4. Stability analysis of discontinuous dynamical systems determined ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005
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Stability Analysis of Discountinuous Dynamical Systems Determined by Semigroups Anthony N. Michel, Life Fellow, IEEE, Ye Sun, and Alexander P. Molchanov, Senior Member, IEEE
Abstract—We present Lyapunov stability results for discontinuous dynamical systems (DDS) determined by linear and nonlinear semigroups defined on Banach space. DDS of the type considered herein arise in the modeling of a variety of finite- and infinitedimensional systems, including certain classes of hybrid systems, discrete-event systems, switched systems, systems subjected to impulse effects, and the like. We apply our results in the analysis of several important specific classes of DDS. Index Terms—Asymptotic stability, 0 -semigroups, discontinuous dynamical systems (DDS), exponential stability, functional differential equations, heat equation, Lyapunov stability, nonlinear semigroups, partial differential equations.
I. INTRODUCTION
D
ISCONTINUOUS dynamical systems (DDS) have motions which are not continuous with respect to time. Such systems arise in the modeling process of a variety of systems, including hybrid dynamical systems, discrete event systems, switched systems, systems subjected to impulse effects, and the like (see, e.g., [2], [3], [6], [12]–[15], [19], and the references cited therein). The stability analysis of specific classes of such systems has thus far been concerned primarily with finite dimensional dynamical systems (defined on ) determined by ordinary differential equations, and more recently, ) with infinite-dimensional dynamical systems (defined on determined by functional differential equations [18]. The results that were established in [14], [15], and [19] are formulated for general dynamical systems in a metric space setting, and as such, are in principle applicable to finite dimensional dynamical systems as well as to infinite dimensional dynamical systems. However, in the latter case, the application of these results to specific classes of infinite dimensional systems is usually not straightforward and frequently requires further analysis [18]. (This is usually also the case for continuous dynamical systems (see, e.g., [8], [15], and [20]). In this paper, we establish (Lyapunov) stability results for DDS determined by linear and nonlinear semigroups defined on Banach spaces. We present general results which do not require determination of Lyapunov functions, as well as results which do involve Lyapunov functions. Our results are very general Manuscript received April 3, 2003; revised February 26, 2004. Recommended by Associate Editor S.-I. Niculescu. A. N. Michel is with the Department of Electrical Engineering, the University of Notre Dame, Notre Dame, IN 46554 USA (e-mail: [email protected]). Y. Sun is with Credit Suisse First Boston, New York, NY 10010 USA (e-mail: [email protected]). A. P. Molchanov, deceased. Digital Object Identifier 10.1109/TAC.2005.854582
and are applicable to large classes of finite- and infinite-dimensional DDS. We demonstrate the applicability of our results in the analysis of DDS determined by linear and nonlinear retarded functional differential equations and a specific initial-value and boundary-value problem governed by the heat equation. The remainder of this paper is organized as follows. In Section II, we provide essential background material on linear and nonlinear semigroups. In Section III, we formulate the classes of DDS determined by linear and nonlinear semigroups considered in this paper. In Section IV, we establish stability results for the classes of DDS considered herein and in Section V we apply these results in the analysis of several important specific classes of DDS. We conclude the paper in Section VI with appropriate comments. II. NOTATION AND BACKGROUND MATERIAL In this section, we provide essential background material concerning dynamical systems determined by semigroups. A. Notation , let Let denote real -space, and let denote any one of the . For a real matrix (i.e., equivalent norms on ) and , let denote the norm of induced by the vector norm . Let and be Banach spaces and let denote norms on Banach spaces. Let be a linear operator defined on a dowith range in . We call bounded if it main into a bounded subset of , or maps each bounded set in equivalently, if it is continuous. Given a bounded linear oper, its norm is defined by ator . We let denote the identity transformation. . Then, signifies that Finally, let belongs to the set of continuous functions from into . B.
-Semigroups
We will require the following concepts (see, e.g., [9]–[11]). Definition II.1: A one-parameter family of bounded linear operators , is said to be a -semigroup (or a linear semigroup) if i) ii) for any ; for all . iii) -semigroups are generated by linear operators.
