Stability of Discontinuous Retarded Functional ... - Semantic Scholar
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Stability of Discontinuous Retarded Functional Differential Equations With Applications Ye Sun, Anthony N. Michel, Life Fellow, IEEE, and Guisheng Zhai, Senior Member, IEEE
Abstract—We present Lyapunov stability results, including converse theorems, for a class of discontinuous dynamical systems (DDS) determined by the solutions of retarded functional differential equations. We demonstrate the applicability of these results in the analysis of several specific important classes of DDS determined by functional differential equations and differential-difference equations. Index Terms—Asymptotic stability, differential-difference equations, discontinuous dynamical systems (DDS), exponential stability, hybrid systems, Lyapunov stability, retarded functional differential equations, switched systems, systems with delays.
I. INTRODUCTION
A
dynamical system is a four-tuple where denotes time set, is the state–space (a metric space with metric ), is the set of initial states and denotes a family of motions. When , we speak of a continuous-time dynamical system and when we speak of a discrete-time dynamical system. (For any motion , we have and for all , where may be finite or infinite. The set of motions is obtained by over .) When is a finite-dimensional varying normed linear space, we speak of finite-dimensional dynamical systems, and otherwise, of infinite-dimensional dynamical systems. Also, when all motions in a continuous-time dynamical system are continuous with respect to (relative to the metric for ), we speak of a continuous dynamical system and when one or more of the motions are not continuous with respect to , we speak of a discontinuous dynamical system (DDS). Finite-dimensional dynamical systems may be determined, e.g., by the solutions of ordinary differential equations, ordinary differential inequalities, difference equations, difference inequalities, and the like, while infinite-dimensional dynamical systems may be determined, e.g., by the solutions of differential-difference equations, functional differential equations, Volterra integrodifferential equations, various classes of partial differential equations, and so forth. Additionally, there are dynamical systems Manuscript received October 6, 2003; revised January 10, 2005. Recommended by Associate Editor S.-I. Niculescu. Y. Sun is with Credit Suisse First Boston, New York, NY 10010 USA, and also with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]). A. N. Michel is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46554 USA (e-mail: [email protected]). G. Zhai is with the Department of Mechanical Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.852563
whose motions are not determined by classical equations or inequalities of the type enumerated above (e.g., certain classes of discrete event systems whose motions are characterized by Petri nets, Boolean logic elements, and the like). The stability analysis of discrete-time dynamical systems and continuous dynamical systems of the type enumerated above is a mature subject and is addressed, e.g., in [1]–[3]. DDSs arise in the modeling process of a variety of systems, including hybrid dynamical systems, discrete event systems, switched systems, intelligent control systems, systems subjected to impulsive effects, and the like (see., e.g., [3]–[10]). The stability analysis of such systems has thus far been concerned mostly with finite dimensional dynamical systems with metric generated by the Euclidean (defined on norm) determined by ordinary differential equations; however, stability results for general DDS defined on metric space (i.e., is an arbitrary metric space) have also been established [3], [4], [6], [7]. In principle, these results provide a general basis for the analysis of DDS determined by the various types of equations and inequalities enumerated earlier. However, the applications of these results to specific classes of DDS, especially infinite-dimensional systems, are normally not entirely straightforward, and usually require further analysis. (This is also the case for continuous dynamical systems (see, e.g., [1]–[3])). Thus far, only a few general stability results for specific classes of infinite dimensional DDS (determined by the solutions of equations of the type enumerated earlier) appear to have been addressed [16]. In this paper, we establish Lyapunov stability results, including converse theorems, for DDS determined by the solutions of a class of retarded functional differential equations (RFDEs) and we apply these results in the analysis of four important specific classes of DDS. II. CONTINUOUS DYNAMICAL SYSTEMS DETERMINED BY RFDES We will find it convenient to employ the following notation. , let , let , let Let denote real n-space, and let denote any one of the equiv. Let denote the space of real alent norms on continuous functions from the interval to with norm for all . defined by , we denote by When . For , we define for , by the relation . Also, we let denote the space of real piecewise to with continuous functions from the interval and defined similarly as for .
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Now assume that connected subset of
is a domain in ) and let
(i.e., an open and . We call
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Finally, a special case of (2.4) is the class of linear differential-difference equations given by (2.5)
an RFDE, where denotes the right-hand derivative of with respect to (i.e., ). For given , we have, associated with , the initial value problem
and . We note that (2.5) is also a where special case of (2.2). To see this, we let in (2.3) (2.6)
We will denote a continuous solution of , over an interval , if it exists, by , or equivalently, by . For details concerning existence, uniqueness, continuation and continuity with respect to initial conditions of , refer, e.g., to [3] and [11]–[14]. solutions of To simplify matters, we will assume throughout this paper possesses for every a unique conthat which depends continuously on its tinuous solution initial conditions and exists for all . In this case, is well posed. We note that the state at we will say that is and the state–space is . Also, time for system determine a continuous dynamical system the solutions of ( , and the metric on is deteron ). In the interests of brevity, we mined by the norm will sometimes refer to this dynamical system simply as “system .” A special case of are initial-value problems involving time-invariant functional differential equations (2.1)
Remark 2.1: In the case of functional differential equations , many of the qualitative results (such as existence and uniqueness of solutions, stability, and so forth) have been generalized in a straightforward manner to the case when (refer to [11, Ch. 6]). In this case, still possesses soluwhich are continuous with respect to time over tions , and as such, still determines a continuous dynamical system. Furthermore, for the state of we have in the , and . present case is in this case . The state space for system III. DDSS DETERMINED BY RFDES To motivate the class of DDSs which we will consider, and to fix some of the ideas involved in our subsequent presentation, we first consider a specific case. A. An Example Perhaps the simplest specific class of DDS determined by functional differential equations (respectively, differential difference equations) are digital control systems with delays, described by equations of the form
where and . The most general class of linear time-invariant functional differential equations [a special case of (2.1)] is given by (2.2) where gral
is defined by the Riemann–Stieltjes inte-
where , and dimensions,
, are real matrices of appropriate , denote sampling instants, denotes time delay, and (3.2)
(2.3) where is an matrix whose entries are . assumed to be functions of bounded variation on Another special case of (2.1) is the class of time-invariant differential-difference equations (2.4) where and case of (2.1), we let
(3.1)
. To show that (2.4) is a special , where .
We will find it convenient to use the notation . Then (3.3) In order that (3.1) be well posed, we specify the initial data, and , where it is assumed that . We now define (respectively, choose) the initial system state as
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Then, noting that have for all
. Consistent with the previous notation (and , since is continuous) we now
and the state transitions at the points of discontinuity, , are determined by the functions . Remark 3.1: In the following variant to the previous example, we first assume that . Given the and , we define initial data , as the initial system state,
(3.1) is
(3.4) (3.9) In the design of the digital controller for (3.1), one has to specify to state the manner in which the transition from state takes place. Consistent with (3.4), we define by
As before, . We now define the state tran, by sitions at the points of discontinuity, , where
(3.5) Let norm
denote the norm of . Then
induced by the vector
(3.6)
(3.10) It is easily verified that (3.11)
Next, we let for all
and define the linear operator
by [see (2.3)] (3.7)
where . of (3.1) are still In the present case, the solutions . However, the state continuous for all for all , and for all . The state–space for this system is . The , state transitions at the points of discontinuity, . are determined by the functions Finally, if we remove the assumption that , then we must allow that , and , possibly for all . B. DDS Determined by RFDEs
Then, (3.1) can be expressed as
We first consider a family of initial-value problems described by continuous RFDEs of the form (3.8)
where
is defined in (3.5) and . For each , we assume that , is Lipschitz continuous in on each compact subset , and that is well posed, so that for every possesses a unique continuous sowhich exists for all and which lution is continuous with respect to initial conditions. In addition, for , we assume that . This ensures each that of
for all , where [see (3.6)]. We conclude by noting that the unique solution of (3.1) is continuous for all , the state of for all , the state–space for system (3.1), , belongs to
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the existence of the zero solution , , which means that is an equilibrium with . of We now consider discontinuous RFDEs of the form
where for each is assumed to possess the identical properties given in , and . Thus, at , the mapping assigns to every state unambiguously a state . For each , we assume that . The , denoting the set of discontinuities, is set with assumed to be an unbounded infinite discrete subset of . Under the previous assumptions for and , it is now clear that for every has which exists for all . a unique solution This solution is made up of a sequence of continuous solution , defined over the intervals segments , with initial conditions , where and where are given. At the points , the solutions of have possible jumps ]. Furthermore, admits the zero so[determined by lution for (with ) and, therefore, is an equilibrium for . of are continSummarizing, the solutions , the state of at uous for all , belongs to for all , is , and at , the tranthe state space of to are given by . sitions of Remark 3.2: We note that the digital control system (3.3) addressed in the previous subsection, is a specific case of system . Remark 3.3: Consistent with the characterization of DDSs given in Section I, it is clear from the above that determines an infinite dimensional discontinuous dynamical system, , where , the metric on is determined by the norm defined on (i.e., ), and denotes the set of all the solutions of cor. responding to all possible initial conditions In the interests of brevity, we will refer to this DDS simply as ,” or simply as “ .” “system Remark 3.4: Assume that . Consistent with Remarks 2.1 and 3.1, we may also consider in the formuthat and/or (resp., lation of ), . In this case, possesses unique solutions which are still continuous with respect to for . For the state of we have that all , and . The state–space of is in this case . The previous observations are still valid if we remove the . However, in this case assumption that . The unique solutions we have to allow of are still continuous for all
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, the state for (possibly) all and the is . state–space of Remark 3.5: If in the continuous dynamical system , and if at has a discontinuity, and the discontinuity is relative to the norm then of . On the other hand, the discontinuities of the soluof the discontinuous dynamical system , tions occurring at , are relative to the norm of (respectively, ). Remark 3.6: If we allow that , then the DDS admits a variety of variants. For example, we may consider and define the transition functions in
by (see also Remark 3.1) (3.12)
Definition 3.1: a) The zero solution of is stable if for every and there exists a such that for all , whenever . The zero solution of is . uniformly stable if is attractive if for any b) The zero solution of there exists an such that whenever . is asymptotically stable if it c) The zero solution of is stable and attractive. is uniformly attractive if d) The zero solution of there exists a and for every there exists such that for all a whenever . is uniformly asymptotically e) The zero solution of stable if it is uniformly stable and uniformly attractive. is exponentially stable if f) The zero solution of , and for every and every there exists , there exists a such that for all , whenever . are uniformly bounded if for g) The solutions of and for every , there exists a every (independent of ) such that if , then for all . is uniformly asymptotically h) The zero solution of stable in the large if it is uniformly stable, uniformly bounded, and uniformly attractive in the large, i.e., for , for every and for every , every , independent of , such there exists a , then for all . that if is exponentially stable in i) The zero solution of the large if there exists and for every , there exists such that
for all
, whenever
.
