∆ t imply that of the CPS/the SiDE?Our proposed systematic framework and theory remove the principal barrier to de-veloping CPS and establish a theoretic foundation for the study of systems numericsand a systems science for CPS.The rest of this paper is organized as follows. Section 2 introduces notation and thenew, general class of SiDEs that can used to represent CPS. In Section 3, we developby Lyapunov methods a stability theory for the general class of SiDEs, which hasvarious applications. In Section 4, we represent the integrated dynamics of a physical5ystem/an SDE and its cyber counter/the Eular-Maruyama method as a CPS in theform of the proposed SiDE, and study the test problem (Q1) with the stability theoryproposed in Section 3. Since the general class of SiDEs are now used to representCPS, the converse problem (Q2) arises, and we proceed to study it for CPS (SDEsand the Eular-Maruyama method). Section 5 is devoted to applications of the resultswe establish in this paper, where, taking linear sytems as an application example, wepresent the cyber-physical/numerical Lyapunov inequality (109) to the problems (Q1)and (Q2) of linear CPS (linear SDEs and the Eular-Maruyama method), and showthat the cyber-physical/numerical Lyapunov inequality (109) is the necessary andsufficient condition for mean-square exponential stability of the linear SDE, the linearCPS or the Eular-Maruyama method. Concluding remarks and some future worksare given in Section 6. The new research fields started in this work are highlighted,which are not only more general than but also essentially different from the classicaltheory of impulsive systems in the literature. Throughout this paper, unless otherwise specified, we shall employ the following nota-tion. Let us denote by (Ω , F , {F t } t ≥ , P ) a complete probability space with a filtration {F t } t ≥ satisfying the usual conditions (i.e. it is right continuous and F containsall P -null sets) and by E [ · ] the expectation operator with respect to the probabilitymeasure. Let B ( t ) = (cid:2) B ( t ) · · · B m ( t ) (cid:3) T be an m -dimensional Brownian motiondefined on the probability space. If x, y are real numbers, then x ∨ y denotes themaximum of x and y , and x ∧ y stands for the minimum of x and y . If A is a vector ora matrix, its transpose is denoted by A T . If P is a square matrix, P >
P < P ≥ P ≤
0) is a symmetric positive (resp. negative)semidefinite matrix. Denote by λ M ( · )and λ m ( · ) the maximum and minimum eigen-values of a matrix, respectively. Let | · | denote the Euclidean norm of a vector andthe trace (or Frobenius) norm of a matrix. Let C , ( R n × R + ; R + ) be the family of allnonnegative functions V ( x, t ) on R n × R + that are continuously twice differentiablein x and once in t . Let M p ([ a, b ]; R n ) be the family of R n -valued adapted process { x ( t ) : a ≤ t ≤ b } such that E R ba | x ( t ) | p d t < ∞ . Denote by I n the n × n identitymatrix and by O n × m the n × m zero matrix. Let { ξ ( k ) } k ∈ N , N = { , , , · · · } , be anindependent and identically distributed sequence with ξ ( k ) = (cid:2) ξ ( k ) · · · ξ m ( k ) (cid:3) T ,and ξ j ( k ), j = 1 , , · · · , m , obeying standard normal distribution while { t k } k ∈ N is astrictly increasing sequence with 0 = t < t < t < · · · and t k → ∞ as k → ∞ .Let us consider the following ( n + q )-dimensional stochastic impulsive differential6quations (SiDEs)d x ( t ) = f ( x ( t ) , t )d t + g ( x ( t ) , t )d B ( t ) (1a)d y ( t ) = ˜ f ( x ( t ) , y ( t ) , t )d t + ˜ g ( x ( t ) , y ( t ) , t )d B ( t ) , t ∈ [ t k , t k +1 ) (1b)∆( x ( t − k +1 ) , k + 1) := x ( t k +1 ) − x ( t − k +1 )= h f ( x ( t − k +1 ) , k + 1) + h g ( x ( t − k +1 ) , k + 1) ξ ( k + 1) (1c)˜∆( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) := y ( t k +1 ) − y ( t − k +1 )= ˜ h f ( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) + ˜ h g ( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) ξ ( k + 1) (1d)for k ∈ N with initial data x (0) = x ∈ R n and y (0) = y ∈ R q , where ξ ( k + 1)is independent of { x ( t ) , y ( t ) : 0 ≤ t < t k +1 } , and functions f : R n × R + → R n , g : R n × R + → R n × m , h f : R n × N → R n , h g : R n × N → R n × m , ˜ f : R n × R q × R + → R q ,˜ g : R n × R q × R + → R q × m , ˜ h f : R n × R q × N → R q , ˜ h g : R n × R q × N → R q × m obey f (0 , t ) = 0, g (0 , t ) = 0, h f (0 , k ) = 0 and h g (0 , k ) = 0, ˜ f (0 , , t ) = 0, ˜ g (0 , , t ) = 0,˜ h f (0 , , k ) = 0 and ˜ h g (0 , , k ) = 0 for all t ∈ R + and k ∈ N , and satisfy the globalLipschitz continuous conditions. Assumption 2.1.
There is
L > such that | f ( x, t ) − f (¯ x, t ) |∨| g ( x, t ) − g (¯ x, t ) |∨| h f ( x, k ) − h f (¯ x, k ) |∨| h g ( x, k ) − h g (¯ x, k ) | ≤ L | x − ¯ x | for all ( x, ¯ x ) ∈ R n × R n , t ∈ R + , k ∈ N ; and there is ˜ L > such that | ˜ f ( x, y, t ) − ˜ f (˜ x, ˜ y, t ) | ∨ | ˜ g ( x, y, t ) − ˜ g (˜ x, ˜ y, t ) |∨ | ˜ h f ( x, y, k ) − ˜ h f (˜ x, ˜ y, k ) | ∨ | ˜ h g ( x, y, k ) − ˜ h g (˜ x, ˜ y, k ) | ≤ ˜ L ( | x − ˜ x | ∨ | y − ˜ y | ) for all ( x, y, ˜ x, ˜ y ) ∈ R n × R q × R n × R q , t ∈ R + , k ∈ N . Strongly motivated by the increasing role of information-based systems in ourinformation rich world [6] such as those in numerical simulations and CPS, we initiatethe study of SiDE (1a-1d), which is used to represent, in Sections 4 and 5, a CPSthat is a seamless, fully synergistic integration of the physical system/the differentialequation and its cyber counterpart/a numerical method. It is observed that SiDEs(1a-1d) are a more general class of stochastic impulsive systems that includes thoseconsidered in the literature. For example, the stochastic impulsive systems in [45, 51,63] are a specific case of SiDE (1a-1d) with q = 0 and h g ( · , · ) ≡
0. Obviously, thetrivial solution is the equilibrium of system (1a-1d). For a function V ∈ C , ( R n × R + ; R + ), the infinitesimal generator L V : R n × R + → R associated with system (1a)is defined by L V ( x, t ) = V t ( x, t ) + V x ( x, t ) f ( x, t ) + 12 trace (cid:2) g T ( x, t ) V xx ( x, t ) g ( x, t ) (cid:3) , (2)where V t ( x, t ) = ∂V ( x, t ) ∂t , V x ( x, t ) = h ∂V ( x,t ) ∂x · · · ∂V ( x,t ) ∂x n i , V xx ( x, t ) = h ∂ V ( x,t ) ∂x i ∂x j i n × n . V ∈ C , ( R q × R + ; R + ), one can define the generator ˜ L ˜ V : R n × R q × R + → R associated with system (1b) by˜ L ˜ V ( x, y, t ) = ˜ V t ( y, t ) + ˜ V y ( y, t ) ˜ f ( x, y, t ) + 12 trace h ˜ g T ( x, y, t ) ˜ V yy ( y, t )˜ g ( x, y, t ) i . (3)Let z ( t ) = (cid:2) x T ( t ) y T ( t ) (cid:3) T ∈ R n + q , C = (cid:2) I n O n × q (cid:3) and D = (cid:2) O q × n I q (cid:3) , then x ( t ) = Cz ( t ) and y ( t ) = Dz ( t ). Stochastic impulsive system (1a-1d) can be writtenin a compact form as followsd z ( t ) = F ( z ( t ) , t )d t + G ( z ( t ) , t )d B ( t ) , t ∈ [ t k , t k +1 ) (4a)∆ z ( z ( t − k +1 ) , k + 1) := z ( t k +1 ) − z ( t − k +1 )= H F ( z ( t − k +1 ) , k + 1) + H G ( z ( t − k +1 ) , k + 1) ξ ( k + 1) (4b)for k ∈ N with initial data z (0) = z = (cid:2) x T y T (cid:3) T , where functions F : R n + q × R + → R n + q , G : R n + q × R + → R ( n + q ) × m , H F : R n + q × N → R n + q , H G : R n + q × N → R ( n + q ) × m are given by F ( z, t ) = (cid:20) f ( Cz, t )˜ f ( Cz, Dz, t ) (cid:21) , G ( z, t ) = (cid:20) g ( Cz, t )˜ g ( Cz, Dz ) , t (cid:21) ,H F ( z, k ) = (cid:20) h f ( Cz, k )˜ h f ( Cz, Dz, k ) (cid:21) , H G ( z, k ) = (cid:20) h g ( Cz, k )˜ h g ( Cz, Dz, k ) (cid:21) . Obviously, stochastic impulsive system (4a-4b) obeys F (0 , t ) = 0, G (0 , t ) = 0, H F (0 , k ) = 0, H G (0 , k ) = 0 for all t ∈ R + , k ∈ N , and, by Assumption 2.1, satisfiesthe global Lipschitz continuous condition, that is, there is L z > | F ( z, t ) − F (˜ z, t ) | ∨ | G ( z, t ) − G (˜ z, t ) |∨| H F ( z, k ) − H F (˜ z, k ) | ∨ | H G ( z, k ) − H G (˜ z, k )) | ≤ L z | z − ˜ z | (5)for all t ∈ R + , k ∈ N . It is easy to obtain the following result on existence anduniqueness of solutions for SiDE (4a-4b). Lemma 2.1.