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Definition II.2: The infinitesimal generator -semigroup is defined as the opof a for erator where is the domain of given by . The Hille–Yoshida–Phillips Theorem provides necessary and sufficient conditions for a linear operator to be the infinitesimal generator of some -semigroup. We refer the reader to [10] or to [15, p. 76] for a statement of this theorem. The following results which provide some of the basic properties of -semigroups will be of particular interest to us. , there exists an Theorem II.1 [10]: For a -semigroup and an such that (2.1) for all The qualitative properties of a -semigroup in terms of its infinitesimal generator is given of the spectrum by the following result. is called differentiable Definition II.3: A -semigroup if for each is continuously differentiable for . on Theorem II.2 [17]: If is a -semigroup which is differentiable for , if is its infinitesimal generator, and if for all , then given any positive , such that there is a constant (2.2) for all
.
C. Nonlinear Semigroups is a family of linear opIn the case of -semigroups, erators. This restriction is removed in the following. Definition II.4 [4], [5], [10]: Assume that is a subset of a Banach space . A family of one-parameter (nonlinear) operators , is said to be a nonlinear semigroup defined on if i) for ii) for ; iii) is continuous in on . As in the case of -semigroups, nonlinear semigroups are also generated by operators. Definition II.5: A (possibly multivalued) operator is said on if to generate a nonlinear semigroup
Of particular importance in applications are quasicontractive is and contraction semigroups. A nonlinear semigroup called a quasicontractive semigroup if there is a number such that (2.3) and for all . If in (2.3), , then for all is called a contraction semigroup. For a set of sufficient conditions under which an operator generates a quasi-contractive semigroup, refer to [10] or to [15, p. 78]. D. Continuous Dynamical Systems Determined by Semigroups -semigroup For a given with initial state by
, we define the motion and initial time (2.4)
and we define the dynamical system determined by a as the family of motions group
-semi-
(2.5) , and in particular, when We note that then for all . We will call an and we will call the equilibrium for the dynamical system the trivial motion. corresponding motion For a given nonlinear semigroup , we define the motion with initial state and initial time by (2.6) and we define the dynamical system determined by a nonlinear as the family of motions semigroup
(2.7) is in the interior Henceforth, we will always assume that of . . Throughout Once more we note that this paper, we will assume that for any nonlinear semigroup for all if . We will an equilibrium for the dynamcall ical system and we will call the corresponding motion , a trivial motion. III. DDS DETERMINED BY SEMIGROUPS
for all semigroup
. The infinitesimal generator is defined by
of a nonlinear
To motivate the class of DDS 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
for all such that this limit exists. The operator and the inare generally different operators. finitesimal generator
In Fig. 1, we depict in block diagram form a configuration which is applicable to many classes of DDS, including hybrid
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Fig. 1. DDS configuration.
systems and switched systems. There is a block which contains continuous-time dynamics, a block which contains phenomena that evolve at discrete points in time (discrete-time dynamics) or at discrete events, and a block which contains interface elements for the above two system components. The block which contains the continuous-time dynamics is usually characterized in the existing current literature by ordinary differential equations while the block on the right in Fig. 1 is usually characterized by difference equations, or it may involve other types of discrete characterizations, e.g., Petri nets, logic commands, various types of discrete event systems, and the like. The block labeled interface elements may vary from the very simple to the very complicated. At the simplest level, this block involves samplers and sample and hold elements. The sampling process may involve only one uniform rate, or it may be nonuniform (variable rate sampling), or there may be several different (uniform or nonuniform) sampling rates occurring simultaneously (multi-rate sampling). Perhaps the simplest specific example of the above class of systems are sampled-data control systems described by the equations
(3.1) where sampling instants, priate dimensions,
denotes are real matrices of approare interface variables, and
and .
Now, define at
and
, and . Then, for all
at . Let
,
.
, let Next, for solution of the initial-value problem
, denote the unique
(3.3) by . Using the properties of the so, is a lutions of (3.3), it is easily shown that -semigroup. Given a set of initial conditions , the unique solution of (3.2) [and, hence, of (3.1)] can now be expressed by the function and define the mapping
(3.4) Consistent with the terminology used later, we refer to as a motion. By varying over , we generate a family of motions, , the dynamical system , and in determined by (3.4). We note that , then for all . We particular, when an equilibrium and the trivial motion. call We conclude the present example with an observation. As noted earlier, in the current literature, the continuous-time dynamics in Fig. 1 are almost always assumed to be determined by finite dimensional systems (lumped parameter systems) described by ordinary differential equations. However, in many applications, the continuous dynamics may more appropriately be determined by infinite dimensional systems, to capture the effects of time delays, hysteresis phenomena, distributed parameters, and the like. We will present in the present section system models which allow the continuous-time dynamics in Fig. 1 to be represented by finite-dimensional as well as infinite-dimensional systems. B. DDS Determined by Linear and Nonlinear Semigroups
Letting
where denotes the identity matrix, then system (3.1) can be described by the discontinuous differential equation
(3.2)
In the following, we will require a given collection of linear or defined on a Banach space nonlinear semigroups , or on a set , respectively; and a given collection of linear and continuous operators , or of nonlinear and continuous operators ; and a given discrete, infinite, and unbounded set . We assume that when consists of linear semigroups, then consists of linear mappings. The number of elements in and may be finite or infinite.