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The above definitions are adaptations of corresponding sta(see, e.g., [3] bility and boundedness definitions for system and [11]–[14]). IV. MAIN STABILITY RESULTS In this section, we establish stability results for system in the sense of the definitions given in Section III. In these results, we will make use of comparison functions (Kamke functions), defined as follows. Definition 4.1: A function (respectively, for some ) is said to belong to class , if and if is strictly increasing on K, i.e., (respectively, on ). If belongs to class and is defined , and if , we will say that belongs on , i.e., . to class Theorem 4.1: Assume that there exists a function and functions defined on such that
b)
b)
. Therefore, by definition, the zero solution of is uniformly stable. , we obtain from the Letting assumptions of this theorem that for . If we denote , then and the aforementioned inequality . Since becomes is nonincreasing and , it follows that for all . Thus, we for all obtain that . It follows that (4.5) . For any given Now, consider a fixed such that can choose a
(4.1) for all a)
is nonincreasing. Furthermore, , we can conclude that and
for any
, we
and . Assume that for every is continuous for all except on a set of dis. Also, assume continuities of the origin that there exists a neighborhood such that is nonincreasing for all and all , and assume that there , independent of exists a function , such that (4.2), as shown at the bottom of is the page, holds. Then, the zero solution of uniformly stable. If in addition to the assumptions in part a), there exists defined on such that a function
, and . For any since with and any , we are now able whenever . The to show that previous statement is true because for any must belong to some interval for some . Therefore, we know that . It follows , which implies from (4.5) that that
(4.3)
(4.7)
for all
, where
(4.6)
and
(4.4)
(4.8)
then the zero solution of is uniformly asymptotically stable. Proof: a) Since is continuous and , then for there exists such that any as long as . We assume that , and . , as long as the initial Thus, for any is satisfied, we have condition , and for , since
In the case when , it follows from (4.7) that , noticing that (4.6) holds. In the case , we can conclude from (4.7) that when . This proves that the zero solution of is uniformly asymptotically stable. Theorem 4.2: Assume that there exists a function and functions such that (4.9) for all
and
.
(4.2)
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i)
Assume that for every continuous for all continuities there exists a function , such that of any
is except on a set of dis. Assume that , independent and such that for
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and (respectively, for all in some for all and ). neighborhood of the origin is i) Assume that for every except on a set of discontinuous for all . Furthermore, continuities , assume that there exists a function independent of , such that and such that for any
(4.10) ii)
Assume that there exists a function such that for all
defined on
(4.13) and such that for some positive constant
(4.11) is defined in (4.4). where Then, the zero solution of is uniformly asymptotically stable in the large. Proof: From a) of Theorem 4.1, the zero solution of is uniformly stable. We need to show that the solutions of system are uniformly bounded and that the zero solution of is uniformly attractive in the large. , and choose Consider arbitrary such that . Let and . Repeating the first few steps in the proof of Theorem 4.1(b), we can easily show that (4.5) is still true, i.e.,
. It is easily seen that whenever . are uniformly bounded. Therefore, the solutions of For any fixed and , we can choose a and a such that . Let and . For any with and any , we can show that whenever . This , we can find some such is true since for any . Therefore, and that
as ii)
Assume that there exists a constant
satisfies (4.14) such that
(4.15) and all (respectively, for all with in some neighborhood of the origin ), where is defined in (4.4). Then the zero solution of is exponentially stable in the large (respectively, exponentially stable). Proof: We only prove the case of global exponential stability. The case of local exponential stability can be proved similarly. and . We Let obtain from (4.12) and assumption ii) that for all
where
which yields
where . If is true for some , then and for all and all . Thus, for all . In the for all . following, we assume that , it follows that Since
which implies that Hence
Thus, we have and when . In the case when , we can conclude from before that . This proves that the is uniformly asymptotically stable in the zero solution of large. Theorem 4.3: Assume that there exists a function and three positive constants , and such that (4.12)
is true for all
. It now follows from (4.12) that
In the last step, we have made use of the fact that
.
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Since
as . Let
Then, for all hypothesis i) that for all
, it is easily seen that
. It follows from , it is true that
The last inequality follows since Thus
For any
and any
such that
.
, let
Then
for all and . Therefore, the zero solution of is exponentially stable in the large. system Remark 4.1: In the following, we give an interpretation of the results of the present section and we relate them to existing Lyapunov stability results for continuous functional differential equations (see, e.g., [3] and [11]–[14]). In the interests of brevity, we consider only the case of uniform stability and uniform asymptotic stability (Theorem 4.1). , is an efficient First, we note that (4.1), involving is positive definite and decrescent. way of stating that (This is also a requirement for the existing stability results of continuous dynamical systems determined by functional differential equations [3], [11]–[14]). Next, for the case of uniform stability, hypothesis a) of Theorem 4.1 requires that when evaluated along the solutions be nonincreasing only at the points of discontiof (i.e., is required to be nuity nonincreasing) and that between the points of discontinuity, be required to be only bounded in some sense [see (4.2)]. In contrast to this, the usual existing results for uniform stability of systems determined by continuous funcbe negative tional differential equations demand that semidefinite [3], [11]–[14], which means that along a system’s is required to be nonincreasing for solutions, (a considerably more stringent hypothesis than all what is required in Theorem 4.1). For the case of uniform asymptotic stability, hypotheses a) and b) require that when evaluated along the solutions of , be strictly decreasing only at the points of discon(i.e., , is required to be tinuity strictly decreasing) and that between the points of discontinuity,
be required to be only bounded in some sense [see (4.2)]. In contrast, the usual existing results for uniform asymptotic stability of systems determined by (continuous) be negafunctional differential equations demand that tive definite [3], [11]–[14], which means that along a system’s is required to be strictly decreasing solutions, (which again, is a considerably more for all stringent hypothesis than what is required in Theorem 4.1). To simplify the remainder of this discourse, we choose in par. From the previous discusticular sion, it is clear that under the hypotheses of Theorem 4.1, the may exhibit “unstable behavior” over some solutions of (i.e., the norm of the soluor all of the time intervals may actually increase over some or all of the time tions of ) as long as the transition functions intervals evaluated at the points of discontinuity , provide to sufficient attenuation or contraction of the solutions of ensure uniform stability (or uniform asymptotic stability). Remark 4.2: In the next section we establish Converse Theorems which ensure the existence of Lyapunov functions for the stability results of the present section. However, as in the case of continuous functional differential equations (and for that matter, all the other types of equations enumerated in Section I), there are no nice and tidy rules for the construction of Lyapunov functions for the stability analysis of specific examples. Rather, the power of the Lyapunov theory lies in its ability to establish stability results for important special classes of systems. Never, we are able theless, for the case of DDS determined by to construct Lyapunov functions (at least in part) for the results of the present section in a systematic manner. (To simplify our discussion, we confine ourselves to Theorem 4.1.) In this apfor the proach, we choose multiple Lyapunov functions , and we construct the Lyainitial-value problems for system as punov function (4.16) . (Recall from Section III that every solution of is made up of solution segments valid .) To satisfy the hypotheses of Theorem over by 4.1, we determine the required properties for each bringing to bear the considerable body of the existing literature concerning the Lyapunov theory for systems determined (see., by continuous functional differential equations e.g., [3] and [11]–[14]). In Section VI, we will demonstrate the applicability of the previous methodology in the stability analysis of four important special classes of systems described by functional differential equations and differential-difference equations. Remark 4.3: The results of the present section apply, with obvious changes, when Remark 3.4 applies, replacing by by (respectively, ), and so forth. V. CONVERSE THEOREMS In the previous section, we established sufficient conditions . It for stability and boundedness properties of system turns out that for most of the results that we presented in
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Section IV, we can establish converse statements (necessary conditions), assuming some additional mild assumptions. We will require the following preliminary results. is stable if and only Lemma 5.1: The zero solution of there exists a function defined on if for each such that
. and
for each solution of and for all whenever , where may depend on . The zero solution of is uniformly stable if and only if the above statement is true for independent of . . We assume that the zero solution of is Proof: , there exists a such stable. For every that whenever . It is also clear from . We let the definition that whenever . Then we know that is strictly increasing with respect to and whenever . Correspondingly, we can construct the such that if then function . Let . Let for and . It is easy to see that . For every with , we know that . For every , let . Then for any with , we have , which says that the zero solution of is stable. The conclusion for uniform stability can be proved in the same way except that is chosen independent of . Definition 5.1: A continuous function is said to belong to Class if is strictly decreasing on and if where . Lemma 5.2: The zero solution of is asymptotically there exists a destable if and only if for each , such that for every solution , fined on such that if , then there exists a function
Then, we have It is obvious that . Let
is nonincreasing on . Then, . It is obvious that
and . We now have . This completes our proof. Lemma 5.3: The zero solution of is exponentially stable if and only if it is uniformly asymptotically stable and if in the statement for uniform asymptotic stability in Lemma 5.2, for some . Proof: Refer to [2]. Theorem 5.1: Suppose the zero solution of system is uniformly stable. Then, there exist neighborhoods and of such that , and a mapping which satisfies the following conditions. such that i) There exist
for all and . with For every solution is nonincreasing for all . Proof: Since the proof of Theorem 5.1 is identical to the proof of the converse theorem for uniform stability of continuous functional differential equations [3], [11]–[14], it is omitted here. Theorem 5.2: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is uniformly asymptotically stable, then there exist neighborhoods and of such that and a mapping which satisfies the following conditions. i) There exist such that ii)
ii) for all . The zero solution of is uniformly asymptotically stable if and only if the above statement is true for independent of and for independent of and of . Proof: We only give the proof for asymptotic stability. Assume that there exist and satisfying the above assumptions. From the above inequality we know that , . From Lemma 5.1, we know that the zero sowhere lution of is stable. From the previous assumption, letting , it is easy to see that the zero solution is attractive. Assume that the zero solution of is asymptotically , there exists a such that stable. For for all whenever . Also, there exists an such that whenever . Let . Define the function for as
and . Let
iii)
for all and . There exists such that for all solu, we have tions for all , where and DV is defined in (4.4). such that There exists a function and for all and all .