Under Assumption Assumption 2.1, there exists a unique (right-continuous)solution z ( t ) to SiDE (4a-4b) (i.e., (1a-1d)) on t ≥ and the solution belongs to M ([0 , T ]; R n + q ) for all T ≥ .Proof. Since SiDE (4a-4b) satisfies the global Lipschitz continuous condition (5),according to [40, Theorem 3.1, p51], there exists a unique solution z ( t ) to (4a-4b)on t ∈ [ t , t ) and the solution belongs to M ([ t , t ); R n + q ). Note that ξ ( k + 1) isindependent of { z ( t ) : t ∈ [ t , t ) } . By virtue of continuity of functions H F and H G ,there exists a unique solution z ( t ) to (4a-4b) on t = t . Moreover, (4b) and (5) implythat the second moment of z ( t ) is finite. And then, again, according to [40, Theorem3.1, p51], one has that there is a unique right-continuous solution z ( t ) to (4a-4b) on8 ∈ [ t , t ) and the solution belongs to M ([ t , t ]; R n + q ) for t ∈ [ t , t ). Recall that0 = t < t < t < · · · < t k < · · · and t k → ∞ as k → ∞ . By induction, one has thatthere exists a unique (right-continuous) solution z ( t ) to SiDE (4a-4b) for all t ≥ M ([0 , T ]; R n + q ) for all T ≥ Definition 2.1. [40, Definition 4.1, p127] The system (4a-4b) is said to be p th ( p > )moment exponentially stable if there is a pair of positive constants K and c such that E | z ( t ) | p ≤ K | z | p e − ct , t ≥ , for all z ∈ R n + q , which leads to lim sup t →∞ t ln( E | z ( t ) | p ) ≤ − c < . Definition 2.2. [40, Definition 3.1, p119] The system (4a-4b) is said to be almostsurely exponentially stable if lim sup t →∞ t ln | z ( t ) | < for all z ∈ R n + q . Since the existing stability results do not apply to our proposed SiDEs, we will estab-lish by Lyapunov methods a stability theory for the general class of SiDEs (1a-1d).This foundational theory can be applied to numerical study of SDEs as well as controlof stochastic impulsive systems. In above, for simplicity, the compact form (4a-4b)of system (1a-1d) is employed to study the existence and uniqueness of solutions tothe SiDE. In this section, we will make use of the description (1a-1d) of the systemand study the stability of the solution to SiDE (1a-1d) because it would be convientto exploit the structure of the system in the decomposed description in the form of(1a-1d).
Theorem 3.1.
Suppose that Assumption 2.1 holds. Let V ∈ C , ( R n × R + ; R + ) and ˜ V ∈ C , ( R q × R + ; R + ) be a pair of candidate Lyapunov functions for systems (1a,1c) and (1b,1d), respectively, which satisfy c | x | p ≤ V ( x, t ) ≤ c | x | p , ˜ c | y | p ≤ ˜ V ( y, t ) ≤ ˜ c | y | p (6)9 or all ( x, y, t ) ∈ R n × R q × R + and some positive constants p, c , c , ˜ c , ˜ c . Assumethat there are positive constants α , ˜ α , ˜ α , ˜ β , ˜ β , ˜ β such that L V ( x, t ) ≤ − αV ( x, t ) , t ∈ [ t k , t k +1 ) (7)˜ L ˜ V ( x, y, t ) ≤ ˜ α V ( x, t ) + ˜ α ˜ V ( y, t ) , t ∈ [ t k , t k +1 ) (8) E V ( x + ∆( x, k + 1) , t ) ≤ βV ( x, t ) , (9) E ˜ V ( y + ˜∆( x, y, k + 1) , t ) ≤ ˜ β V ( x, t ) + ˜ β ˜ V ( y, t ) (10) for all ( x, y, t ) ∈ R n × R q × R + and k ∈ N . Let the impulse time sequence { t k } satisfy ln βα < ∆ t ≤ ∆ t < − ln ˜ β ˜ α , (11) where < ∆ t := inf k ∈ N { t k +1 − t k } ≤ ∆ t := sup k ∈ N { t k +1 − t k } < ∞ . Then SiDE(4a-4b) (i.e., (1a-1d)) is p th moment exponentially stable.Proof. According to Lemma 2.1, that Assumption 2.1 holds implies there exists aunique solution to SiDE (1a-1d). Let us fix any z (0) = z = (cid:2) x T y T (cid:3) T ∈ R n + q andstart to show the stability of the solution. The proof is rather technical so we devideit into five steps, in which we will show: 1) the exponential stability of x ( t ); 2) somepropeties of y ( t ); 3) the exponential stability of y ( t ) when | x | = 0; 4) the exponentialstability of y ( t ) when | x | >
0; 5) the exponential stability of z ( t ). Some ideas andtechniques in this proof are derived from input-to-state stability of SDEs [25], where x ( t ) is treated as disturbance in the subsystem of y ( t ). Step 1:
By Lemma 2.1 and Itˆo’s formula, one can derive that for t ∈ [ t , t ) E (cid:2) e αt V ( x ( t ) , t ) (cid:3) = V ( x ,
0) + E Z t e αs [ αV ( x ( s ) , s ) + L V ( x ( s ) , s )] d s. (12)Using condition (81) gives E V ( x ( t ) , t ) ≤ V ( x , e − αt (13)for all t ∈ [ t , t ), and, particularly, E V ( x ( t − ) , t − ) ≤ V ( x , e − αt . (14)At t = t , by (1c) and (9), one has E V ( x ( t ) , t ) = E V ( x ( t − ) + ∆( x ( t − ) , , t )= E h E (cid:2) V ( x ( t − ) + ∆( x ( t − ) , , t ) | x ( t − ) (cid:3)i ≤ E [ βV ( x ( t − ) , t − )] = β E V ( x ( t − ) , t − ) ≤ βe − αt V ( x , . (15)But, by condition (11), one observes β e (¯ α − α ) t = β e (¯ α − α ) ( t − t ) ≤ β e ¯ α ∆ t − α ∆ t ≤ , (16)10here ¯ α ∈ (0 , ( α ∆ t − ln β ) / ∆ t ] is a positive constant. Combination of (13)-(16)yields E V ( x ( t ) , t ) ≤ V ( x , e − ¯ αt (17)on t ∈ [0 , t ]. Using the same reasoning, one can obtain E V ( x ( t ) , t ) ≤ E V ( x ( t k ) , t k ) e − ¯ α ( t − t k ) (18)on t ∈ [ t k , t k +1 ] for all k ∈ N . This means that E V ( x ( t ) , t ) is right-continous on t ∈ [0 , ∞ ) and satisfy E V ( x ( t ) , t ) ≤ V ( x , e − ¯ αt (19)for all t ≥
0. This, by condition (6), implies that E | x ( t ) | p ≤ c c | x | p e − ¯ αt (20)for all t ≥
0, that is, part of the system (1a-1d), x ( t ) is p th moment exponentiallystable (with Lypunov exponent no larger than − ¯ α ). Step 2:
Let us now proceed to consider the dynamics of the other part, y ( t ), ofthe system (1a-1d). By 2.1 and Itˆo’s formula, one has E ˜ V ( y ( t ) , t ) = ˜ V ( y ,
0) + Z t E ˜ L ˜ V ( x ( s ) , y ( s ) , s )d s (21)on t ∈ [ t , t ). Using the same reasoning, one can derive that E ˜ V ( y ( t ) , t ) = E ˜ V ( y ( t k ) , t k ) + Z tt k E ˜ L ˜ V ( x ( s ) , y ( s ) , s )d s (22)between any two consecutive impulses t k and t k +1 while condition (82) gives E ˜ L ˜ V ( y ( t ) , t ) ≤ ˜ α E V ( x ( t ) , t ) + ˜ α E ˜ V ( y ( t ) , t ) (23)on t ∈ [ t k , t k +1 ) for all k ∈ N . Obviously, this means that E ˜ V ( y ( t ) , t ) is right-continuous on t ∈ [0 , ∞ ) and could only have jumps at impulse times { t k +1 } k ∈ N .Notice condition (11) implies ˜ β e ˜ α ∆ t < δ ∈ (0 , − ˜ β ) and ¯ δ ∈ (0 , ¯ α ] sufficientlysmall for ( ˜ β + δ ) e (˜ α + δ +¯ δ ) ∆ t ≤ . (24)It is easy to observe, from (23), that E ˜ L ˜ V ( y ( t ) , t ) ≤ ( ˜ α + δ ) E ˜ V ( y ( t ) , t ) (25)11or such t ∈ [ t k , t k +1 ) and k ∈ N where E ˜ V ( y ( t ) , t ) ≥ ˜ α δ E V ( x ( t ) , t ) . Similarly, one can observe, from (10), that E ˜ V ( y ( t k +1 ) , t k +1 ) ≤ ( ˜ β + δ ) E ˜ V ( y ( t − k +1 ) , t − k +1 ) (26)whenever E ˜ V ( y ( t − k +1 ) , t − k +1 ) ≥ ˜ β δ E V ( x ( t − k +1 ) , t − k +1 ) . Step 3: If x = 0, i.e., V ( x ,
0) = 0, then inequality (19) gives E V ( x ( t ) , t ) = 0 forall t ≥
0. Using (82), (21) and (23), one has E ˜ V ( y ( t ) , t ) ≤ ˜ V ( y ,
0) + ˜ α Z t E ˜ V ( y ( s ) , s )d s (27)for all t ∈ [0 , t ). This, by Gronwall’s inequality (see, e.g., [40, Theorem 8.1, p44]),implies E ˜ V ( y ( t ) , t ) ≤ ˜ V ( y , e ˜ α t , ∀ t ∈ [0 , t ) . (28)and, particularly, E ˜ V ( y ( t − ) , t − ) ≤ ˜ V ( y , e ˜ α t . (29)Using this with conditions (10) and (24), one obtains E ˜ V ( y ( t ) , t ) ≤ ˜ β E ˜ V ( y ( t − ) , t − ) ≤ ˜ β ˜ V ( y , e ˜ α t < ˜ V ( y , e − (˜ α + δ +¯ δ ) ∆ t e ˜ α t ≤ ˜ V ( y , e − ( δ +¯ δ ) ∆ t . (30)In fact, one can repeat the derivations (27)-(30) over the interval between any twoconsecutive impulse times [ t k , t k +1 ] and obtain E ˜ V ( y ( t ) , t ) ≤ ˜ V ( y , e ˜ α ( t − t k ) − k ( δ +¯ δ ) ∆ t (31)for all t ∈ [ t k , t k +1 ) and k ∈ N . This implies E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ +¯ δ ) ∆ t ˜ V ( y , e − ( δ +¯ δ ) t . (32)for all t ≥
0. When x = 0, under condition (6), inequality (32) implies that y ( t ) is p th moment exponentially stable (with Lyapunov exponent no larger than − ( δ + ¯ δ )). Step 4:
In this step, we will show the exponential stability of y ( t ) in the case when | x | >
0, i.e., V ( x , >
0. Recall that both E V ( x ( t ) , t ) and E ˜ V ( y ( t ) , t ) are right-continuous on t ∈ [0 , ∞ ), which could only have jumps at impulse times { t k +1 } k ∈ N .Define a function ¯ v : R + → R by¯ v ( t ) = ( ˜ α ∨ ˜ β ) δ E V ( x ( t ) , t ) − E ˜ V ( y ( t ) , t ) (33)12n t ∈ [0 , ∞ ) with initial value ¯ v (0) = (˜ α ∨ ˜ β ) δ V ( x , − ˜ V ( y , E V ( x ( t ) , t ) and E ˜ V ( y ( t ) , t ), right-continuous on t ∈ [0 , ∞ ) and couldonly have jumps at impulse times { t k +1 } k ∈ N . Notice that, given any t ≥
0, either¯ v ( t ) ≥ v ( t ) <