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We now consider dynamical systems whose motions with initial time and initial state (respectively, ) are given by
(3.5) We define the DDS determined by semigroups as
(3.6) is unique, with Note that every motion , exists for all , and is continuous with respect to on , and that at may be discontinuous. We call the set the set of discontinuities for the motion . consists of -semigroups and conWhen in (3.6), sists of linear mappings, we speak of a DDS determined by . Simlinear semigroups, and we denote this system by consists of nonlinear semigroups, we ilarly, when in (3.6), speak of a DDS determined by nonlinear semigroups and we . When the types of the elements in denote this system by and are not specified, we simply speak of a DDS determined by semigroups and we denote this system, as in (3.6), by . , and are linear Since in the case of for all operators, it follows that in particular . We call the equilibrium for the DDS and , the trivial motion. , we assume that is in the interior In the case of for all if and that of and that if for all . From this, it follows that for all if . We call an , a trivial motion for the equilibrium and . DDS Finally, in the case of linear mappings, we use in (3.5) the and notation . We conclude the present section with a few remarks. , reRemark III.1: For different initial conditions , we allow the set of sulting in different motions , the set of semigroups discontinuities , and the set of functions to differ, and accordingly, the notation , and might be more appropriate. However, since in all cases, all meaning will be clear from context, we will not use such superscripts. Remark III.2: To the best of our knowledge, the DDS models and ) are original and have not considered herein ( been considered previously. These models are very general and include large classes of finite dimensional dynamical systems determined by ordinary differential equations and inequalities and by large classes of infinite dimensional dynamical systems
determined by differential-difference equations, functional differential equations, Volterra integrodifferential equations, certain classes of partial differential equations, and more generally, differential equations and inclusions defined on Banach space. In contrast, most of the existing literature concerning stability of DDS (including hybrid systems, switched systems, systems subjected to impulse disturbances, and so forth) is confined to finite-dimensional systems determined by ordinary differential equations (e.g., [2], [3], [6], [12], and [13]). Remark III.3: The dynamical system models and are very flexible, and include as special cases, many of the DDS considered thus far in the literature, as well as general autonomous continuous dynamical systems: a) If for all ( has only one element) and if for all , where denotes the identity transformation, then reduces to an autonomous, linear, continuous dynamical system and to an autonomous, nonlinear, continuous dynamical system; b) in the case of dynamical systems subjected to impulse effects [considered in the literature for finite dimensional systems (see, for all while the e.g., [2])], one would choose impulse effects are captured by an infinite family of functions ; c) in the case of switched systems, frequently only a finite number of systems that are being switched is required, and so in this case one would choose a finite family of (see, e.g., [6] and [12]); and so forth. semigroup Remark III.4: Perhaps it needs pointing out that even though and are determined by families of semisystems groups (and nonlinearities), by themselves they are not semigroups, since in general, they are time-varying and do not satisfy the hypotheses i)–iii) given in Definitions II.1 and II.4. How, used in describing ever, each individual semigroup or , does possess the semigroup properties, albeit, only . over a finite interval IV. STABILITY RESULTS FOR DDS DETERMINED BY SEMIGROUPS In this section, we establish several stability results for discontinuous dynamical systems determined by linear and nonlinear semigroups. Before stating and proving these results, we give the definitions of the various stability concepts that we will employ. A. Qualitative Characterization of DDS Recall that the DDS determined by linear semigroups, is defined on a Banach space while the nonlinear DDS given is defined on . Recall also that the origin 0 is by assumed to be in the interior of and that is an equiand . Since the following definitions librium for both and , we will refer to either one of pertain to both them simply as . Definition IV.1: The equilibrium of is stable if for and every , there exists a every such that for all of for all , (and ). The equilibrium is whenever uniformly stable if is independent of , i.e., . The of is unstable if it is not stable. equilibrium
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
Definition IV.2: The equilibrium such that there exists an
of
is attractive if