In the proof of the previous theorem, we will require the following preliminary result. be defined on . Then there exists Lemma 5.4: Let a function defined on such that for any closed satisfying discrete subset , it is true that . and we define Proof: We define by
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Clearly, is strictly decreasing for all , and for all . Furthermore, is invertible, is strictly increasing and for all . and We now define for . Then it is obvious that and . It follows that . If we denote . Hence, it is true that
, then .
fine
. Hence, if we deis true for all
Consider and the corresponding set . In view of the uniqueness of solutions, it is clear that . It follows that
, we know that
We now proceed to the proof of Theorem 5.2. is uniformly Proof: Since the zero solution of asymptotically stable, we know by Theorem 5.1 that there and of such that exist some neighborhoods and a mapping which satisfies the following conditions. such that a) There exist
for all and . with For every solution is nonincreasing for all From a) and b), we conclude that for any true that
for
. Since
by assumption, it follows that
where we define . so that the summation We now show how to choose (5.4) converges. In view of (5.3), we know that for any , we have
b)
.
(5.5)
, it is Let 5.4, there exists an (5.1)
. Then, . Hence, by Lemma defined on such that . If we define , then
it follows that
which implies that (5.2) and . for all In view of Lemma 5.2, there exists a function on for some and another function that
defined such
(5.6) Hence, we conclude that
(5.3) for all
and all
, where
. Define
and if otherwise Since for any , there exists a unique solution which is continuous everywhere on except on , we define (5.4) where will be specified later in such a manner that the aforementioned summation will converge. Obviously,
If we define by , then it follows that . Thus, we have proved parts i) and ii) of the theorem. . We have already shown To prove part iii), let . Furthermore, since that , from (5.2) we know that (5.7)
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On the other hand, we have also shown that which implies that
For
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, we have
(5.8) Combining (5.7) and (5.8), we obtain that for all . If we now de, . This concludes
and all
, then fine and the proof of Theorem 5.2. Our next result yields necessary conditions for the exponen. tial stability of system Theorem 5.3: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is exponentially stable, then there exist neighborhoods and of such that and a mapping which satisfies the following conditions. i) There exist such that
ii)
iii)
, we obtain
Also, (5.9) and (5.10) imply that for all . By (5.11), we have for every that
for all and . such that for all soThere exists a constant , we have lutions for all , where and is defined in (4.4). with There exists a function and for some constant such that
for all and all . deProof: By Lemma 5.3, there exists a function fined on and a constant such that (5.9) for all
Letting
and all
, whenever and let
. Let
The proof is completed by letting . Our next result yields necessary conditions for the uniform global asymptotic stability of . Theorem 5.4: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is uniformly asymptotically stable in the large, then there exists which satisfies the following a mapping conditions. i) There exist such that
if otherwise For
, define
ii) (5.10)
Now, for
and
iii)
, we have
(5.11)
for all and . There exists a defined on such that , we have for all for all and , where is defined in (4.4). There exists a function with such that
for all
and all
.
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Proof: From the definition of uniform asymptotic stability such in the large, we know that there exists a function that
, we know that . Hence, . From (5.13) and (5.14), we have
(5.12) of system . for all solutions Next, from the definition of uniform attractivity in the large, , for every and for every , there for every , independent of , such that if exists a , then for all . We extend the domain of function by letting for all . Without loss of is strictly generality, we can also assume that the function for each fixed and strictly increasing decreasing in for each fixed , and that it is a continuous function in of which satisfies for all . Indeed, if this is not the case, we can define
Since and is strictly increasing, there exists a such that for all function and . as We now choose a Lyapunov function (5.15) Obviously,
and use as . Let be a fixed positive constant. By Massera’s lemma (see [15, Lemma 2] and [12, Lemma 19.1]), there exist two funcis the set of all contions having derivatives of any order, tinuous functions defined on each of which is uniformly bounded on any bounded subset of ), such that for for , and such that (5.13) for all and for all . The functions and are strictly increasing in and , respectively. , we define mapFor any given positive integer as follows: pings
We also define any , let
in the following way. For
. Since
such that
we have
For
such that
we have
Therefore, there exists a function such that , and, consequently, satisfies condition i) of the theorem. Next, we show that satisfies condition ii). From the aforementioned proof, we have (5.16)
(5.14) Choosing
then for
for all
, we can show that . The previous inequality is true because for any
for all . Since and are arbitrary values from the , we can change to and to for to set obtain (5.17)
with Let
. From (5.17), we have (5.18)
which shows that satisfies (ii). and We let Then, we have
We can also show that for all such that . This is true because when , for any
in (5.17).
(5.19)
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Let . Then, we know that satisfies iii). This completes the proof. Theorem 5.5: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is exponentially stable in the large, then there exists a mapping which satisfies the following conditions. i) There exist such that
, and
for all and There exists a constant , we have
assumes the form
. Assuming that for all satisfies the Lipschitz condition
and that (6.1)
for all e.g., [13])
ii)
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, we obtain for system
, the estimate (see,
(6.2)
. such that for all
and all
for all
. We assume that (6.3)
iii)
and There exists a function and such that
with for some constant
In system that
we assume that for all
and (6.4)
for all
, that (6.5)
for all and all . Proof: The proof of the previous result is similar to the proof given in Theorem 5.4 for uniform asymptotic stability in the large and is omitted. Remark 5.1: As in the case of dynamical systems determined by continuous functional differential equations, the preceding converse theorems are generally not useful in the construction of Lyapunov functions. Nevertheless, their importance cannot be overemphasized. In particular, in the present case, these converse theorems establish (under some reasonable additional hypotheses) the existence of Lyapunov functions for the various results of Section IV, and furthermore, these converse theorems show that under the given assumptions, the stability results of Section IV are as good as one can expect. Remark 5.2: The results of this section apply, with obvious by changes, when Remark 3.4 applies, replacing by (respectively, ), and so forth. VI. APPLICATIONS In this section, we apply the method of constructing Lyapunov functions proposed in Remark 4.2 in the stability analysis of four important special classes of DDS determined by functional differential equations and differential-difference equations. Example 6.1: (Time-invariant nonlinear functional differential equations) , then takes If we let the form
and letting
, that (6.6)
, and be the parameTheorem 6.1: Let ters for system given in (6.1)–(6.6). a) If for all , then the zero solution is uniformly stable. of b) If for all , where is a constant, then is uniformly asymptotically i) the zero solution of stable, in fact, uniformly asymptotically stable in the large; and is exponentially stable, in ii) the zero solution of fact, exponentially stable in the large. Proof: Using the method of constructing Lyapunov functions outlined in Remark 4.2 [see (4.16)], we choose for , which, when evaluated along assumes the form the solutions of
(6.7)
tions that
(i.e., consists of a sequence of multiple Lyapunov func, valid over the time intervals ). (Recall denotes the solution segment of the solution over the interval (see Section III). Clearly (6.8)
, where , i.e., for all . Therefore, (4.1) in Theorem 4.1 is satisfied.
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Along the solutions of
we have, in view of (6.2), that
(6.9) for
. At
, we have, in view of (6.4), that
(6.10)
Next, from (6.8) it is clear that in relation (4.12) of Theorem and from (6.12) it is clear that 4.3 we have . We have in relation (4.15) of Theorem 4.3, already shown that (4.13) of Theorem 4.3 is true, and clearly , we have as for any for . Therefore, all conditions of Theorem 4.3 are satisfied is exponentially and we conclude that the zero solution of stable and exponentially stable in the large. Example 6.2: (Time-invariant linear functional differential equations) , then takes the form If we let
Combining (6.9) and (6.10), we have
(6.13) (6.11)
and since by assumption
and
assumes the form
, we have (6.14) . For each
is defined, as in (2.3), by (6.15)
Since this holds for arbitrary , is nonincreasing. Next, for (6.9) we have, recalling that , that
, it follows that and
We suppose that all assumptions that we made for given in (2.3) hold as well for . Then, is Lipschitz continuous on with Lipschitz constant less or equal to the variation , and as such, condition (6.1) still holds for (6.13). The of consists of all solutions of the equation (see, spectrum of e.g., [13]) (6.16)
, where . Therefore, all conditions of Theorem 4.1(a) are satisfied and we conclude that is uniformly stable. the zero solution of , we have If in (6.11) we assume that
When all solutions of (6.16) satisfy the relation then for any positive , there is a constant such that the solutions of (6.13) allow the estimate
,
(6.17) for all and (see, e.g., [13]). When the aforementioned assumption is not true, then the solutions of (6.13) still allow the estimate (see, e.g., [13])
and
(6.18) for all
and
. Thus, in all cases, we have (6.19)
(6.12) for all
. In (6.12), we have , i.e., . Therefore, all conditions of Theorem 4.1(b) are satisfied, and the zero solution of is uniformly asymptotically stable. and since actually, Since (6.8) holds for all , and since (6.12) is true for all , it follows is uniformly from Theorem 4.2 that the zero solution of asymptotically stable in the large.
for all and when (6.18) applies and , when (6.17) applies. Finally, for each to satisfy the condition
, where
and and in (6.14) is assumed (6.20)
for all
, where
is a constant.