0. The interval [0 , ∞ ) is broken into a disjoint union of subsets T + ∪ T − , where T + = { t ≥ v ( t ) ≥ } , T − = { t ≥ v ( t ) < } . (34)Obviously, one has, from (19), that E ˜ V ( y ( t ) , t ) ≤ ( ˜ α ∨ ˜ β ) δ E V ( x ( t ) , t ) ≤ ( ˜ α ∨ ˜ β ) δ V ( x , e − ¯ αt (35)for all t ∈ T + .Without loss of generality, assume that ¯ v (0) > V ( x , >
0, onecan always choose a sufficiently small δ > v (0) > v ( t ) > , ǫ ) for some number ǫ >
0, i.e., [0 , ǫ ) ⊂ T + . If T + = [0 , ∞ ) (i.e., T − = ∅ ), the proof is complete. Otherwise, let us consider the right-continuous process E ˜ V ( y ( t ) , t ) on the subset T − . Actually, due to the right-continuity of ¯ v ( t ) on [0 , ∞ ),given any ¯ t ∈ T − , one can find an interval [ τ (¯ t ) , τ (¯ t )) such that ( τ (¯ t ) , τ (¯ t )) ⊂ T − ,where τ (¯ t ) = inf { τ ≤ ¯ t : ¯ v ( τ ) < , ∀ τ ∈ [ τ , ¯ t ] } ,τ (¯ t ) = sup { ¯ τ > ¯ t : ¯ v ( τ ) < , ∀ τ ∈ [ ¯ t, ¯ τ ) } . (36)For convenience, we also write τ = τ (¯ t ) and τ = τ (¯ t ) when there is no ambiguity.Now we consider the right-continuous process E ˜ V ( y ( t ) , t ) on the interval [ τ , τ ),which would fall into one of three categories as follows:I)There is no impulse time on [ τ , τ ). Because of the property that ¯ v ( t ) is right-continuous and could only have jumps at at impulse times { t k +1 } k ∈ N , that τ is not animpulse time, namely, τ / ∈ { t k +1 } k ∈ N implies ¯ v ( t ) is continuous on t = τ and hence¯ v ( τ ) = 0. This means E ˜ V ( y ( τ ) , τ ) = ( ˜ α ∨ ˜ β ) δ E V ( x ( τ ) , τ ) ≤ ( ˜ α ∨ ˜ β ) δ V ( x , e − ¯ ατ . (37)Using Gronwall’s inequality, one can derive from equation (22), inequalities (23) and(25) that E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ ) ( t − τ ) E ˜ V ( y ( τ ) , τ ) (38)for all t ∈ [ τ , τ ). Note that, in this case, τ − τ ≤ ∆ t . Substitution of inequality1337) into (38) leads to E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ ) ∆ t E ˜ V ( y ( τ ) , τ ) ≤ ( ˜ α ∨ ˜ β ) δ e (˜ α + δ ) ∆ t V ( x , e − ¯ ατ ≤ ( ˜ α ∨ ˜ β ) δ e (¯ α +˜ α + δ ) ∆ t V ( x , e − ¯ αt (39)for all t ∈ [ τ , τ ).II) There is exactly one impulse time t k on [ τ , τ ). There are two cases: i) τ < t k and ii) τ = t k .i) If τ < t k , which means τ / ∈ { t k +1 } k ∈ N , then, as above, ineqaulity (37) holds for t = τ and inequality (39) holds for all t ∈ [ τ , t k ). But, by (26) and (24), one has E ˜ V ( y ( t k ) , t k ) ≤ ( ˜ β + δ ) E ˜ V ( y ( t − k ) , t − k ) ≤ ( ˜ β + δ ) e (˜ α + δ ) ( t k − τ ) E ˜ V ( y ( τ ) , τ ) ≤ ( ˜ β + δ ) e (˜ α + δ ) ∆ t E ˜ V ( y ( τ ) , τ ) ≤ e − ¯ δ ∆ t E ˜ V ( y ( τ ) , τ ) . (40)By Gronwall’s inequality, (22), (23), (25) and (40) yield E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ )( t − t k ) E ˜ V ( y ( t k ) , t k ) ≤ e (˜ α + δ ) ∆ t E ˜ V ( y ( t k ) , t k ) ≤ e (˜ α + δ − ¯ δ ) ∆ t E ˜ V ( y ( τ ) , τ ) (41)for all t ∈ [ t k , τ ). Therefore, when τ < t k < τ , combination of (37), (39) and (41)yields that inequality (39) holds for all t ∈ [ τ , τ ).ii) By the property of ¯ v ( t ) and definition of τ , that τ = t k implies ¯ v ( t − k ) > E ˜ V ( y ( t − k ) , t − k ) < ( ˜ α ∨ ˜ β ) δ E V ( x ( t − k ) , t − k ) . (42)This, with (10) and (19), gives E ˜ V ( y ( τ ) , τ ) = E ˜ V ( y ( t k ) , t k ) ≤ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1) E V ( x ( t − k ) , t − k ) ≤ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1) V ( x , e − ¯ ατ (43)14nd hence E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ ) ( t − τ ) E ˜ V ( y ( τ ) , τ ) ≤ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1) e (˜ α + δ ) ( t − τ ) E V ( x ( τ − ) , τ − ) ≤ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1) e (˜ α + δ ) ∆ t V ( x , e − ¯ ατ ≤ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1) e (¯ α +˜ α + δ ) ∆ t V ( x , e − ¯ αt (44)for all t ∈ [ τ , τ ).Combining (39) and (44) yields E ˜ V ( y ( t ) , t ) ≤ K V ( x , e − ¯ αt (45)for all t ∈ [ τ , τ ) whenever there is only one impulse time on the interval [ τ , τ ),where K = (cid:16) ( ˜ α ∨ ˜ β ) δ ∨ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1)(cid:17) e (¯ α +˜ α + δ ) ∆ t (46)is a positive constant.III) There are at least two impulse times on [ τ , τ ). For any two consecutiveimpulses t k and t k +1 on [ τ , τ ), using the reasoning as above, one can derive that E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ ) ( t − t k ) E ˜ V ( y ( t k ) , t k ) (47)for all t ∈ [ t k , t k +1 ) and then E ˜ V ( y ( t k +1 ) , t k +1 ) ≤ ( ˜ β + δ ) E ˜ V ( y ( t − k +1 ) , t − k +1 ) ≤ ( ˜ β + δ ) e (˜ α + δ ) ( t k +1 − t k ) E ˜ V ( y ( t k ) , t k ) ≤ ( ˜ β + δ ) e (˜ α + δ ) ∆ t E ˜ V ( y ( t k ) , t k ) ≤ e − ¯ δ ∆ t E ˜ V ( y ( t k ) , t k ) . (48)Suppose that there are impulse times t k < t k +1 < · · · on [ τ , τ ). Let us nowconsider E ˜ V ( y ( t ) , t ) on the interval [ t k , τ ). By (47) and (48), one obtains E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ ) ( t − t ¯ k ) − (¯ k − k ) ¯ δ ∆ t E ˜ V ( y ( t k ) , t k ) (49)for all t ∈ [ t ¯ k , t ¯ k +1 ∧ τ ), where ¯ k ∈ N and ¯ k ≥ k . This implies E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ +¯ δ ) ∆ t − ¯ δ ( t − t k ) E ˜ V ( y ( t k ) , t k ) (50)for all t ∈ [ t k , τ ). Recall that 0 < ¯ δ ≤ ¯ α and 0 ≤ t k − τ ≤ ∆ t .15n the case when τ < t k , from (37), (40) and (50), one has E ˜ V ( y ( t ) , t ) ≤ e (˜ α + δ +¯ δ ) ∆ t − ¯ δ ( t − t k ) e − ¯ δ ∆ t E ˜ V ( y ( τ ) , τ ) ≤ ( ˜ α ∨ ˜ β ) δ e (˜ α + δ +¯ δ ) ∆ t V ( x , e − (¯ ατ +¯ δ ∆ t − ¯ δt k ) − ¯ δ t ≤ ( ˜ α ∨ ˜ β ) δ e (˜ α + δ +¯ δ ) ∆ t V ( x , e − ¯ δ ( τ +∆ t − t k ) − ¯ δ t ≤ ( ˜ α ∨ ˜ β ) δ e (˜ α + δ +¯ δ ) ∆ t V ( x , e − ¯ δ t (51)for all t ∈ [ t k , τ ), and then, by (39) E ˜ V ( y ( t ) , t ) ≤ ( ˜ α ∨ ˜ β ) δ e (˜ α + δ +¯ δ ) ∆ t V ( x , e − ¯ δ t (52)for all t ∈ [ τ , τ ).In the other case when τ = t k , substituting (43) into (50) yields E ˜ V ( y ( t ) , t ) ≤ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1) e (˜ α + δ +¯ δ ) ∆ t V ( x , e − (¯ α − ¯ δ ) τ − ¯ δ t ≤ (cid:0) ˜ β + ( ˜ α ∨ ˜ β ) δ ˜ β (cid:1) e (˜ α + δ +¯ δ ) ∆ t V ( x , e − ¯ δ t (53)for all t ∈ [ τ , τ ).Therefore, combining (52) and (53) gives E ˜ V ( y ( t ) , t ) ≤ K V ( x , e − ¯ δ t (54)for all t ∈ [ τ , τ ) which includes at least two impulse times, where K is the positiveconstant given by (46).From inequlities (39), (45) and (54), one sees E ˜ V ( y ( t ) , t ) ≤ K V ( x , e − ¯ δ t (55)for all t ∈ T − , where K is the positive number given by (46). Combining (35) and(55), one can conclude that E ˜ V ( y ( t ) , t ) ≤ K V ( x , e − ¯ δ t (56)for all t ≥
0. This means that, under condition (6), the other part, y ( t ), of thesystems is also p th moment exponentially stable (with Lyapunov exponent no largerthan − ¯ δ ) when | x | > Step 5:
We have shown the p th moment exponential stability of x ( t ) by inequality(19) and that of y ( t ) by (32) and (56) when | x | = 0 and | x | >
0, respectively.16ote that z ( t ) = (cid:20) x ( t ) y ( t ) (cid:21) = (cid:20) x ( t )0 (cid:21) + (cid:20) y ( t ) (cid:21) and hence | z ( t ) | ≤ | x ( t ) | + | y ( t ) | (57)for all t ≥
0. From (57), it is easy to see that | z ( t ) | p ≤ ( | x ( t ) | + | y ( t ) | ) p ≤ k p ( | x ( t ) | p + | y ( t ) | p ) , (58)where k p = 1 when 0 < p < k p = 2 p − when p ≥
1. Obviously, in thecase when | x | = 0 and hence E | x ( t ) | p = 0 for all t ≥
0, inequalities (58), which is E | z ( t ) | p ≤ k p E | y ( t ) | p in this case, and (32) imply the p th moment exponential stabilityof z ( t ). Let us consider the general case, that is, when | x | >
0. By inequalities (6),(19), (56) and (58), one has E | z ( t ) | p ≤ k p (1 ∨ β ) c c | x | p e − ¯ αt + k p K c c | x | p e − ¯ δ t ≤ ¯ K p | z | p e − ¯ δ t , (59)where ¯ K p = k p c c ((1 ∨ β ) + K ) and K is given by (46). This means that z ( t ) is p thmoment exponentially stable (with Lyapunov exponent no larger than − ¯ δ ).It is noticed that condition (11) imposed on the impulse interval does not explicitlydepend on the growth constant c /c in the exponential stability (20) of subsystem x ( t ). This will be given more specifications in Sections 4 and 5 when 3.1 is applied tothe test problem (Q1) of SDEs. In Theorem 3.1, the continuous dynamics stabilizespart of the system, x ( t ), though the discrete one could destabilize it. But we can alsoestablish a stability criterion for the cases where the discrete dynamics stabilizes x ( t )while the continuous one destabilizes it. This result can be applied to the problemsof impulsive control and stabilization [60, 62]. Theorem 3.2.