(4.1) for all of whenever (and ). We call the set of all such that (4.1) holds the domain . of attraction of of is asymptotiDefinition IV.3: The equilibrium cally stable if it is stable and attractive. Definition IV.4: The equilibrium of is uniformly and every , there exists a attractive if for every , independent of and , and a , independent for all and for all of , such that of , whenever (and ). Definition IV.5: The equilibrium of is uniformly asymptotically stable if it is uniformly stable and uniformly attractive. Definition IV.6: The equilibrium of is exponentially , and for every and every stable if there exists , there exists a such that for all and for all of whenever (and ). The preceding concern local characterizations of an equilibrium. In the following, we address global characterizations. In . this case, we will find it convenient to let Definition IV.7: The equilibrium of is asymptotically stable in the large if i) it is stable, and of and for all ii) for every , (4.1) holds. In this case, the domain of attraction of is all of . of is uniformly Definition IV.8: The equilibrium asymptotically stable in the large if i) it is uniformly stable, and ii) it is uniformly attractive in the large, i.e., for every and every , and for every , there exists (independent of ), such that if a , then for all of for all . Definition IV.9: The equilibrium stable in the large if there exists such that there exists
of is exponentially and for every ,
(4.2) of , for all , whenever . for all Although system and are determined by semigroups, they themselves are not semigroups and in general, they are time-varying systems. However, in the following we identify an assumption under which the motions of these systems exhibit the time-invariance property. or Assumption IV.1: Assume that any two motions (of ) with identical initial states but different initial times, say and , are determined by identical and operators . sequences of semigroups
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Furthermore, assume that if for the motion , then for the motion , we have , where . By applying definitions, it is easily shown that under Assumpand have the proption IV.1, the motions of systems for all and erty that , i.e., the motions possess the time-invariance for every property. It has been shown (see, e.g., [4], [5], [9]–[11], and [15]) that for dynamical systems whose motions have the time-invariance , is stable (respectively, property, an equilibrium, say asymptotically stable, asymptotically stable in the large) if and only if it is uniformly stable (respectively, uniformly asymptotically stable, uniformly asymptotically stable in the large). From the above observations, the following result follows readily. Proposition IV.1: Under Assumption IV.1, the following and : statements are true for systems is stable if and only if it is a) the equilibrium uniformly stable; is attractive if and only if it is b) the equilibrium uniformly attractive; is asymptotically stable if and c) the equilibrium only if it is uniformly asymptotically stable; is asymptotically stable in the d) the equilibrium large if and only if it is uniformly asymptotically stable in the large. . We emphasize that in all subsequent results, Assumption IV.1 is not required, and the distinction between stability and uniform stability (respectively, between asymptotic stability and uniform asymptotic stability) is required. B. Principal Stability Results In our first results, we establish sufficient conditions for various stability properties for system . We will assume in there exist these results that for each nonlinear semigroup and and for each mapping constants there exists a constant such that (4.3) and (4.4) for all . We recall from Section 2-C (see (2.3)) that in particular, (4.3) is always satisfied for a quafor some computable paramesicontractive semigroup , and , while for a contractive ters , inequality (4.3) is satisfied with and semigroup . We will require the following additional notation. Let , for any given and , we let , and we let and denote the finite products (4.5)
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Theorem IV.1: a) For system , under (4.3) and (4.4), assume that for there exists a constant such that any
is true for . and Therefore, by (4.5) and (4.10), we have
(4.6)
b)
c)
d)
e)
(4.11)
, where is defined in (4.5). Then the for all equilibrium of is stable. , i.e., in (4.6) can be chosen If in part a), , then the equilibrium of independent of is uniformly stable. If in part a), (4.6) is replaced by (4.7) for all , then the equilibrium of is asymptotically stable. If the conditions of part b) are satisfied and if in part c) relation (4.7) is satisfied uniformly with respect to (i.e., for every and every there , independent of , such that exists a for all ), then the equilibrium of is uniformly asymptotically stable. Assume that in part a) (4.6) is replaced by
b)
c)
(4.8) where
and
. Assume also that (4.9)
is a constant. Then, the equilibrium where of is exponentially stable. f) If in parts c)–e), respectively, conditions (4.3) and (4.4) , then the equilibrium of hold for all is asymptotically stable in the large, uniformly asymptotically stable in the large and exponentially stable in the large, respectively. Proof: a) For system , with , we associate each interval with the index . We will find it convenient to employ a relabeling of , where indexes. To this end, let denotes the integer part of , and let . Then, we can relabel as and as . and , If we have for . Therefore, in view of (4.5) (4.10) is
true.