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In the following result, we assume that the relations (6.3) and . (6.6) still hold, and we assume that , and be the paramTheorem 6.2: Let eters for (6.14) defined previously. a) If for all , then the zero solution of (6.14) is uniformly stable. , where b) If for all is a constant, then i) the zero solution of (6.14) is uniformly asymptotically stable, in fact, uniformly asymptotically stable in the large; and ii) the zero solution of (6.14) is exponentially stable, in fact, exponentially stable in the large. Proof: The proof is similar to the proof of Theorem 6.1. Remark 6.1: System (3.1) (see the example in Section III-A) is clearly a special case of system (6.13), and as such, Theorem 6.2 is directly applicable in the analysis of (3.1). Example 6.3: (Time-invariant nonlinear differential-difference equations) We now consider a family of initial-value problems described by differential-difference equations of the form
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Similarly as in (6.1)–(6.6), we assume that , and , where
.
Theorem 6.3: Let , and be the parameters for (6.22). , then the zero a) If for all solution of (6.22) is uniformly stable. , where b) If for all is a constant, then i) the zero solution of (6.22) is uniformly asymptotically stable, in fact, uniformly asymptotically stable in the large; and ii) the zero solution of (6.22) is exponentially stable, in fact, exponentially stable in the large. Proof: As in the proof of Theorem 6.1, we choose . Then, (6.8) still holds with . Also, in accordance with (6.25), we have along the solutions of (6.22), the estimate (6.27) . From (6.22) and (6.26), we have
(6.21) , and a DDS determined by (6.22) where
, and
(6.28) and, therefore, using (6.27) and (6.28) and assuming that , we have
(6.23) Defining
and ( and ), and , then (6.21) and (6.22) are special cases of in (6.21) satisfies the respectively. If we assume that each Lipschitz condition
, i.e., , is nonincreasing. Also, in the present case we have, in view of (6.27)
(6.24) for all , then, by invoking the Gronwall Inequality (see, e.g., [3]), we obtain for (6.21) the estimate (6.25) and for all for all We will assume that for each and
.
(6.26) for all
.
, where , and and . The rest of the proof is identical to the Proof of Theorem 4.1. We conclude from Theorem 6.1 that the zero solution of (6.22) , and when is uniformly stable when , we conclude from Theorem 4.1 that the zero solution of (6.22) is uniformly asymptotically , we conclude stable. Also, when from Theorem 4.2 that the zero solution of (6.22) is uniformly asymptotically stable in the large, and from Theorem 4.3, that the zero solution of (6.22) is exponentially stable and exponentially stable in the large.
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Example 6.4: (Time-invariant linear differential-difference equations) A special case of (6.21) are differential-difference equations of the form (6.29) , where and case of (6.22) are DDSs determined by
. Also, a special
b) i)
ii)
Proof: The proof is similar to the proof of Theorem 6.1. Remark 6.2: a)
(6.33)
It is emphasized that the results in Theorems 6.1–6.4 involve available system parameters, including Lipschitz constants, time intervals between discontinuities, properties of the spectrum of the linear operators describing the systems, and properties of the transition (e.g., their “gains”) at the instants of functions discontinuity. , are not preWhen the instants of discontinuity, cisely known, then the results of this section may be modified, involving estimates of the largest time interval between instants of discontinuity. For example, in the case of Theorem 6.1, we would in this case reby ( are defined place in Theorem 6.1). The results of the present section allow that stable and asymptotically stable discontinuous dynamical systems may exhibit unstable behavior over some or all of the open time intervals between the points of disconti. Specifically, if e.g., in Example 6.2, nuity, condition (6.17) is applicable over , then the norm of the system’s solution will decrease exponentially, while if condition (6.17) is not applicable (so that condition (6.18) is applicable), the norm of the system’s solution may actually increase, yet the zero solution of system (6.14) will still be uniformly stable, respectively, uniformly asymptotically stable, as long as the conditions of Theorem 6.2 are satisfied (refer also to the last paragraph in Remark 4.1). This type of behavior is not possible in continuous dynamical systems.
. Thus, we have in all
Remark 6.3: Consistent with the variant to system given in (3.12) (refer to Remark 3.6), we modify system (6.30) as
(6.34)
(6.35)
(6.30)
where and is defined similarly as in (6.23). As shown in Section II, (6.29) is also a special case of the time-invariant linear functional differential (6.13), and (6.30) is a special case of the DDS described by the functional differential equations given in (6.14). In the present case, the spectrum for (6.29) is determined by the equation (6.31) When all solutions of (6.31) satisfy the relation , there is a constant then for any positive such that the solutions of (6.29) allow the estimate
,
(6.32) and (see, e.g., [2]). When the previous for all assumption is not true, then (noting that (6.29) is Lipschitz con( is the norm of induced by the tinuous with vector norm )), we obtain the estimate
and for all for all cases the estimate
for all
Let let
, where If for all is a constant, then the zero solution of (6.30) is uniformly asymptotically stable, in fact, uniformly asymptotically stable in the large; and the zero solution of (6.30) is exponentially stable, in fact, exponentially stable in the large.
and , where when (6.33) applies and , when (6.32) applies.
and and
b)
c)
where all symbols are defined as in (6.30). A simple computation shows that in this case for each
, , and assume that and .
Theorem 6.4: Let and be the parameters for (6.30). a) If for all , then the zero solution of (6.30) is uniformly stable.
Letting , it is easily shown that all conclusions of Theorem 6.4 apply to system (6.35) if we replace in by , assuming that for Theorem 6.4 , the estimate (6.32) applies. all
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REFERENCES [1] V. I. Zubov, Methods of A. M. Lyapunov and their Applications. Groningen, The Netherlands: Noordhoff, 1964. [2] W. Hahn, Stability of Motion. Berlin, Germany: Springer-Verlag, 1967. [3] A. N. Michel, K. Wang, and B. Hu, Qualitative Analysis of Dynamical Systems, 2nd ed. New York: Marcel Dekker, 2001. [4] H. Ye, A. N. Michel, and L. Hou, “Stability theory for hybrid dynamical systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 461–474, Apr. 1998. [5] M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 475–482, Apr. 1998. [6] A. N. Michel, “Recent trends in the stability analysis of hybrid dynamical systems,” IEEE Trans. Circuits Syst I, Fundam. Theory Appl., vol. 46, no. 1, pp. 120–134, Jan. 1999. [7] A. N. Michel and B. Hu, “Toward a stability theory of general hybrid dynamical systems,” Automatica, vol. 35, pp. 371–384, Apr. 1999. [8] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Syst. Mag., vol. 19, no. 5, pp. 59–70, 1999. [9] R. DeCarlo, M. Branicky, S. Pettersson, and B. Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proc. IEEE, vol. 88, no. 7, pp. 1069–1082, Jul. 2000. [10] D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effects: Stability, Theory and Applications. New York: Halsted, 1989. [11] N. N. Krasovskii, Stability of Motion. Stanford, CA: Stanford Univ. Press, 1963. [12] T. Yoshizawa, Stability Theory by Lyapunov’s Second Method. Tokyo, Japan: Math. Soc. Japan, 1966. [13] J. K. Hale, Functional Differential Equations. Berlin, Germany: Springer-Verlag, 1971. [14] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Cambridge, MA: Academic, 1993. [15] J. L. Massera, “Contributions to stability theory,” Ann. Math., vol. 64, pp. 182–206, 1956. [16] A. N. Michel and Y. Sun, “Stability analysis of discontinuous dynamical systems determined by semigroups,” IEEE Trans. Autom. Control, 2005, to be published.
Ye Sun received the B.S. degree in mathematics from the University of Science and Technology of China, and the M.S. and Ph.D. degrees from the University of Notre Dame, Notre Dame, IN, in 1999, 2002, and 2004, respectively. She is currently a Systems Analyst with Credit Suisse First Boston, New York, working on fixed-income models. Her research interests include systems modeling and qualitative analysis of discontinuous dynamical systems.
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Anthony N. Michel (S’55–M’59–SM’79–F’82) received the Ph.D. degree in electrical engineering from Marquette University, Milwaukee, WI, and the D.Sc. degree in applied mathematics from the Technical University of Graz, Graz, Austria. He has seven years of industrial experience and was on the Electrical Engineering Faculty at Iowa State University, Ames, for 16 years. In 1984, he joined the University of Notre Dame, Notre Dame, IN, as Chair of the Department of Electrical Engineering. In 1988, he became Dean of the College of Engineering, a position he held for ten years. He is currently Frank M. Freimann Professor Emeritus and Matthew H. McCloskey Dean of Engineering Emeritus at Notre Dame. He has also held visiting faculty positions at the Technical University of Vienna, Vienna, Austria, the Johannes Kepler University, Linz, Austria, and Ruhr University, Bochum, Germany. He is the author or coauthor of eight books and numerous archival publications. His more recent work is concerned with stability analysis of finite- and infinite-dimensional dynamical systems and qualitative analysis and synthesis of recurrent neural networks. Dr. Michel has rendered substantial service to several professional organizations, especially the IEEE Circuits and Systems Society and the IEEE Control Systems Society. He has been honored by a number of prestigious awards for his work as an educator and scholar.
Guisheng Zhai (M’98–SM’04) was born in Hubei, China, in 1967. He received the B.S. degree from Fudan University, China, in 1988, and the M.E. and the Ph.D. degrees, both in system science, from Kobe University, Kobe, Japan, in 1993 and 1996, respectively. From 1996 to 1998, he worked in the Kansai Laboratory of OKI Electric Industry Co., Ltd., Japan. From 1998 to 2004, he was a Research Associate in the Department of Opto-Mechatronics, Wakayama University, Japan. He also held a visiting research position in the Department of Electrical Engineering, the University of Notre Dame, Notre Dame, IN, from August 2001 to July 2002. In April 2004, he joined the faculty of Osaka Prefecture University, Osaka, Japan, where he currently is an Associate Professor of Mechanical Engineering. His research interests include large scale and decentralized control systems, robust control, switched systems and switching control, networked control systems, neural networks and signal processing, etc. He is a Member of ISCIE, SICE, and JSME.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 8, AUGUST 2005
Stability of Discontinuous Retarded Functional Differential Equations With Applications Ye Sun, Anthony N. Michel, Life Fellow, IEEE, and Guisheng Zhai, Senior Member, IEEE
Abstract—We present Lyapunov stability results, including converse theorems, for a class of discontinuous dynamical systems (DDS) determined by the solutions of retarded functional differential equations. We demonstrate the applicability of these results in the analysis of several specific important classes of DDS determined by functional differential equations and differential-difference equations. Index Terms—Asymptotic stability, differential-difference equations, discontinuous dynamical systems (DDS), exponential stability, hybrid systems, Lyapunov stability, retarded functional differential equations, switched systems, systems with delays.