Suppose that Assumption 2.1 holds. Let V ∈ C , ( R n × R + ; R + ) and ˜ V ∈ C , ( R q × R + ; R + ) be a pair of candidate Lyapunov functions that satisfycondition (6). Assume there are positive constants α , ˜ α , ˜ α , ˜ β , ˜ β , ˜ β such that L V ( x, t ) ≤ αV ( x, t ) , t ∈ [ t k , t k +1 ) (60)˜ L ˜ V ( x, y, t ) ≤ ˜ α V ( x, t ) + ˜ α ˜ V ( y, t ) , t ∈ [ t k , t k +1 ) (61) E V ( x + ∆( x, k + 1) , t ) ≤ βV ( x, t ) , (62) E ˜ V ( y + ˜∆( x, y, k + 1) , t ) ≤ ˜ β V ( x, t ) + ˜ β ˜ V ( y, t ) (63) for all ( x, y ) ∈ R n × R q and k ∈ N . Let the impulse time sequence { t k } satisfy ∆ t < − ln βα ∧ − ln ˜ β ˜ α ! . (64) Then SiDE (4a-4b) (i.e., (1a-1d)) is p th moment exponentially stable. roof. By Itˆo’s formula and the reasonging as above, one can obtain E V ( x ( t ) , t ) = V ( x ,
0) + Z t E L V ( x ( s ) , s )d s ≤ V ( x ,
0) + Z t α E V ( x ( s ) , s )d s (65)and then, by Gronwall’s inequality, E V ( x ( t ) , t ) ≤ V ( x , e α t (66)on t ∈ [0 , t ), which gives E V ( x ( t − ) , t − ) ≤ V ( x , e α t . Therefore, by (62), at t = t ,one has E V ( x ( t ) , t ) ≤ β E V ( x ( t − ) , t − ) ≤ βV ( x , e α t . (67)But condition (64) means that βe α ∆ t < α > βe ( α +ˆ α ) ∆ t ≤ . (68)This with (67) implies that E V ( x ( t ) , t ) ≤ V ( x , e − ( α +ˆ α ) ∆ t e α t ≤ V ( x , e − ˆ α ∆ t . (69)Similarly, on the interval between any two consecutive impulse times, one can obtain E V ( x ( t ) , t ) ≤ E V ( x ( t k ) , t k ) e α ( t − t k ) ≤ V ( x , e − k ˆ α ∆ t e α ( t − t k ) (70)for all t ∈ [ t k , t k +1 ) and k ∈ N , which implies E V ( x ( t ) , t ) ≤ e ( α +ˆ α )∆ t V ( x , e − ˆ α t (71)for all t ≥
0. This means that the subsystem of x ( t ) is p th moment exponentially sta-ble (with Lyapunov exponent no larger than − ˆ α ). Under the conditions (6), (61), (63)and (71), one can show the p th moment exponential stability of y ( t ) and, therefore,that of z ( t ) as in the proof of 3.1.Furthermore, one can show that, under Assumption 2.1, the p th moment expo-nential stability of SiDE (4a-4b) implies it is also almost surely exponentially stable.The proof is similar to that of [40, Theorem 4.2, p128] and hence is omitted. Theorem 3.3.
Suppose that Assumption 2.1 holds. Then the p th ( p > ) moment ex-ponential stability of SiDE (4a-4b) (i.e., (1a-1d)) implies the almost sure exponentialstability. Exponential stability of CPS
In this section, we formulate a hybrid model in the form of SiDE (1a-1d) to representCPS that is a seamless, fully synergistic integration of a physical system/an SDE andits cyber counterpart/a numerical method. In this systematic framework, we applythe stability theory for SiDEs established in Section 3, specifically, Theorem 3.1 to thetest problem (Q1) for CPS. It should be pointed out that the link between the theoryfor the general class of impulsive systems and the test problem (Q1) is not clear untilwe formulate a hybrid model of SiDE to represent CPS. Moreover, although it appliesto more general class of impulsive systems than the classical theory in the literatureand hence to the test problem (Q1), the stability theory presented in Section 3 isnot prepared for the converse problem (Q2) that arises (only) when a cyber model/anumerical method is involved. So we proceed to study the problem (Q2) for CPSand prove a positive result. In this section, we provide a systems perspective andestablish a foundational theory for CPS.Let us consider a physical system described by the following SDEd x ( t ) = f ( x ( t ))d t + g ( x ( t ))d B ( t ) (72)on t ≥ x (0) = x ∈ R n , where f : R n → R n and g : R n → R n × m satisfy the global Lipschitz condition | f ( x ) − f (¯ x ) | ∨ | g ( x ) − g (¯ x ) | ≤ L | x − ¯ x | (73)for all ( x, ¯ x ) ∈ R n × R n . Given a free parameter θ ∈ [0 , X k +1 = X k + (1 − θ ) f ( X k )∆ t + θf ( X k +1 )∆ t + g ( X k ) √ ∆ t ξ ( k + 1) , k ∈ N (74)with the initial value X = x , where ∆ t > √ ∆ t ξ ( k +1)is the implementation of ∆ B k = B (( k + 1)∆ t ) − B ( k ∆ t ), see [17]. This is a cybersystem/the stochastic theta method for the pysical system/the SDE (72 ). When θ =0, numerical scheme (74) gives the widely-used Euler-Maruyama method. Therefore,the Euler-Maruyama method applied to SDE (72) computes approximations X k ≈ x ( t k ), k ∈ N , where t k = k ∆ t , by setting X = x and forming X k +1 = X k + f ( X k )∆ t + g ( X k ) √ ∆ t ξ ( k + 1) , k ∈ N . (75)Stochastic difference equation (75), also called discrete-time stochastic systems [29],has been intensively studied over the past a few decades. In practical applications, itis natural to form and use some continuous-time extensions of the discrete approxi-mation { X k } such as X ( t ) defined by [20, 43] X ( t ) = ∞ X k =0 X k [ t k ,t k +1 ) ( t ) , t ≥ T is the indicator function of set T . This is a simple step process of theequidistant Euler-Maruyama approximations so its sample paths are right-continuouson [0 , ∞ ). Note that (75), alternatively, (76) is the cyber system/the Euler-Maruyamamethod considered in this work for the pysical system/the SDE (72).Now we formulate a hybrid system of SiDE to represent the CPS that integratesthe cyber subsystem/the Euler-Maruyama method (75) for computations with thepysical subsystem/the SDE (72), and, applying our theory established, address thekey questions (Q1) and (Q2) for the resulting CPS/SiDE. Consider the process y ( t )of difference between the solution x ( t ) to the physical subsystem/the SDE and itscyber counterpart/the numerical solution X ( t ) defined by (76) above y ( t ) = x ( t ) − X ( t ) , t ≥ y (0) = x (0) − X (0) = 0. Notice that x ( t ) is a process of continuouspaths and X ( t ) is a simple step process, which imply that y ( t ) is right-continuous on t ∈ R + and could only have jumps at the sequence of times { t k +1 } k ∈ N . According tothe approximation scheme (75-76), the jump of y ( t ) at t = t k +1 y ( t k +1 ) − y ( t − k +1 ) = x ( t k +1 ) − X ( t k +1 ) − (cid:0) x ( t − k +1 ) − X ( t − k +1 ) (cid:1) = X ( t − k +1 ) − X ( t k +1 ) = − f ( X k )∆ t − g ( X k ) √ ∆ t ξ ( k + 1)= − f ( X ( t − k +1 ))∆ t − g ( X ( t − k +1 )) √ ∆ t ξ ( k + 1)= − f ( x ( t − k +1 ) − y ( t − k +1 ))∆ t − g ( x ( t − k +1 ) − y ( t − k +1 )) √ ∆ t ξ ( k + 1) (78)for all k ∈ N since X ( t ) = x ( t ) − y ( t ) = X k (79)for all t ∈ [ t k , t k +1 ) and k ∈ N .A seamless integration of the solution x ( t ) to physical subsystem/SDE (72) andthe process y ( t ) of difference (77), which is a typical CPS, is described by the followingSiDEd x ( t ) = f ( x ( t ))d t + g ( x ( t ))d B ( t ) (80a)d y ( t ) = f ( x ( t ))d t + g ( x ( t ))d B ( t ) , t ∈ [ t k , t k +1 ) (80b)˜∆( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) := y ( t k +1 ) − y ( t − k +1 )= − f ( x ( t − k +1 ) − y ( t − k +1 ))∆ t − g ( x ( t − k +1 ) − y ( t − k +1 )) √ ∆ t ξ ( k + 1) (80c)for all k ∈ N with initial data x (0) = x ∈ R n and y (0) = x (0) − X (0) = 0 ∈ R n .Clearly, CPS/SiDE (80) is a specific case of (1) where