It
is
clear
that
. Similarly, for , if
, then
,
d)
e)
For any and , let . From (4.6) and (4.11), it now follows that , whenever and . Since and since for all and we can equate , all it follows that the equilibrium of is stable. can In proving part b), note that , and consequently, be chosen independent of can also be chosen independent of . Therefore, the equiof is uniformly stable. librium it From the assumption on follows that . Hence, as . Since for any we have for some , then when . Hence, it follows from (4.7) and (4.11) that (4.1) holds for all of whenever . of is attractive Therefore, the equilibrium and its domain of attraction coincides with the entire . Since (4.6) follows from (4.7), then, as in set part a), of is stable. Hence, the equilibrium of is asymptotically stable. Since the conditions of part b) are satisfied, the equilibof system is uniformly stable. Thererium fore, we only need to prove that is uniformly attractive. in such a way that Choose . Since (4.7) is satisfied uniformly with respect to , then for every and every there exists a (independent of ) for all . Hence, from such that for all (4.11), we have and for all . Let . and for Then, . If we let , then all for all and we have that for all of , whenever . Hence, of is uniformly attractive the equilibrium and uniformly asymptotically stable. To prove part e), note that as was shown in the proofs of and any , there parts a) and c), for any and such that and exists (4.11) holds. Since and in view of (4.9), , and therefore, we have . Hence, in . view of (4.8), we have , let . Then, for any For any with , we have , where . Therefore, the equilibrium of is exponentially stable.
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
f)
We note that if the estimates (4.3) and (4.4) hold for all , then inequality (4.11) is valid for all . i) Repeating the reasoning in the proof of part c) for any and any , we can conclude that in this of whenever case (4.1) holds for all and . Therefore, the equilibrium of is asymptotically stable in the large. ii) Similarly as in the proof of part d), for every and for every there exists a (independent of ), such that for all . If we let , then we have that for all and for all of , whenever . of is uniformly Hence, the equilibrium asymptotically stable in the large. and for every we have iii) For every similarly as in the proof of part e) that for , where . Let all . It now follows that the equilibrium of is exponentially stable in the large. This completes the proof. Corollary IV.1: a) For system , assume that the following statements are true. ). i) Condition (4.3) holds (with parameters ii) Condition (4.4) holds (with parameter ): ; iii) for all and iv) for all where and are constants; v) for all
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b)
c)
In view of (4.13) the estimate (4.8) is true with and . Therefore, the limit relation (4.7) is satisfied uniformly with respect to . This proves part b) of the corollary. The conclusions of part c) of this corollary follow directly from part f) of Theorem IV.1.
From Theorem II.1, we recall that for any -semigroup , there will exist and such that (4.14) is a Furthermore, in accordance with Theorem II.2, if -semigroup which is differentiable for , if is its infor all , finitesimal generator, and if , there is a constant then given any positive such that (4.15) Similarly as in Theorem IV.1, we will utilize in our next result the relation (4.16) where, depending on the situation on hand the constants are obtained from either (4.14) or (4.15). Similarly as in (4.5), we define in the case of DDS finite products
and the
(4.17) (4.12)
b)
of is stable and Then, the equilibrium uniformly stable. If in part a), hypothesis v) is replaced by (4.13)
c)
a)
, where , then the equilibrium for all of is asymptotically stable, uniformly asymptotically stable and exponentially stable. If in part a) it is assumed that inequalities (4.3) and and inequality (4.12) is re(4.4) hold for all of placed by (4.13), then the equilibrium is asymptotically stable in the large, uniformly asymptotically stable in the large, and exponentially stable in the large. Proof: It is easily shown that in part a) the estimate (4.6) is , indepensatisfied with . Therefore, the conditions in part a) and dent of b) of Theorem IV.1 are satisfied. This proves part a) of the corollary.