I. INTRODUCTION
A
dynamical system is a four-tuple where denotes time set, is the state–space (a metric space with metric ), is the set of initial states and denotes a family of motions. When , we speak of a continuous-time dynamical system and when we speak of a discrete-time dynamical system. (For any motion , we have and for all , where may be finite or infinite. The set of motions is obtained by over .) When is a finite-dimensional varying normed linear space, we speak of finite-dimensional dynamical systems, and otherwise, of infinite-dimensional dynamical systems. Also, when all motions in a continuous-time dynamical system are continuous with respect to (relative to the metric for ), we speak of a continuous dynamical system and when one or more of the motions are not continuous with respect to , we speak of a discontinuous dynamical system (DDS). Finite-dimensional dynamical systems may be determined, e.g., by the solutions of ordinary differential equations, ordinary differential inequalities, difference equations, difference inequalities, and the like, while infinite-dimensional dynamical systems may be determined, e.g., by the solutions of differential-difference equations, functional differential equations, Volterra integrodifferential equations, various classes of partial differential equations, and so forth. Additionally, there are dynamical systems Manuscript received October 6, 2003; revised January 10, 2005. Recommended by Associate Editor S.-I. Niculescu. Y. Sun is with Credit Suisse First Boston, New York, NY 10010 USA, and also with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: [email protected]). A. N. Michel is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46554 USA (e-mail: [email protected]). G. Zhai is with the Department of Mechanical Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.852563
whose motions are not determined by classical equations or inequalities of the type enumerated above (e.g., certain classes of discrete event systems whose motions are characterized by Petri nets, Boolean logic elements, and the like). The stability analysis of discrete-time dynamical systems and continuous dynamical systems of the type enumerated above is a mature subject and is addressed, e.g., in [1]–[3]. DDSs arise in the modeling process of a variety of systems, including hybrid dynamical systems, discrete event systems, switched systems, intelligent control systems, systems subjected to impulsive effects, and the like (see., e.g., [3]–[10]). The stability analysis of such systems has thus far been concerned mostly with finite dimensional dynamical systems with metric generated by the Euclidean (defined on norm) determined by ordinary differential equations; however, stability results for general DDS defined on metric space (i.e., is an arbitrary metric space) have also been established [3], [4], [6], [7]. In principle, these results provide a general basis for the analysis of DDS determined by the various types of equations and inequalities enumerated earlier. However, the applications of these results to specific classes of DDS, especially infinite-dimensional systems, are normally not entirely straightforward, and usually require further analysis. (This is also the case for continuous dynamical systems (see, e.g., [1]–[3])). Thus far, only a few general stability results for specific classes of infinite dimensional DDS (determined by the solutions of equations of the type enumerated earlier) appear to have been addressed [16]. In this paper, we establish Lyapunov stability results, including converse theorems, for DDS determined by the solutions of a class of retarded functional differential equations (RFDEs) and we apply these results in the analysis of four important specific classes of DDS. II. CONTINUOUS DYNAMICAL SYSTEMS DETERMINED BY RFDES We will find it convenient to employ the following notation. , let , let , let Let denote real n-space, and let denote any one of the equiv. Let denote the space of real alent norms on continuous functions from the interval to with norm for all . defined by , we denote by When . For , we define for , by the relation . Also, we let denote the space of real piecewise to with continuous functions from the interval and defined similarly as for .
0018-9286/$20.00 © 2005 IEEE
SUN et al.: STABILITY OF DISCONTINUOUS RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
Now assume that connected subset of
is a domain in ) and let
(i.e., an open and . We call
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Finally, a special case of (2.4) is the class of linear differential-difference equations given by (2.5)
an RFDE, where denotes the right-hand derivative of with respect to (i.e., ). For given , we have, associated with , the initial value problem
and . We note that (2.5) is also a where special case of (2.2). To see this, we let in (2.3) (2.6)
We will denote a continuous solution of , over an interval , if it exists, by , or equivalently, by . For details concerning existence, uniqueness, continuation and continuity with respect to initial conditions of , refer, e.g., to [3] and [11]–[14]. solutions of To simplify matters, we will assume throughout this paper possesses for every a unique conthat which depends continuously on its tinuous solution initial conditions and exists for all . In this case, is well posed. We note that the state at we will say that is and the state–space is . Also, time for system determine a continuous dynamical system the solutions of ( , and the metric on is deteron ). In the interests of brevity, we mined by the norm will sometimes refer to this dynamical system simply as “system .” A special case of are initial-value problems involving time-invariant functional differential equations (2.1)
Remark 2.1: In the case of functional differential equations , many of the qualitative results (such as existence and uniqueness of solutions, stability, and so forth) have been generalized in a straightforward manner to the case when (refer to [11, Ch. 6]). In this case, still possesses soluwhich are continuous with respect to time over tions , and as such, still determines a continuous dynamical system. Furthermore, for the state of we have in the , and . present case is in this case . The state space for system III. DDSS DETERMINED BY RFDES To motivate the class of DDSs which we will consider, and to fix some of the ideas involved in our subsequent presentation, we first consider a specific case. A. An Example Perhaps the simplest specific class of DDS determined by functional differential equations (respectively, differential difference equations) are digital control systems with delays, described by equations of the form
where and . The most general class of linear time-invariant functional differential equations [a special case of (2.1)] is given by (2.2) where gral
is defined by the Riemann–Stieltjes inte-
where , and dimensions,
, are real matrices of appropriate , denote sampling instants, denotes time delay, and (3.2)
(2.3) where is an matrix whose entries are . assumed to be functions of bounded variation on Another special case of (2.1) is the class of time-invariant differential-difference equations (2.4) where and case of (2.1), we let
(3.1)
. To show that (2.4) is a special , where .
We will find it convenient to use the notation . Then (3.3) In order that (3.1) be well posed, we specify the initial data, and , where it is assumed that . We now define (respectively, choose) the initial system state as
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Then, noting that have for all
. Consistent with the previous notation (and , since is continuous) we now
and the state transitions at the points of discontinuity, , are determined by the functions . Remark 3.1: In the following variant to the previous example, we first assume that . Given the and , we define initial data , as the initial system state,
(3.1) is
(3.4) (3.9) In the design of the digital controller for (3.1), one has to specify to state the manner in which the transition from state takes place. Consistent with (3.4), we define by
As before, . We now define the state tran, by sitions at the points of discontinuity, , where
(3.5) Let norm
denote the norm of . Then
induced by the vector
(3.6)
(3.10) It is easily verified that (3.11)
Next, we let for all
and define the linear operator
by [see (2.3)] (3.7)
where . of (3.1) are still In the present case, the solutions . However, the state continuous for all for all , and for all . The state–space for this system is . The , state transitions at the points of discontinuity, . are determined by the functions Finally, if we remove the assumption that , then we must allow that , and , possibly for all . B. DDS Determined by RFDEs
Then, (3.1) can be expressed as
We first consider a family of initial-value problems described by continuous RFDEs of the form (3.8)
where
is defined in (3.5) and . For each , we assume that , is Lipschitz continuous in on each compact subset , and that is well posed, so that for every possesses a unique continuous sowhich exists for all and which lution is continuous with respect to initial conditions. In addition, for , we assume that . This ensures each that of
for all , where [see (3.6)]. We conclude by noting that the unique solution of (3.1) is continuous for all , the state of for all , the state–space for system (3.1), , belongs to
SUN et al.: STABILITY OF DISCONTINUOUS RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
the existence of the zero solution , , which means that is an equilibrium with . of We now consider discontinuous RFDEs of the form
where for each is assumed to possess the identical properties given in , and . Thus, at , the mapping assigns to every state unambiguously a state . For each , we assume that . The , denoting the set of discontinuities, is set with assumed to be an unbounded infinite discrete subset of . Under the previous assumptions for and , it is now clear that for every has which exists for all . a unique solution This solution is made up of a sequence of continuous solution , defined over the intervals segments , with initial conditions , where and where are given. At the points , the solutions of have possible jumps ]. Furthermore, admits the zero so[determined by lution for (with ) and, therefore, is an equilibrium for . of are continSummarizing, the solutions , the state of at uous for all , belongs to for all , is , and at , the tranthe state space of to are given by . sitions of Remark 3.2: We note that the digital control system (3.3) addressed in the previous subsection, is a specific case of system . Remark 3.3: Consistent with the characterization of DDSs given in Section I, it is clear from the above that determines an infinite dimensional discontinuous dynamical system, , where , the metric on is determined by the norm defined on (i.e., ), and denotes the set of all the solutions of cor. responding to all possible initial conditions In the interests of brevity, we will refer to this DDS simply as ,” or simply as “ .” “system Remark 3.4: Assume that . Consistent with Remarks 2.1 and 3.1, we may also consider in the formuthat and/or (resp., lation of ), . In this case, possesses unique solutions which are still continuous with respect to for . For the state of we have that all , and . The state–space of is in this case . The previous observations are still valid if we remove the . However, in this case assumption that . The unique solutions we have to allow of are still continuous for all
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, the state for (possibly) all and the is . state–space of Remark 3.5: If in the continuous dynamical system , and if at has a discontinuity, and the discontinuity is relative to the norm then of . On the other hand, the discontinuities of the soluof the discontinuous dynamical system , tions occurring at , are relative to the norm of (respectively, ). Remark 3.6: If we allow that , then the DDS admits a variety of variants. For example, we may consider and define the transition functions in
by (see also Remark 3.1) (3.12)
Definition 3.1: a) The zero solution of is stable if for every and there exists a such that for all , whenever . The zero solution of is . uniformly stable if is attractive if for any b) The zero solution of there exists an such that whenever . is asymptotically stable if it c) The zero solution of is stable and attractive. is uniformly attractive if d) The zero solution of there exists a and for every there exists such that for all a whenever . is uniformly asymptotically e) The zero solution of stable if it is uniformly stable and uniformly attractive. is exponentially stable if f) The zero solution of , and for every and every there exists , there exists a such that for all , whenever . are uniformly bounded if for g) The solutions of and for every , there exists a every (independent of ) such that if , then for all . is uniformly asymptotically h) The zero solution of stable in the large if it is uniformly stable, uniformly bounded, and uniformly attractive in the large, i.e., for , for every and for every , every , independent of , such there exists a , then for all . that if is exponentially stable in i) The zero solution of the large if there exists and for every , there exists such that
for all
, whenever
.