q = n , f ( x, t ) = f ( x ), g ( x, t ) = g ( x ), ˜ f ( x, y, t ) = f ( x ), ˜ g ( x, y, t ) = g ( x ), h f ( x, k ) = 0, h g ( x, k ) = 0,˜ h f ( x, y, k ) = − f ( x − y )∆ t , ˜ h g ( x, y, k ) = − g ( x − y ) √ ∆ t and t k = k ∆ t . Conse-quently, the infinitesimal generators (2) and (3) associated with (80a) and (80b) areof the specific forms L V ( x ) = V x ( x ) f ( x, t ) + 12 trace (cid:2) g T ( x ) V xx ( x ) g ( x ) (cid:3) (81)20nd ˜ L ˜ V ( x, y ) = ˜ V y ( y ) f ( x ) + 12 trace h g T ( x ) ˜ V yy ( y ) g ( x ) i , (82)respectively. It is easy to see that Assumption 2.1 holds since both f and g satisfythe global Lipschitz condition (73).It is stressed that the typical CPS/SiDE (80) represent the synergistic mechanismof physical subsystem x ( t ) and cyber-subsystem X ( t ) = x ( t ) − y ( t ). Usually, in acontrol system, the state X ( t ) = x ( t ) − y ( t ) of cyber subsystem can be synthesizedto control/steer the physical-subsystem and hence the CPS has a general form, forall k ∈ N ,d x ( t ) = ˆ f ( x ( t ) , y ( t ))d t + ˆ g ( x ( t ) , y ( t ))d B ( t ) (83a)d y ( t ) = ˆ f ( x ( t ) , y ( t ))d t + ˆ g ( x ( t ) , y ( t ))d B ( t ) , t ∈ [ t k , t k +1 ) (83b)∆( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) := x ( t k +1 ) − x ( t − k +1 )= h f ( x ( t − k +1 ) , y ( t − k +1 ) , ∆ t ) + h g ( x ( t − k +1 ) , y ( t − k +1 ) , ∆ t ) ξ ( k + 1) (83c)˜∆( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) := y ( t k +1 ) − y ( t − k +1 )= ˆ h f ( x ( t − k +1 ) , y ( t − k +1 ))∆ t + ˆ h g ( x ( t − k +1 ) , y ( t − k +1 )) √ ∆ t ξ ( k + 1) (83d)with initial data x (0) = x ∈ R n and y (0) = x (0) − X (0) = 0 ∈ R n , where ˆ f : R n × R n → R n , ˆ g : R n × R n → R n × m , ˆ h f : R n × R n → R n and ˆ h g : R n × R n → R n × m are continuous functions while h f : R n × R n × R + → R n and h g : R n × R n × R + → R n × m are set for implusive control of the physical-subsystem.We highlight that this systematic representation provides a holistic view for CPSthat comprehends cyber and physical resources in a single unified framework. Thisremoves the principal barrier to developing CPS, see [7, 35]. In our systematicframework, the physical subsystem/the SDE and its cyber counterpart/the numeri-cal method, unlike in the literature where they are linked in some moment sense by(more or less restrictive) inequalities, are coherently integrated in the CPS describedby the hybrid system of SiDE. This typical CPS/SiDE shows how the the cyber sub-system/the numerical method is driven by both the physical subsystem/the SDE andthe sequence { ξ ( k ) } k ∈ N of simulations while the physical subsystem/the SDE is, ofcourse, conducted by itself only. But it has been pointed out above that informationfrom the cyber subsystem/the numerical method could be synthesized to control thephysical subsystem/the SDE and thereby the resulting CPS would be in a generalform.Under some conditions (see [13, 33]), a seminal converse Lyapunov theorem [33,Theorem 5.12, p172] states that, if SDE (72) is p th moment exponentially stable,there is a Lypunov function that proves the exponential stability of the dynamicalsystem. Therefore, the Lyapunov function for physical subsystem/SDE (72) could beused to construct a candidate Lyapunov function for p th moment exponential stabilityof CPS/SiDE (80). Applying Theorem 3.1 to CPS/SiDE (80), one can obtain the fol-lowing result, which ensures that the cyber subsystem/the Euler-Maruyama method21 ( t ) defined by (75-76) shares the p th moment and hence almost sure exponentialstability with physical subsystem/SDE (80a). Theorem 4.1.
Let V ∈ C ( R n ; R + ) be a candidate Lyapunov function for both sub-systems (80a) and (80b, 80c), which satisfies c | x | p ≤ V ( x ) ≤ c | x | p (84) for all x ∈ R n and some positive constants p, c , c . Assume that there are positiveconstants α , ˜ α , ˜ α , ˜ β , ˜ β such that L V ( x ) ≤ − αV ( x ) , t ∈ [ t k , t k +1 ) (85)˜ L V ( x, y ) ≤ ˜ α V ( x ) + ˜ α V ( y ) , t ∈ [ t k , t k +1 ) (86) E V ( y + ˜∆( x, y, k + 1)) ≤ ˜ β V ( x ) + ˜ β V ( y ) (87) for all ( x, y ) ∈ R n × R n and k ∈ N . Let the stepsize ∆ t < − ln ˜ β ˜ α . (88) Then CPS/SiDE (80) is p th moment exponentially stable and hence is also almostsurely exponentially stable. Moreover, cyber subsystem/the Euler-Maruyama method X ( t ) given by (75-76) with stepsize ∆ t shares the p th moment exponential stabilitywith physical subsystem/SDE (72) and hence it is also almost surely exponentiallystable.Proof. That CPS/SiDE (80) is p th moment exponentially stable and is also almostsurely exponentially stable is just a result from Theorems 3.1 and 3.3. Notice that(77), that is, X ( t ) = x ( t ) − y ( t ) for all t ≥ | X ( t ) | ≤ | x ( t ) | + | y ( t ) | and hence | X ( t ) | p ≤ k p ( | x ( t ) | p + | y ( t ) | p )where k p = 1 when 0 < p < k p = 2 p − when p ≥
1. Recall that X (0) = x (0) = x and y (0) = x (0) − X (0) = 0. If x = 0, by (19), (32) and (84), one has E V ( x ( t )) = 0, E ˜ V ( y ( t )) = 0 and hence E | X ( t ) | p = 0 for all t ≥
0. In the general casewhen | x | >
0, similar to the Step 5 in the proof for Theorem 3.1, one can derive E | X ( t ) | p ≤ ¯ K p | x | p e − ¯ δ t , (89)where ¯ K p is given by (59). That is, the continuous-time approximation { X ( t ) } t ≥ defined by (76) is p th moment exponentially stable. Moreover, since | X ( t ) | ≤ | x ( t ) | + | y ( t ) | ≤ | z ( t ) | z ( t ) = (cid:2) x T ( t ) y T ( t ) (cid:3) T , the almost sure exponential stability of CPS/SiDE(80a-80c) implies thatlim sup t →∞ t ln | X ( t ) | ≤ lim sup t →∞ t ln | z ( t ) | < . Therefore, the continuous-time approximation { X ( t ) } t ≥ defined by (75-76) is alsoalmost surely exponentially stable.This means that, if physical subsystem/SDE (72) is p th moment exponentiallystable, CPS/SiDE (80) and hence its cyber counterpart/numerical solution X ( t )generated by the widely-used Euler-Maruyama method reproduce the p th momentexponential stability of the physical subsystem/the SDE when the conditions in The-orem 4.1 hold. The ability of a cyber system/numerical simulations to reproduce themean-square exponential stability of the physical system/the SDE has been studiedin [21], where, under some conditions, the mean-square exponential stability of thephysical system/the SDE and that of its cyber counterpart/a numerical method (forsufficiently small step sizes) are shown to be equivalent. In this work, we investigateon the ability of the cyber subsystem/numerical simulations to reproduce the mean-square exponential stability of the physical subsystem/the SDE (72) in our proposedframework of CPS/SiDE (80). A result on on mean-square exponential stability isthen derived from Theorem 4.1 as follows, in which the Lyapunov function for mean-square exponential stability of physical subsystem/SDE (72) also guarantees that ofits cyber counterpart/itsnumerical solution (75-76) and thereby that of the resultingCPS/SiDE (80). Theorem 4.2.
Let the candidate Lyapunov function V ∈ C ( R n ; R + ) for physicalsystem/SDE (72) be a quadratic function V ( x ) = x T P x, (90) where P ∈ R n × n is a positive definite matrix. Assume that there exist positive con-stants ¯ α and ∆ t with ¯ α ∆ t < such that L V ( x ) + ∆ t V ( f ( x )) ≤ − ¯ αV ( x ) (91) for all x ∈ R n . Then CPS/SiDE (80) with ∆ t ∈ (0 , ∆ t ] is mean-square exponentiallystable and hence is also almost surely exponentially stable. Moreover, the cyber sub-system/numerical solution X ( t ) given by (75-76) with stepsize ∆ t ∈ (0 , ∆ t ] shares themean-square exponential stability with physical subsystem/the SDE (72) and hence itis also almost surely exponentially stable.Proof. It will follow the conclusion from Theorem 4.1 if one shows that conditions(84)-(88) in Theorem 4.1 are satisfied with p = 2 for CPS/SiDE (80).Since (90) gives λ m ( P ) | x | ≤ V ( x ) ≤ λ M ( P ) | x | , condition (84) holds with posi-tive constants p = 2 , c = λ m ( P ) , c = λ M ( P ). Moreover, inequalities (85) and (91)23re equivalent. Obviously, inequality (91) implies that (85) holds with positive con-stant α ≥ ¯ α . But inequality (85) implies that there is a sufficiently small positivenumber ∆ t such that (91) holds with some positive constant ¯ α < α ∧ (1 / ∆ t ). By [41,Theorem 4.4, p130], one has E | x ( t ) | ≤ λ M ( P ) λ m ( P ) | x | e − ¯ α t (92)for all t ≥
0. That is, stochastic system (72) is mean-square exponentially stable andhence, by [41, Theorem 4.2, p128], it is also almost surely exponentially stable.On the other hand, using Itˆo’s lemma, [27, Lemmas 1 and 2] and global Lipschitzcondition (73), one has˜ L V ( x, y ) = 2 y T P f ( x ) + trace (cid:2) g T ( x ) P g ( x ) (cid:3) ≤ ˜ α y T P y + ˜ α − f T ( x ) P f ( x ) + λ M ( P ) trace (cid:2) g T ( x ) g ( x ) (cid:3) ≤ ˜ α − λ M ( P ) | f ( x ) | + λ M ( P ) L | x | + ˜ α ˜ V ( y ) ≤ ( ˜ α − + 1) λ M ( P ) L | x | + ˜ α ˜ V ( y ) ≤ ˜ α x T P x + ˜ α ˜ V ( y ) = ˜ α V ( x ) + ˜ α ˜ V ( y ) , (93)where ˜ α = (1 + ˜ α ) λ M ( P ) L ˜ α λ m ( P )and ˜ α given by (101) below are both positive numbers. So condition (86) in Theorem4.1 is satisfied.Note that, given any ∆ t ∈ (0 , ∆ t ], inequality (91) implies L V ( x ) + ∆ t V ( f ( x )) ≤ − ¯ αV ( x ) , (94)and (80c) gives y + ˜∆( x, y, k + 1) = y − f ( x − y ) ∆ t − g ( x − y ) √ ∆ t ξ ( k + 1)= x − ( x − y ) − f ( x − y ) ∆ t − g ( x − y ) √ ∆ t ξ ( k + 1) . (95)Using inequality (94), one obtains E V ( y + ˜∆( x, y, k + 1))= x T P x − x T P ( x − y ) + ( x − y ) T P ( x − y ) − tx T P f ( x − y )+ ∆ t (cid:2) x − y ) T P f ( x − y )+ trace [ g T ( x − y ) P g ( x − y )] + ∆ t f T ( x − y ) P f ( x − y ) (cid:3) ≤ (1 + c − ) x T P x + (1 + c )( x − y ) T P ( x − y ) − tx T P f ( x − y )+ ∆ t (cid:2) L V ( x − y ) + ∆ t V ( f ( x − y )) (cid:3) ≤ (1 + c − ) V ( x ) + (1 + c )( x − y ) T P ( x − y ) − tx T P f ( x − y ) − ¯ α ∆ t ( x − y ) T P ( x − y ) ≤ (1 + c − ) V ( x ) + (1 + c − ¯ α ∆ t ) ( x − y ) T P ( x − y ) − tx T P f ( x − y ) (96)24or all ( x, y ) ∈ R n × R n and k ∈ N , where c > c − ¯ α ∆ t < x − y ) T P ( x − y ) ≤ x T P x − x T P y + y T P y ≤ (1 + b − ) x T P x + (1 + b ) y T P y, (97) − x T P f ( x − y ) ≤ b − x T P x + bf ( x − y ) T P f ( x − y ) ≤ b − x T P x + bλ M ( P ) L ( x − y ) T ( x − y ) ≤ (cid:0) b − + (1 + b ) λ M ( P ) L λ m ( P ) (cid:1) x T P x + b (1 + b ) λ M ( P ) L λ m ( P ) y T P y (98)where b is a positive constant sufficiently small for˜ β := (1 + c − ¯ α ∆ t )(1 + b ) + ∆ t b (1 + b ) λ M ( P ) L λ m ( P ) < . (99)Substitution of (97) and (98) into (96) yields E V ( y + ˜∆( x, y, k + 1)) ≤ ˜ β V ( x ) + ˜ β V ( y ) , (100)where ˜ β = (1 + c − ) + (1 + c − ¯ α ∆ t )(1 + b − ) + ∆ t (cid:18) b − + (1 + b ) λ M ( P ) L λ m ( P ) (cid:19) and ˜ β given by (99) above are both positive constants. This is the condition (87) inTheorem 4.1.Let ˜ α be a positive number such that˜ α < − ln ˜ β ∆ t . (101)For instance, let ˜ α = − ln ˜ β / (2 ∆ t ). Then∆ t ≤ ∆ t = − ln ˜ β α < − ln ˜ β ˜ α , which means that condition (88) in Theorem 4.1 is also satisfied.According to Theorem 4.1, it follows the conclusion.In order to ensure that the cyber system/the numerical method shares the ex-ponential stability with the physical system/the SDE, the choice of stepsizes ∆ t isexplicitly and heavily limited by both the growth and the rate constants of the phys-ical system/the SDE in the literature [21, 42]. Although the both are related, it isonly the rate constant that plays a key role in the definitions of exponential stability.It makes sense to lessen the dependence of stepsizes ∆ t on the growth constant, which25tself includes conservativeness from condition (84). In Theorem 4.2, we manage to re-move the explicit dependence of stepsizes ∆ t on the growth constant λ M ( P ) /λ m ( P ).Instead, we show that the influence of the growth constant λ M ( P ) /λ m ( P ) on thechoice of stepsizes ∆ t is through the rate-like constant ˜ β by equation (99). Thiscould reduce much the restriction introduced by the growth constant. As is shown inSection 5 below, it significantly improves the upper bound ∆ t of stepsizes and easeits computation in linear systems.We have applied Theorem 3.1 and proved the posive results Theorems 4.1-4.2to the test problem (Q1) for CPS. Recall that, in Theorem 3.1, SiDE (4a-4b) (i.e.,(1a-1d)) is a general class of impulsive systems that is appropriately constructed (forexpression of integrated dynamics of the exact and numerical solutions of SDEs),where the continuous dynamics stabilizes one part x ( t ) of the system and the discreteone the other part y ( t ) so that the whole system z ( t ) = [ x T ( t ) y T ( t )] T is p th momentexponentially stable. Since this implies the exponential stabililty of the differenceprocess | x ( t ) − y ( t ) | p ≤ p | z ( t ) | p , the fundamental Theorem 3.1 applies to the testproblem (Q1) for stochastic dynamical systems and yields positive results. However,the stability theory presented in Section 3 would not consider the problem (Q2) dueto the fact that a cyber model/a numerical method is not defined/involved in SiDE(4a-4b) and hence problem (Q2) is irrelevant. But the key question (Q2) naturallyarises when we express CPS in the form of SiDEs that, clearly, is composed of thecyber/the numerical and the physical parts/the exact solutions. This shows that theresults established in this work are not only more general than but also substantiallydifferent from the classical theory for impulsive systems in the literature.Let us proceed to study the converse problem (Q2), namely, whether one caninfer that the physical system/the SDE (72) is mean-square exponentially stable if itscyber counterpart/its numerical solution (75-76) is mean-square exponentially stablefor some small stepsize ∆ t . Similarly, the converse Lyapunov theorem (see [33, 56])gives that, if discrete-time stochastic system (75) is mean-square exponentially stable,there is a Lyapunov function that proves the exponential stability of the dynamicalsystem. One can apply Theorem 4.2 and find that mean-square exponential stabilityof physical system/SDE (72) can be inferred from that of its cyber counterpart/itsnumerical simulation (75) if the Lyapunov function is in a quadratic form. Theorem 4.3.
Assume that there is a candidate Lyapunov function V ∈ C ( R n ; R + ) of the form (90) for cyber subsystem/numerical method (75) with stepsizes ∆ t = ∆ t > such that E (cid:2) V ( X k +1 ) (cid:12)(cid:12) X k (cid:3) ≤ ¯ c V ( X k ) (102) for some positive constant ¯ c < and all X k ∈ R n . Then CPS/SiDE (80) with ∆ t ∈ (0 , ∆ t ] is mean-square exponentially stable and hence is also almost surely ex-ponentially stable, which implies that physical subsystem/SDE (72) is mean-squareexponentially stable and hence is also almost surely exponentially stable.Proof. By Lyapunov stability theory (see [33, 56]), conditions (90) and (102) imme-diately imply that discrete-time stochastic system (72) with ∆ t = ∆ t is mean-square26xponentially stable. Let function V ( x ) given by (90) be also the candidate Lyapunovfunction for physical system/SDE (72). But condition (102) E (cid:2) V ( X k +1 ) (cid:12)(cid:12) X k (cid:3) = E (cid:2) X Tk +1 P X Tk +1 (cid:12)(cid:12) X k (cid:3) = E h(cid:0) X k + f ( X k )∆ t + g ( X k ) p ∆ t ξ ( k + 1) (cid:1) T P · (cid:0) X k + f ( X k )∆ t + g ( X k ) p ∆ t ξ ( k + 1) (cid:1)(cid:12)(cid:12) X k i = V ( X k ) + ∆ t h X Tk P f ( X k ) + f T ( X k ) P X k + trace (cid:2) g T ( X k ) P g ( X k ) (cid:3) + ∆ tf T ( X k ) P f ( X k ) i ≤ ¯ c V ( X k )yields X Tk P f ( X k ) + f T ( X k ) P X k + trace (cid:2) g T ( X k ) P g ( X k ) (cid:3) + ∆ tf T ( X k ) P f ( X k ) ≤ − ¯ αV ( X k ) (103)for all X k ∈ R n , where ¯ α ∆ t = 1 − ¯ c . This gives L V ( x ) + ∆ t V ( f ( x )) = x T P f ( x ) + f T ( x ) P x + trace (cid:2) g T ( x ) P g ( x ) (cid:3) + ∆ tf T ( x ) P f ( x ) ≤ − ¯ αV ( x )for all x ∈ R n , that is, condition (91) in Theorem 4.2. According to Theorem 4.2,CPS/SiDE (80) with ∆ t ∈ (0 , ∆ t ] is mean-square exponentially stable and hence isalso almost surely exponentially stable, which implies that physical subsystem/theSDE (72) is mean-square exponentially stable and hence is also almost surely expo-nentially stable. Let us consider linear systemsd x ( t ) = F x ( t )d t + m X j =1 G j x ( t )d B j ( t ) (104)on t ≥ x (0) = x ∈ R n , where F ∈ R n × n , G j ∈ R n × n , j =1 , , · · · , m , are constant matrices. Obviously, the linear pysical system/SDE (104)satisfies the global Lipschitz continuous condition and has a unique solution x ( t )on t ∈ [0 , ∞ ). It is well known that linear stochastic system (104) is mean-squareexponentially stable (or, equivalently, asymptotically stable) if and only if there existsa positive definite matrix P ∈ R n × n such that [4, 10] F T P + P F + m X j =1 G Tj P G j < . (105)27his is the well-known Lyapunov-Itˆo inequality [4, 10], the linear matrix inequality(LMI) [11] equivalent of the classic Lyapunov-Itˆo equation [39]. By [40, Theorem 4.2,p128], the mean-square exponential stability of physical system/SDE (104) impliesthat it is also almost surely exponentially stable.According to numerical scheme (76), one can compute the continuous-time ap-proximation X ( t ) of the solution x ( t ) and X k ≈ x ( t k ) at t k = k ∆ t , k ∈ N , by theEuler-Maruyama method X k +1 = X k + F X k ∆ t + m X j =1 G j X k √ ∆ t ξ j ( k + 1) , k ∈ N (106)with X = x , where ∆ t > √ ∆ t ξ j ( k + 1) is theimplementation of ∆ B j,k = B j (( k + 1)∆ t ) − B j ( k ∆ t ). As is also well known, lineardiscrete-time stochastic system (106) is mean-square exponentially stable if and onlyif there exists a positive definite matrix P ∈ R n × n such that [4]( I + ∆ t F ) T P ( I + ∆ t F ) + ∆ t m X j =1 G Tj P G j < P. (107)Let y ( t ) be the process of difference between x ( t ) and X ( t ) defined by (77) above.The integrated dynamics of x ( t ) and y ( t ) are described by a liner SiDE that is atypical CPSd x ( t ) = F x ( t )d t + m X j =1 G j x ( t )d B j ( t ) (108a)d y ( t ) = F x ( t )d t + m X j =1 G j x ( t )d B j ( t ) , t ∈ [ t k , t k +1 ) (108b)˜∆( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) := y ( t k +1 ) − y ( t − k +1 )= − F (cid:0) x ( t − k +1 ) − y ( t − k +1 ) (cid:1) ∆ t − m X j =1 G j (cid:0) x ( t − k +1 ) − y ( t − k +1 ) (cid:1) √ ∆ t ξ j ( k + 1) (108c)with initial data x (0) = x ∈ R n and y (0) = x (0) − X (0) = 0 ∈ R n , where t k = k ∆ t and k ∈ N . Obviously, this linear CPS/SiDE (108) satisfies the global Lipschitzcondition (73).Our theory immediately provides positive results to the key questions (Q1) and(Q2), which presents the upper bound ∆ t of stepsizes to the test problem (Q1). Theorem 5.1.