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The above definitions are adaptations of corresponding sta(see, e.g., [3] bility and boundedness definitions for system and [11]–[14]). IV. MAIN STABILITY RESULTS In this section, we establish stability results for system in the sense of the definitions given in Section III. In these results, we will make use of comparison functions (Kamke functions), defined as follows. Definition 4.1: A function (respectively, for some ) is said to belong to class , if and if is strictly increasing on K, i.e., (respectively, on ). If belongs to class and is defined , and if , we will say that belongs on , i.e., . to class Theorem 4.1: Assume that there exists a function and functions defined on such that
b)
b)
. Therefore, by definition, the zero solution of is uniformly stable. , we obtain from the Letting assumptions of this theorem that for . If we denote , then and the aforementioned inequality . Since becomes is nonincreasing and , it follows that for all . Thus, we for all obtain that . It follows that (4.5) . For any given Now, consider a fixed such that can choose a
(4.1) for all a)
is nonincreasing. Furthermore, , we can conclude that and
for any
, we
and . Assume that for every is continuous for all except on a set of dis. Also, assume continuities of the origin that there exists a neighborhood such that is nonincreasing for all and all , and assume that there , independent of exists a function , such that (4.2), as shown at the bottom of is the page, holds. Then, the zero solution of uniformly stable. If in addition to the assumptions in part a), there exists defined on such that a function
, and . For any since with and any , we are now able whenever . The to show that previous statement is true because for any must belong to some interval for some . Therefore, we know that . It follows , which implies from (4.5) that that
(4.3)
(4.7)
for all
, where
(4.6)
and
(4.4)
(4.8)
then the zero solution of is uniformly asymptotically stable. Proof: a) Since is continuous and , then for there exists such that any as long as . We assume that , and . , as long as the initial Thus, for any is satisfied, we have condition , and for , since
In the case when , it follows from (4.7) that , noticing that (4.6) holds. In the case , we can conclude from (4.7) that when . This proves that the zero solution of is uniformly asymptotically stable. Theorem 4.2: Assume that there exists a function and functions such that (4.9) for all
and
.
(4.2)
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i)
Assume that for every continuous for all continuities there exists a function , such that of any
is except on a set of dis. Assume that , independent and such that for
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and (respectively, for all in some for all and ). neighborhood of the origin is i) Assume that for every except on a set of discontinuous for all . Furthermore, continuities , assume that there exists a function independent of , such that and such that for any
(4.10) ii)
Assume that there exists a function such that for all
defined on
(4.13) and such that for some positive constant
(4.11) is defined in (4.4). where Then, the zero solution of is uniformly asymptotically stable in the large. Proof: From a) of Theorem 4.1, the zero solution of is uniformly stable. We need to show that the solutions of system are uniformly bounded and that the zero solution of is uniformly attractive in the large. , and choose Consider arbitrary such that . Let and . Repeating the first few steps in the proof of Theorem 4.1(b), we can easily show that (4.5) is still true, i.e.,
. It is easily seen that whenever . are uniformly bounded. Therefore, the solutions of For any fixed and , we can choose a and a such that . Let and . For any with and any , we can show that whenever . This , we can find some such is true since for any . Therefore, and that
as ii)
Assume that there exists a constant
satisfies (4.14) such that
(4.15) and all (respectively, for all with in some neighborhood of the origin ), where is defined in (4.4). Then the zero solution of is exponentially stable in the large (respectively, exponentially stable). Proof: We only prove the case of global exponential stability. The case of local exponential stability can be proved similarly. and . We Let obtain from (4.12) and assumption ii) that for all
where
which yields
where . If is true for some , then and for all and all . Thus, for all . In the for all . following, we assume that , it follows that Since
which implies that Hence
Thus, we have and when . In the case when , we can conclude from before that . This proves that the is uniformly asymptotically stable in the zero solution of large. Theorem 4.3: Assume that there exists a function and three positive constants , and such that (4.12)
is true for all
. It now follows from (4.12) that
In the last step, we have made use of the fact that
.
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Since
as . Let
Then, for all hypothesis i) that for all
, it is easily seen that
. It follows from , it is true that
The last inequality follows since Thus
For any
and any
such that
.
, let
Then
for all and . Therefore, the zero solution of is exponentially stable in the large. system Remark 4.1: In the following, we give an interpretation of the results of the present section and we relate them to existing Lyapunov stability results for continuous functional differential equations (see, e.g., [3] and [11]–[14]). In the interests of brevity, we consider only the case of uniform stability and uniform asymptotic stability (Theorem 4.1). , is an efficient First, we note that (4.1), involving is positive definite and decrescent. way of stating that (This is also a requirement for the existing stability results of continuous dynamical systems determined by functional differential equations [3], [11]–[14]). Next, for the case of uniform stability, hypothesis a) of Theorem 4.1 requires that when evaluated along the solutions be nonincreasing only at the points of discontiof (i.e., is required to be nuity nonincreasing) and that between the points of discontinuity, be required to be only bounded in some sense [see (4.2)]. In contrast to this, the usual existing results for uniform stability of systems determined by continuous funcbe negative tional differential equations demand that semidefinite [3], [11]–[14], which means that along a system’s is required to be nonincreasing for solutions, (a considerably more stringent hypothesis than all what is required in Theorem 4.1). For the case of uniform asymptotic stability, hypotheses a) and b) require that when evaluated along the solutions of , be strictly decreasing only at the points of discon(i.e., , is required to be tinuity strictly decreasing) and that between the points of discontinuity,
be required to be only bounded in some sense [see (4.2)]. In contrast, the usual existing results for uniform asymptotic stability of systems determined by (continuous) be negafunctional differential equations demand that tive definite [3], [11]–[14], which means that along a system’s is required to be strictly decreasing solutions, (which again, is a considerably more for all stringent hypothesis than what is required in Theorem 4.1). To simplify the remainder of this discourse, we choose in par. From the previous discusticular sion, it is clear that under the hypotheses of Theorem 4.1, the may exhibit “unstable behavior” over some solutions of (i.e., the norm of the soluor all of the time intervals may actually increase over some or all of the time tions of ) as long as the transition functions intervals evaluated at the points of discontinuity , provide to sufficient attenuation or contraction of the solutions of ensure uniform stability (or uniform asymptotic stability). Remark 4.2: In the next section we establish Converse Theorems which ensure the existence of Lyapunov functions for the stability results of the present section. However, as in the case of continuous functional differential equations (and for that matter, all the other types of equations enumerated in Section I), there are no nice and tidy rules for the construction of Lyapunov functions for the stability analysis of specific examples. Rather, the power of the Lyapunov theory lies in its ability to establish stability results for important special classes of systems. Never, we are able theless, for the case of DDS determined by to construct Lyapunov functions (at least in part) for the results of the present section in a systematic manner. (To simplify our discussion, we confine ourselves to Theorem 4.1.) In this apfor the proach, we choose multiple Lyapunov functions , and we construct the Lyainitial-value problems for system as punov function (4.16) . (Recall from Section III that every solution of is made up of solution segments valid .) To satisfy the hypotheses of Theorem over by 4.1, we determine the required properties for each bringing to bear the considerable body of the existing literature concerning the Lyapunov theory for systems determined (see., by continuous functional differential equations e.g., [3] and [11]–[14]). In Section VI, we will demonstrate the applicability of the previous methodology in the stability analysis of four important special classes of systems described by functional differential equations and differential-difference equations. Remark 4.3: The results of the present section apply, with obvious changes, when Remark 3.4 applies, replacing by by (respectively, ), and so forth. V. CONVERSE THEOREMS In the previous section, we established sufficient conditions . It for stability and boundedness properties of system turns out that for most of the results that we presented in
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Section IV, we can establish converse statements (necessary conditions), assuming some additional mild assumptions. We will require the following preliminary results. is stable if and only Lemma 5.1: The zero solution of there exists a function defined on if for each such that
. and
for each solution of and for all whenever , where may depend on . The zero solution of is uniformly stable if and only if the above statement is true for independent of . . We assume that the zero solution of is Proof: , there exists a such stable. For every that whenever . It is also clear from . We let the definition that whenever . Then we know that is strictly increasing with respect to and whenever . Correspondingly, we can construct the such that if then function . Let . Let for and . It is easy to see that . For every with , we know that . For every , let . Then for any with , we have , which says that the zero solution of is stable. The conclusion for uniform stability can be proved in the same way except that is chosen independent of . Definition 5.1: A continuous function is said to belong to Class if is strictly decreasing on and if where . Lemma 5.2: The zero solution of is asymptotically there exists a destable if and only if for each , such that for every solution , fined on such that if , then there exists a function
Then, we have It is obvious that . Let
is nonincreasing on . Then, . It is obvious that
and . We now have . This completes our proof. Lemma 5.3: The zero solution of is exponentially stable if and only if it is uniformly asymptotically stable and if in the statement for uniform asymptotic stability in Lemma 5.2, for some . Proof: Refer to [2]. Theorem 5.1: Suppose the zero solution of system is uniformly stable. Then, there exist neighborhoods and of such that , and a mapping which satisfies the following conditions. such that i) There exist
for all and . with For every solution is nonincreasing for all . Proof: Since the proof of Theorem 5.1 is identical to the proof of the converse theorem for uniform stability of continuous functional differential equations [3], [11]–[14], it is omitted here. Theorem 5.2: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is uniformly asymptotically stable, then there exist neighborhoods and of such that and a mapping which satisfies the following conditions. i) There exist such that ii)
ii) for all . The zero solution of is uniformly asymptotically stable if and only if the above statement is true for independent of and for independent of and of . Proof: We only give the proof for asymptotic stability. Assume that there exist and satisfying the above assumptions. From the above inequality we know that , . From Lemma 5.1, we know that the zero sowhere lution of is stable. From the previous assumption, letting , it is easy to see that the zero solution is attractive. Assume that the zero solution of is asymptotically , there exists a such that stable. For for all whenever . Also, there exists an such that whenever . Let . Define the function for as
and . Let
iii)
for all and . There exists such that for all solu, we have tions for all , where and DV is defined in (4.4). such that There exists a function and for all and all .