The following are equivalent.( A ) If there exists a positive definite matrix P ∈ R n × n such that cyber-physical/numericalLyapunov inequality F T P + P F + m X j =1 G Tj P G j + ∆ tF T P F < olds for some positive number ∆ t .( B ) Physical system/SDE (104) is mean-square exponentially stable.( C ) CPS/SiDE (108) with ∆ t ∈ (0 , ∆ t ] is mean-square exponentially stable.( D ) Cyber system/numerial method (106) with ∆ t ∈ (0 , ∆ t ] is mean-square expo-nentially stable.That is, ( A ) ⇔ ( B ) ⇔ ( C ) ⇔ ( D ) . Proof. ( A ) ⇔ ( B ). We only need to show that the classic Lyapunov inequality (105)and the cyber-phyisical/numerical Lyapunov inequality (109) are equivalent, whichis implied by the equivalence of the inequalities (85) and (91) shown in the proofof Theorem 4.2. Alternatively, it can be easily proved in the same way as follows.Clearly, (109) implies (105). But inequality (105) implies that there is a sufficientlysmall positive number ∆ t such that (109) holds. Therefore, LMI (105) ⇔ LMI (109).( A ) ⇒ ( C ) & ( D ). Let us consider the quadratic Lyapunov function (90), V ( x ) = x T P x, for linear system (104). LMI (109) implies there is a positve number ¯ α < / ∆ t sufficiently small for F T P + P F + m X j =1 G Tj P G j + ∆ tF T P F ≤ − ¯ αP, (110)which means that condition (91) in Theorem 4.2 holds. By Theorem 4.2, CPS/SiDE(108) and hence cyber subsystem/numerial method (106) with ∆ t ∈ (0 , ∆ t ] are mean-square exponentially stable.( C ) ⇒ ( B ). Notice that subsystem (108a) is self-conducted, which is not affectedby the other parts of CPS/SiDE (108). Therefore, mean-square exponential stabilityof CPS/SiDE (108) implies that the subsystem (108a) itself is mean-square exponen-tially stable.( D ) ⇒ ( C ) & ( B ). Let ∆ t = ∆ t . Due to mean-square exponential stability ofcyber subsystem/numerial method (106), there is positive definite matrix P ∈ R n × n such that Lyapunov inequality (107) with ∆ t = ∆ t > c < − ¯ c sufficiently small for( I + ∆ t F ) T P ( I + ∆ t F ) + ∆ t m X j =1 G Tj P G j ≤ ¯ c P. (111)Let the quadratic function V ( x ) = x T P x
29e the candidate Lyapunov function for discrete-time stochastic system (106) with∆ t = ∆ t . It is observed that condition (102) in Theorem 4.3 is implied by inequal-ity (111). By Theorem 4.3, CPS/SiDE (80) with ∆ t ∈ (0 , ∆ t ] and hence physicalsubsystem/SDE (72) are mean-square exponentially stable.The proof is complete.It is easy to obtain the upper bound ∆ t of stepsizes for the ability of the cy-ber system/numerical method to reproduce the exponential stability of the linearphysical system/SDE by solving the cyber-physical/numerical Lyapunov inequality(109). Particularly, let us consider a scalar SDE, which is the linear SDE (104) with n = m = 1, d x ( t ) = λx ( t )d t + µx ( t )d B ( t ) , t ≥ , x (0) = x = 0 (112)where λ and µ are both constants. The cyber-physical/numerical Lyapunov inequality(109) immediately gives2 λ + µ + λ ∆ t < ⇔ ∆ t < − (2 λ + µ ) λ . (113)According to Theorem 5.1, this is the necessary and sufficent condition for mean-square exponential stability of the linear scalar physical system/SDE, the cybersystem/Euler-Maruyama method, and the linear CPS/SiDE (108) with n = m = 1, F = λ and G = µ . It is observed that inequality (113) is exactly the inequality(4.3) in [17] with θ = 0 for the Euler-Maruyama method. Note that the inequality(113) in [17] is the very special sclar case of our result (109) that applies to generalmulti-dimensional linear systems. This paper has established a systematic framework for numerical study of SDEs,in which a hybrid system of SiDE has been formulated to describe the integrateddynamics of the exact and numerical solutions of an SDE, unlike in the literaturewhere they are linked in some moment sense by inequalities. The proposed system-atic representation reveals the intrinsic relationship between the exact and numericalsolutions of the SDE. This provides a holistic view of the continuous-time systemsin nature/practice and their discrete-time models in computers. As a foundation ofthe proposed framework, this paper has presented a Lyapunov stability theory fora general class of SiDEs that can be used to represent a seamless, fully synergisticintegration of the exact and numerical solutions of SDEs. Applying the establishedtheory, it studied the test problem (Q1) and the converse problem (Q2) of an SDEfor the widely-used Euler-Maruyama method. As was shown in above, it significantly30mproved the upper bound of stepsizes and facilitated its computation, e.g., in linearSDEs.Consequently, our proposed framework has provided a novel approach to con-vergence analysis of numerical methods for SDEs as well, where the implimenta-tion √ ∆ t ξ ( k + 1) of SiDE (80) should be replaced with the increment ∆ B k = B (( k + 1)∆ t ) − B ( k ∆ t ), that is,d x ( t ) = f ( x ( t ))d t + g ( x ( t ))d B ( t ) (114a)d y ( t ) = f ( x ( t ))d t + g ( x ( t ))d B ( t ) , t ∈ [ t k , t k +1 ) (114b)˜∆( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) := y ( t k +1 ) − y ( t − k +1 )= − f ( x ( t − k +1 ) − y ( t − k +1 ))∆ t − g ( x ( t − k +1 ) − y ( t − k +1 ))∆ B k (114c)with x (0) = x and y (0) = 0, where t k = k ∆ t and k ∈ N . According to Lemma2.1, SiDE (114) has a unique (right-continuous) solution z ( t ) = [ x T ( t ) y T ( t )] T , whichbelongs to M ([0 , T ]; R n + q ) for all T ≥
0. In particular, [40, Lemma 3.2, p51] gives E (cid:20) sup ≤ t ≤ T | x ( t ) | (cid:21) ≤ (1 + 3 | x | ) e LT ( T +4) =: C T . (115)On every interval [ t k , t k +1 ], k ∈ N , one has y ( t k +1 ) − y ( t k ) = Z t − k +1 t k f ( x ( t ))d t + Z t − k +1 t k g ( x ( t ))d B ( t ) − f ( x ( t − k +1 ) − y ( t − k +1 ))∆ t − g ( x ( t − k +1 ) − y ( t − k +1 ))∆ B k = Z t − k +1 t k (cid:2) f ( x ( t )) − f ( x ( t k ) − y ( t k )) (cid:3) d t + Z t − k +1 t k (cid:2) g ( x ( t )) − g ( x ( t k ) − y ( t k )) (cid:3) d B ( t )and hence, due to y (0) = 0, y ( t k +1 ) = Z t k +1 (cid:2) f ( x ( t )) − f ( x ([ t ]) − y ([ t ])) (cid:3) d t + Z t k +1 (cid:2) g ( x ( t )) − g ( x ([ t ]) − y ([ t ])) (cid:3) d B ( t ) , (116)where [ t ] = sup { t j : t j ≤ t, j ∈ N } for t ≥
0. By Cauchy-Schwaz inequality, this yields | y ( t k +1 ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z t k +1 (cid:2) f ( x ( t )) − f ( x ([ t ]) − y ([ t ])) (cid:3) d t + Z t k +1 (cid:2) g ( x ( t )) − g ( x ([ t ]) − y ([ t ])) (cid:3) d B ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20) t k +1 Z t k +1 (cid:12)(cid:12) f ( x ( t )) − f ( x ([ t ]) − y ([ t ])) (cid:12)(cid:12) d t + (cid:12)(cid:12)(cid:12)(cid:12)Z t k +1 (cid:2) g ( x ( t )) − g ( x ([ t ]) − y ([ t ])) (cid:3) d B ( t ) (cid:12)(cid:12)(cid:12)(cid:12) (117)31o, from Itˆo’s isometry and the global Lipschitz condition (73), one obtains E | y ( t k +1 ) | ≤ (cid:20) t k +1 E Z t k +1 (cid:12)(cid:12) f ( x ( t )) − f ( x ([ t ]) − y ([ t ])) (cid:12)(cid:12) d t + E Z t k +1 (cid:12)(cid:12) g ( x ( t )) − g ( x ([ t ]) − y ([ t ])) (cid:12)(cid:12) d t (cid:21) ≤ L ( t k +1 + 1) E Z t k +1 (cid:12)(cid:12) x ( t ) − x ([ t ]) + y ([ t ]) (cid:12)(cid:12) d t. (118)Notice that (114a) and (114b) imply x ( t ) − x ([ t ]) = y ( t ) − y ([ t ]) for all t ≥
0. Substi-tuting this into (118) gives E | y ( t k +1 ) | ≤ L ( t k +1 + 1) E Z t k +1 | y ( t ) | d t. (119)Given any T ≥
0, using Itˆo’s lemma, (73), (82) and (119), one has E | y ( T ) | = E | y ([ T ]) | + E Z T [ T ] (cid:2) y T ( s ) f ( x ( s )) + | g ( x ( s ) | (cid:3) d s ≤ L ([ T ] + 1) E Z [ T ]0 | y ( s ) | d s + E Z T [ T ] | y ( s ) | d s + E Z T [ T ] (cid:2) | f ( x ( s )) | + | g ( x ( s ) | (cid:3) d s ≤ K T E Z T | y ( s ) | d s + 2 L E Z T [ T ] | x ( s ) | d s, (120)where constant K T = 2 L ([ T ] + 1) ∨