In the proof of the previous theorem, we will require the following preliminary result. be defined on . Then there exists Lemma 5.4: Let a function defined on such that for any closed satisfying discrete subset , it is true that . and we define Proof: We define by
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Clearly, is strictly decreasing for all , and for all . Furthermore, is invertible, is strictly increasing and for all . and We now define for . Then it is obvious that and . It follows that . If we denote . Hence, it is true that
, then .
fine
. Hence, if we deis true for all
Consider and the corresponding set . In view of the uniqueness of solutions, it is clear that . It follows that
, we know that
We now proceed to the proof of Theorem 5.2. is uniformly Proof: Since the zero solution of asymptotically stable, we know by Theorem 5.1 that there and of such that exist some neighborhoods and a mapping which satisfies the following conditions. such that a) There exist
for all and . with For every solution is nonincreasing for all From a) and b), we conclude that for any true that
for
. Since
by assumption, it follows that
where we define . so that the summation We now show how to choose (5.4) converges. In view of (5.3), we know that for any , we have
b)
.
(5.5)
, it is Let 5.4, there exists an (5.1)
. Then, . Hence, by Lemma defined on such that . If we define , then
it follows that
which implies that (5.2) and . for all In view of Lemma 5.2, there exists a function on for some and another function that
defined such
(5.6) Hence, we conclude that
(5.3) for all
and all
, where
. Define
and if otherwise Since for any , there exists a unique solution which is continuous everywhere on except on , we define (5.4) where will be specified later in such a manner that the aforementioned summation will converge. Obviously,
If we define by , then it follows that . Thus, we have proved parts i) and ii) of the theorem. . We have already shown To prove part iii), let . Furthermore, since that , from (5.2) we know that (5.7)
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On the other hand, we have also shown that which implies that
For
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, we have
(5.8) Combining (5.7) and (5.8), we obtain that for all . If we now de, . This concludes
and all
, then fine and the proof of Theorem 5.2. Our next result yields necessary conditions for the exponen. tial stability of system Theorem 5.3: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is exponentially stable, then there exist neighborhoods and of such that and a mapping which satisfies the following conditions. i) There exist such that
ii)
iii)
, we obtain
Also, (5.9) and (5.10) imply that for all . By (5.11), we have for every that
for all and . such that for all soThere exists a constant , we have lutions for all , where and is defined in (4.4). with There exists a function and for some constant such that
for all and all . deProof: By Lemma 5.3, there exists a function fined on and a constant such that (5.9) for all
Letting
and all
, whenever and let
. Let
The proof is completed by letting . Our next result yields necessary conditions for the uniform global asymptotic stability of . Theorem 5.4: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is uniformly asymptotically stable in the large, then there exists which satisfies the following a mapping conditions. i) There exist such that
if otherwise For
, define
ii) (5.10)
Now, for
and
iii)
, we have
(5.11)
for all and . There exists a defined on such that , we have for all for all and , where is defined in (4.4). There exists a function with such that
for all
and all
.
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Proof: From the definition of uniform asymptotic stability such in the large, we know that there exists a function that
, we know that . Hence, . From (5.13) and (5.14), we have
(5.12) of system . for all solutions Next, from the definition of uniform attractivity in the large, , for every and for every , there for every , independent of , such that if exists a , then for all . We extend the domain of function by letting for all . Without loss of is strictly generality, we can also assume that the function for each fixed and strictly increasing decreasing in for each fixed , and that it is a continuous function in of which satisfies for all . Indeed, if this is not the case, we can define
Since and is strictly increasing, there exists a such that for all function and . as We now choose a Lyapunov function (5.15) Obviously,
and use as . Let be a fixed positive constant. By Massera’s lemma (see [15, Lemma 2] and [12, Lemma 19.1]), there exist two funcis the set of all contions having derivatives of any order, tinuous functions defined on each of which is uniformly bounded on any bounded subset of ), such that for for , and such that (5.13) for all and for all . The functions and are strictly increasing in and , respectively. , we define mapFor any given positive integer as follows: pings
We also define any , let
in the following way. For
. Since
such that
we have
For
such that
we have
Therefore, there exists a function such that , and, consequently, satisfies condition i) of the theorem. Next, we show that satisfies condition ii). From the aforementioned proof, we have (5.16)
(5.14) Choosing
then for
for all
, we can show that . The previous inequality is true because for any
for all . Since and are arbitrary values from the , we can change to and to for to set obtain (5.17)
with Let
. From (5.17), we have (5.18)
which shows that satisfies (ii). and We let Then, we have
We can also show that for all such that . This is true because when , for any
in (5.17).
(5.19)
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Let . Then, we know that satisfies iii). This completes the proof. Theorem 5.5: Assume that for each solution of system , its set of discontinuity points satisfies and . If the zero solution of is exponentially stable in the large, then there exists a mapping which satisfies the following conditions. i) There exist such that
, and
for all and There exists a constant , we have
assumes the form
. Assuming that for all satisfies the Lipschitz condition
and that (6.1)
for all e.g., [13])
ii)
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, we obtain for system
, the estimate (see,
(6.2)
. such that for all
and all
for all
. We assume that (6.3)
iii)
and There exists a function and such that
with for some constant
In system that
we assume that for all
and (6.4)
for all
, that (6.5)
for all and all . Proof: The proof of the previous result is similar to the proof given in Theorem 5.4 for uniform asymptotic stability in the large and is omitted. Remark 5.1: As in the case of dynamical systems determined by continuous functional differential equations, the preceding converse theorems are generally not useful in the construction of Lyapunov functions. Nevertheless, their importance cannot be overemphasized. In particular, in the present case, these converse theorems establish (under some reasonable additional hypotheses) the existence of Lyapunov functions for the various results of Section IV, and furthermore, these converse theorems show that under the given assumptions, the stability results of Section IV are as good as one can expect. Remark 5.2: The results of this section apply, with obvious by changes, when Remark 3.4 applies, replacing by (respectively, ), and so forth. VI. APPLICATIONS In this section, we apply the method of constructing Lyapunov functions proposed in Remark 4.2 in the stability analysis of four important special classes of DDS determined by functional differential equations and differential-difference equations. Example 6.1: (Time-invariant nonlinear functional differential equations) , then takes If we let the form
and letting
, that (6.6)
, and be the parameTheorem 6.1: Let ters for system given in (6.1)–(6.6). a) If for all , then the zero solution is uniformly stable. of b) If for all , where is a constant, then is uniformly asymptotically i) the zero solution of stable, in fact, uniformly asymptotically stable in the large; and is exponentially stable, in ii) the zero solution of fact, exponentially stable in the large. Proof: Using the method of constructing Lyapunov functions outlined in Remark 4.2 [see (4.16)], we choose for , which, when evaluated along assumes the form the solutions of
(6.7)
tions that
(i.e., consists of a sequence of multiple Lyapunov func, valid over the time intervals ). (Recall denotes the solution segment of the solution over the interval (see Section III). Clearly (6.8)
, where , i.e., for all . Therefore, (4.1) in Theorem 4.1 is satisfied.
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Along the solutions of
we have, in view of (6.2), that
(6.9) for
. At
, we have, in view of (6.4), that
(6.10)
Next, from (6.8) it is clear that in relation (4.12) of Theorem and from (6.12) it is clear that 4.3 we have . We have in relation (4.15) of Theorem 4.3, already shown that (4.13) of Theorem 4.3 is true, and clearly , we have as for any for . Therefore, all conditions of Theorem 4.3 are satisfied is exponentially and we conclude that the zero solution of stable and exponentially stable in the large. Example 6.2: (Time-invariant linear functional differential equations) , then takes the form If we let
Combining (6.9) and (6.10), we have
(6.13) (6.11)
and since by assumption
and
assumes the form
, we have (6.14) . For each
is defined, as in (2.3), by (6.15)
Since this holds for arbitrary , is nonincreasing. Next, for (6.9) we have, recalling that , that
, it follows that and
We suppose that all assumptions that we made for given in (2.3) hold as well for . Then, is Lipschitz continuous on with Lipschitz constant less or equal to the variation , and as such, condition (6.1) still holds for (6.13). The of consists of all solutions of the equation (see, spectrum of e.g., [13]) (6.16)
, where . Therefore, all conditions of Theorem 4.1(a) are satisfied and we conclude that is uniformly stable. the zero solution of , we have If in (6.11) we assume that
When all solutions of (6.16) satisfy the relation then for any positive , there is a constant such that the solutions of (6.13) allow the estimate
,
(6.17) for all and (see, e.g., [13]). When the aforementioned assumption is not true, then the solutions of (6.13) still allow the estimate (see, e.g., [13])
and
(6.18) for all
and
. Thus, in all cases, we have (6.19)
(6.12) for all
. In (6.12), we have , i.e., . Therefore, all conditions of Theorem 4.1(b) are satisfied, and the zero solution of is uniformly asymptotically stable. and since actually, Since (6.8) holds for all , and since (6.12) is true for all , it follows is uniformly from Theorem 4.2 that the zero solution of asymptotically stable in the large.
for all and when (6.18) applies and , when (6.17) applies. Finally, for each to satisfy the condition
, where
and and in (6.14) is assumed (6.20)
for all
, where
is a constant.