1, which with (115) implies E h sup ≤ t ≤ T | y ( t ) | i ≤ L Z ∆ t E h sup ≤ t j ≤ [ T ] | x ( t j + s ) | i d s + K T Z T E h sup ≤ s ≤ t | y ( s ) | i d t ≤ C T L ∆ t + K T Z T E h sup ≤ s ≤ t | y ( s ) | i d t. (121)In view of the Gronwall inequality ([34, Lemma 4.5.1, p129], [40, Theorem 8.1, p45]),this yields E (cid:20) sup ≤ t ≤ T | y ( t ) | (cid:21) ≤ C T L e K T T ∆ t. (122)Using the dyanimcs of the discretization error y ( t ), we have shown this classicalfinite-time convergence result E h sup ≤ t ≤ T | y ( t ) | i = O (∆ t ) for the Euler-Maruyamamethod by an approach that is different from those found in the literature [20, 34, 40].32nitiated by this systematic framework, there are many open and challengingproblems to be investigated in numerical study of dynamical systems. For example,it would be interesting to extend this systematic framework and theory to some moregeneral classes of SDEs like those with time delays, switching, and/or nonglobalLipschitz continuous coefficients as well as to apply the framework to other (explicitor implicit) numerical schemes [48, 50] such as some variants of the Euler-Maruyamamethod [30, 43]. Our proposed hybrid system of SiDEs is exactly a CPS, which is composed of a phys-ical subsystem and its cyber counterpart, described by the SDE and the nuericalmethod, respectively, and whose behaviour is defined by both the physical and cyberparts of the system. Our systematic representation and theory has established a the-oretic foundation for CPS that comprehends cyber and physical resources in a singleunified framework. This provides a systems perspective and removes the principalbarrier to developing CPS, see [6, 7, 35], which initiates the study of a new systemsscience for CPS and arouses lots of open and challenging problems. Virtually allmodern control systems are implemented using digital computers. It has been shownthat the cyber-physical/numerical Lyapunov inequality (109) is the necessary andsufficient condition for mean-square exponential stability of linear CPS/SiDE (108).But, as a control system (see Section 4), the physical subsystem admits control input,see, e.g., [10, 24],d x ( t ) = (cid:16) F x ( t ) + ˆ Du ( t ) (cid:17) d t + m X j =1 (cid:16) G j x ( t ) + ˆ D j u ( t ) (cid:17) d B j ( t ) , (123)where D ∈ R n × r , D j ∈ R n × r , j = 1 , , · · · , m , are constant matrices and u : R + → R r is the control inut. When the uncontrolled system (123) with u ( t ) ≡ X ( t ) from the cyber-part, say,letting u ( t ) = u ( x ( t ) , X ( t )) = ˆ K ( t ) X ( t ) = ˆ K ( t ) ( x ( t ) − y ( t )) = ˆ K ( t ) x ( t ) − ˆ K ( t ) y ( t ) (124)where ˆ K : R + → R r × n is the cyber-physical feedback gain matrix and may use someinformation of the pair x ( t ) and X ( t ) (or say, of x ( t ) and y ( t )), to stabilize the physicalsubsystem and hence the whole system for some sufficiently small stepsize? In this33ase, the resulting CPS is a specific form of (83) as followsd x ( t ) = (cid:16)(cid:0) F + ˆ D ˆ K ( t ) (cid:1) x ( t ) − ˆ D ˆ K ( t ) y ( t ) (cid:17) d t + m X j =1 (cid:16)(cid:0) G j + ˆ D j ˆ K ( t ) (cid:1) x ( t ) − ˆ D j ˆ Ky ( t ) (cid:17) d B j ( t ) (125a)d y ( t ) = (cid:16)(cid:0) F + ˆ D ˆ K ( t ) (cid:1) x ( t ) − ˆ D ˆ Ky ( t ) (cid:17) d t + m X j =1 (cid:16)(cid:0) G j + ˆ D j ˆ K ( t ) (cid:1) x ( t ) − ˆ D j ˆ K ( t ) y ( t ) (cid:17) d B j ( t ) , t ∈ [ k ∆ t, ( k + 1)∆ t )(125b)˜∆( x ( t − k +1 ) , y ( t − k +1 ) , k + 1) := y ( t k +1 ) − y ( t − k +1 )= − (cid:0) F ∆ t + ˆ D ˆ K C (∆ t ) (cid:1)(cid:0) x ( t − k +1 ) − y ( t − k +1 ) (cid:1) − m X j =1 (cid:0) G j √ ∆ t + ˆ D j ˆ K C (∆ t ) (cid:1)(cid:0) x ( t − k +1 ) − y ( t − k +1 ) (cid:1) ξ j ( k + 1) , (125c)where ˆ K C ∈ R r × n is the cyber feedback gain matrix that is usually a function of∆ t . Notice that this works in a simple case. Let us consider a scalar control systemdescribed by an ordinary differential equation (ODE)˙ x ( t ) = ax ( t ) + u ( t ) (126)with initial value x (0) = x ∈ R \{ } , constant a > u : R + → R .The uncontrolled ODE (126) with u ( t ) ≡ x ( t ) = x e at and thereby | x ( t ) | → ∞ as t → ∞ . The classical control theory givesthat, if the state x ( t ) of the system is observed, a state-feedback controller u ( t ) = − kx ( t ) with constant k > a can be synthesized so that the closed-loop system isexponentially stable. It is possible to design a cyber system, instead of the state-feedback control law (in case that the state observer is unavailable), that generatescontrol input to stabilize the physical system (126). A cyber-physical controller u ( t ) = − k p X ( t ) (127)leads to CPS ˙ x ( t ) = ax ( t ) − k p X ( t ) , (128)where k p > a is a constant and X ( t ) = X k , for all t ∈ [ k ∆ t, ( k + 1)∆ t ) and k ∈ N , isthe state of a cyber system X k +1 = (cid:16) k p a − k p − aa e a ∆ t (cid:17) X k , k ∈ N (129)with X = x . Let stepsize ∆ t < a ln k p k p − a . x ( t ) has the same sign as x and | x ( t ) | ≤ | x | on [0 , ∆ t ]. Moreover, {| X k |} isdecreasing and every X k has the same sign as X = x . In fact, the cyber system(129) is expoentially stable. Let v ( t ) = v ( x ( t )) = | x ( t ) | . It is easy to observe thatd v ( t )d t = 2 a | x ( t ) | − k p | x ( t ) | · | X | = 2 a | x ( t ) | − k p | x ( t ) | · | x | ≤ − k p − a ) v ( t )for t ∈ [0 , ∆ t ), which implies v ( t ) ≤ v (0) e − k p − a ) t on [0 , ∆ t ] and hence | x (∆ t ) | ≤ | x | e − ( k p − a )∆ t . Actually, it is found x (∆ t ) = X and, from Taylor’s theorem, | X | ≤ | x | e − ( k p − a )∆ t .By induction, one can show thatd v ( t )d t ≤ − k p − a ) v ( t ) , ∀ t ≥ . (130)Therefore, CPS (128) is exponentially stable. Let y ( t ) = x ( t ) − X ( t ), then the CPS(128) can be expressed by impulsive systems in the form of (83), that is,˙ x ( t ) = − ( k p − a ) x ( t ) + k p y ( t ) (131a)˙ y ( t ) = − ( k p − a ) x ( t ) + k p y ( t ) , t ∈ [ k ∆ t, ( k + 1)∆ t ) (131b) y ( t k +1 ) − y ( t − k +1 ) = k p − aa (cid:0) e a ∆ t − (cid:1)(cid:0) x ( t − k +1 ) − y ( t − k +1 ) (cid:1) (131c)with x (0) = x and y (0) = 0. Alternatively, CPS (128) can be described as a linearimpulsive system z ( t ) = [ x T ( t ) X T ( t )] T of the compact form˙ z ( t ) = (cid:20) a − k p (cid:21) z ( t ) , t ∈ [ k ∆ t, ( k + 1)∆ t ) (132a) z ( t k +1 ) − z ( t − k +1 ) = (cid:20) − e a ∆ t )( k p − a ) /a (cid:21) z ( t − k +1 ) (132b)with z (0) = [ x T x T ] T . It is easy to check that neither CPS (131) nor (132) satisfiesthe necessary and sufficient condition for asymptotic stability of linear impulsivesystems (with any initial values) in the literature [49, Theorem 11, p57], [61, Theorem2.1.2, p19]. But it has been shown above that CPS (128) is carefully designed to beexponentially stable by making use of information X (0) = x (0) = x . This helpshighlight not only the increasing role of computing in control systems but also that thestudy of systems science for CPS initiated in this work is substantially different fromthe classical theory of impulsive systems in the literature. The study of information-based impulsive systems including (83) and (125) is among the future works of thesystems science for CPS. 35 cknowledgement The author was supported in part by the National Natural Science Foundation ofChina (No.61877012) and the starter grant from Guangdong University of Technology.The author gratefully acknowledges Prof. Xuerong Mao (the author’s PhD supervisorat University of Strathclyde, UK) for his comments and suggestions, which helpimprove the quality of this work. The author did some of this work during hisvisit to The Centre for Stochastic & Scientific Computations at Harbin Institute ofTechnology and would like to thank Prof. Minghui Song, Prof. Mingzhu Liu andtheir group for helpful discusssions.
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