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In the following result, we assume that the relations (6.3) and . (6.6) still hold, and we assume that , and be the paramTheorem 6.2: Let eters for (6.14) defined previously. a) If for all , then the zero solution of (6.14) is uniformly stable. , where b) If for all is a constant, then i) the zero solution of (6.14) is uniformly asymptotically stable, in fact, uniformly asymptotically stable in the large; and ii) the zero solution of (6.14) is exponentially stable, in fact, exponentially stable in the large. Proof: The proof is similar to the proof of Theorem 6.1. Remark 6.1: System (3.1) (see the example in Section III-A) is clearly a special case of system (6.13), and as such, Theorem 6.2 is directly applicable in the analysis of (3.1). Example 6.3: (Time-invariant nonlinear differential-difference equations) We now consider a family of initial-value problems described by differential-difference equations of the form
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Similarly as in (6.1)–(6.6), we assume that , and , where
.
Theorem 6.3: Let , and be the parameters for (6.22). , then the zero a) If for all solution of (6.22) is uniformly stable. , where b) If for all is a constant, then i) the zero solution of (6.22) is uniformly asymptotically stable, in fact, uniformly asymptotically stable in the large; and ii) the zero solution of (6.22) is exponentially stable, in fact, exponentially stable in the large. Proof: As in the proof of Theorem 6.1, we choose . Then, (6.8) still holds with . Also, in accordance with (6.25), we have along the solutions of (6.22), the estimate (6.27) . From (6.22) and (6.26), we have
(6.21) , and a DDS determined by (6.22) where
, and
(6.28) and, therefore, using (6.27) and (6.28) and assuming that , we have
(6.23) Defining
and ( and ), and , then (6.21) and (6.22) are special cases of in (6.21) satisfies the respectively. If we assume that each Lipschitz condition
, i.e., , is nonincreasing. Also, in the present case we have, in view of (6.27)
(6.24) for all , then, by invoking the Gronwall Inequality (see, e.g., [3]), we obtain for (6.21) the estimate (6.25) and for all for all We will assume that for each and
.
(6.26) for all
.
, where , and and . The rest of the proof is identical to the Proof of Theorem 4.1. We conclude from Theorem 6.1 that the zero solution of (6.22) , and when is uniformly stable when , we conclude from Theorem 4.1 that the zero solution of (6.22) is uniformly asymptotically , we conclude stable. Also, when from Theorem 4.2 that the zero solution of (6.22) is uniformly asymptotically stable in the large, and from Theorem 4.3, that the zero solution of (6.22) is exponentially stable and exponentially stable in the large.
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Example 6.4: (Time-invariant linear differential-difference equations) A special case of (6.21) are differential-difference equations of the form (6.29) , where and case of (6.22) are DDSs determined by
. Also, a special
b) i)
ii)
Proof: The proof is similar to the proof of Theorem 6.1. Remark 6.2: a)
(6.33)
It is emphasized that the results in Theorems 6.1–6.4 involve available system parameters, including Lipschitz constants, time intervals between discontinuities, properties of the spectrum of the linear operators describing the systems, and properties of the transition (e.g., their “gains”) at the instants of functions discontinuity. , are not preWhen the instants of discontinuity, cisely known, then the results of this section may be modified, involving estimates of the largest time interval between instants of discontinuity. For example, in the case of Theorem 6.1, we would in this case reby ( are defined place in Theorem 6.1). The results of the present section allow that stable and asymptotically stable discontinuous dynamical systems may exhibit unstable behavior over some or all of the open time intervals between the points of disconti. Specifically, if e.g., in Example 6.2, nuity, condition (6.17) is applicable over , then the norm of the system’s solution will decrease exponentially, while if condition (6.17) is not applicable (so that condition (6.18) is applicable), the norm of the system’s solution may actually increase, yet the zero solution of system (6.14) will still be uniformly stable, respectively, uniformly asymptotically stable, as long as the conditions of Theorem 6.2 are satisfied (refer also to the last paragraph in Remark 4.1). This type of behavior is not possible in continuous dynamical systems.
. Thus, we have in all
Remark 6.3: Consistent with the variant to system given in (3.12) (refer to Remark 3.6), we modify system (6.30) as
(6.34)
(6.35)
(6.30)
where and is defined similarly as in (6.23). As shown in Section II, (6.29) is also a special case of the time-invariant linear functional differential (6.13), and (6.30) is a special case of the DDS described by the functional differential equations given in (6.14). In the present case, the spectrum for (6.29) is determined by the equation (6.31) When all solutions of (6.31) satisfy the relation , there is a constant then for any positive such that the solutions of (6.29) allow the estimate
,
(6.32) and (see, e.g., [2]). When the previous for all assumption is not true, then (noting that (6.29) is Lipschitz con( is the norm of induced by the tinuous with vector norm )), we obtain the estimate
and for all for all cases the estimate
for all
Let let
, where If for all is a constant, then the zero solution of (6.30) is uniformly asymptotically stable, in fact, uniformly asymptotically stable in the large; and the zero solution of (6.30) is exponentially stable, in fact, exponentially stable in the large.
and , where when (6.33) applies and , when (6.32) applies.
and and
b)
c)
where all symbols are defined as in (6.30). A simple computation shows that in this case for each
, , and assume that and .
Theorem 6.4: Let and be the parameters for (6.30). a) If for all , then the zero solution of (6.30) is uniformly stable.
Letting , it is easily shown that all conclusions of Theorem 6.4 apply to system (6.35) if we replace in by , assuming that for Theorem 6.4 , the estimate (6.32) applies. all
SUN et al.: STABILITY OF DISCONTINUOUS RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
REFERENCES [1] V. I. Zubov, Methods of A. M. Lyapunov and their Applications. Groningen, The Netherlands: Noordhoff, 1964. [2] W. Hahn, Stability of Motion. Berlin, Germany: Springer-Verlag, 1967. [3] A. N. Michel, K. Wang, and B. Hu, Qualitative Analysis of Dynamical Systems, 2nd ed. New York: Marcel Dekker, 2001. [4] H. Ye, A. N. Michel, and L. Hou, “Stability theory for hybrid dynamical systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 461–474, Apr. 1998. [5] M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 475–482, Apr. 1998. [6] A. N. Michel, “Recent trends in the stability analysis of hybrid dynamical systems,” IEEE Trans. Circuits Syst I, Fundam. Theory Appl., vol. 46, no. 1, pp. 120–134, Jan. 1999. [7] A. N. Michel and B. Hu, “Toward a stability theory of general hybrid dynamical systems,” Automatica, vol. 35, pp. 371–384, Apr. 1999. [8] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Syst. Mag., vol. 19, no. 5, pp. 59–70, 1999. [9] R. DeCarlo, M. Branicky, S. Pettersson, and B. Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proc. IEEE, vol. 88, no. 7, pp. 1069–1082, Jul. 2000. [10] D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effects: Stability, Theory and Applications. New York: Halsted, 1989. [11] N. N. Krasovskii, Stability of Motion. Stanford, CA: Stanford Univ. Press, 1963. [12] T. Yoshizawa, Stability Theory by Lyapunov’s Second Method. Tokyo, Japan: Math. Soc. Japan, 1966. [13] J. K. Hale, Functional Differential Equations. Berlin, Germany: Springer-Verlag, 1971. [14] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Cambridge, MA: Academic, 1993. [15] J. L. Massera, “Contributions to stability theory,” Ann. Math., vol. 64, pp. 182–206, 1956. [16] A. N. Michel and Y. Sun, “Stability analysis of discontinuous dynamical systems determined by semigroups,” IEEE Trans. Autom. Control, 2005, to be published.
Ye Sun received the B.S. degree in mathematics from the University of Science and Technology of China, and the M.S. and Ph.D. degrees from the University of Notre Dame, Notre Dame, IN, in 1999, 2002, and 2004, respectively. She is currently a Systems Analyst with Credit Suisse First Boston, New York, working on fixed-income models. Her research interests include systems modeling and qualitative analysis of discontinuous dynamical systems.
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Anthony N. Michel (S’55–M’59–SM’79–F’82) received the Ph.D. degree in electrical engineering from Marquette University, Milwaukee, WI, and the D.Sc. degree in applied mathematics from the Technical University of Graz, Graz, Austria. He has seven years of industrial experience and was on the Electrical Engineering Faculty at Iowa State University, Ames, for 16 years. In 1984, he joined the University of Notre Dame, Notre Dame, IN, as Chair of the Department of Electrical Engineering. In 1988, he became Dean of the College of Engineering, a position he held for ten years. He is currently Frank M. Freimann Professor Emeritus and Matthew H. McCloskey Dean of Engineering Emeritus at Notre Dame. He has also held visiting faculty positions at the Technical University of Vienna, Vienna, Austria, the Johannes Kepler University, Linz, Austria, and Ruhr University, Bochum, Germany. He is the author or coauthor of eight books and numerous archival publications. His more recent work is concerned with stability analysis of finite- and infinite-dimensional dynamical systems and qualitative analysis and synthesis of recurrent neural networks. Dr. Michel has rendered substantial service to several professional organizations, especially the IEEE Circuits and Systems Society and the IEEE Control Systems Society. He has been honored by a number of prestigious awards for his work as an educator and scholar.
Guisheng Zhai (M’98–SM’04) was born in Hubei, China, in 1967. He received the B.S. degree from Fudan University, China, in 1988, and the M.E. and the Ph.D. degrees, both in system science, from Kobe University, Kobe, Japan, in 1993 and 1996, respectively. From 1996 to 1998, he worked in the Kansai Laboratory of OKI Electric Industry Co., Ltd., Japan. From 1998 to 2004, he was a Research Associate in the Department of Opto-Mechatronics, Wakayama University, Japan. He also held a visiting research position in the Department of Electrical Engineering, the University of Notre Dame, Notre Dame, IN, from August 2001 to July 2002. In April 2004, he joined the faculty of Osaka Prefecture University, Osaka, Japan, where he currently is an Associate Professor of Mechanical Engineering. His research interests include large scale and decentralized control systems, robust control, switched systems and switching control, networked control systems, neural networks and signal processing, etc. He is a Member of ISCIE, SICE, and JSME.