, denotes the norm of the bounded linear where used in defining the DDS in (3.5). operator By invoking Theorem IV.1, we obtain in a straightforward . manner the following result for system Corollary IV.2: In Theorem IV.1, replace (4.3) by (4.14)–(4.16) and (4.5) by (4.17). Then, all conclusions of . Theorem IV.1 are true for system Corollary IV.3: In Corollary IV.1, replace (4.12) and (4.13) and , respecby denotes operator norm and where and tively, where are given in (4.16). Then, the conclusions of Corollary IV.1 are . true for system Remark IV.1: Corollaries IV.1 and IV.3 are more conservative than Theorem IV.1 and Corollary IV.2 since in the case of the latter we put restrictions on partial products [see, e.g., (4.6)] while in the case of the former, we put corresponding restrictions on the individual members of the partial products [see, e.g., (4.12)]. However, Corollaries IV.1 and IV.3 are easier to apply than Theorem IV.1 and Corollary IV.2. Remark IV.2: For linear, finite-dimensional DDS given by
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where and , Corollary IV.3 can be improved somewhat by replacing the inequalities in that corollary, e.g., by the inequalities
for all
, where
(4.21)
and
assuming that , where [see, e.g., [7] for the definition of the measure of by the inequalities
denotes ], or
c)
d)
We omit the proofs of the aforementioned assertions due to space limitations. In the preceding results, we relied on fundamental definitions and to establish various stability properties for systems . It turns out that we can also obtain stability results for and , making use of Lyapunov functions. In systems doing so, we employ the following comparison functions. is said to beDefinition IV.10: A function ), if and if is strictly long to class , (i.e., . If and if , we say increasing on that belongs to class (i.e., ). Since the following results are applicable to both system and , we will use the term “system ” to indicate either system. Theorem IV.2: Assume that there exists a function and functions such that
then the equilibrium of system is uniformly asymptotically stable. Assume that all assumptions in parts a) and (b) are true and . Then, the equiwith of system is uniformly asymptotically librium stable in the large. Assume that all assumptions in parts a) and b) are true , and , with , and are positive constants. Furtherwhere more, assume that the function in (4.19) satisfies as
(4.22)
where is some positive constant. Then, the equilibrium of system is exponentially stable. e) Assume that all assumptions in part d) are true with . Then, the equilibrium of system is exponentially stable in the large. Proof: , then for any a) Since is continuous and there exists such that as long as . We assume that , and . Thus, for any , is satas long as the initial condition isfied, then
(4.18) and , where is a neighborhood of for all the origin . a) Assume that there exists a neighborhood of such that for every motion the origin of , with for all and is continuous for all except on a set . Also, assume that is nonincreasing for all and all , and assume that there exists a function , independent of , such that
and since more, for any
for , is nonincreasing. Further, we can conclude that
and
b)
Therefore, by definition, the equilibrium of system is uniformly stable. , we obtain from the Letting assumption of the theorem, that for
(4.19)
b)
. for all of system is uniThen, the equilibrium formly stable. If in addition to the assumptions in part a), there exists such that a function (4.20)
If we let , then inequality becomes Since is nonincreasing and thus obtain that . It follows that all
and the previous . , it follows that for all . We for
(4.23)
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
Now, consider a fixed . For any given such that can choose a
, we
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and
Let We obtain from
. and that
(4.24) since and . For any with and any , we are now able to show that whenever . The aforementioned statement is true because for any must belong to some interval for some . Therefore, we know that . It follows from (4.23) that , which implies that
(4.25)
which yields (4.27) . If where , then and and all . Thus, following, we assume that Since
is true for some for all for all
. In the for all . , it follows from (4.27) that
Hence
and
(4.26) , it follows from (4.25) that , noticing that (4.19) holds. In the , we can conclude from (4.26) . This proves that the equilibof system is uniformly asymptotically
In the case when
c)
case when that rium stable. of system is From part a), the equilibrium uniformly stable. We need to show that the equilibrium is uniformly attractive in the large. For any fixed and , we can choose and a a such that . Let and . For any with and any , we can show that whenever . This is true , we can find some since for any such that . Therefore, and , which implies that
is true for all that
. It now follows from . In the last step, we . From (4.22), it is . Let
have made use of the fact that easily seen that
Then, for all relations (4.19) and (4.22) that for all that
. It follows from , it is true
The last inequality follows since . Thus, . For any
and any
such that
, let
and Thus, we have
d)
and when . , we can conclude In the case when . This proves that from above that the equilibrium of system is uniformly asymptotically stable in the large. We only prove the case of global exponential stability. The case of local exponential stability can be proved similarly.
Then
for all of system
of and . Therefore, the equilibrium is exponentially stable in the large.
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We close by noting that in the previous proofs, we did not require the semigroup property ii) in Definitions II.1 and II.4 and, therefore, the present results will also be applicable to appropriately formulated DDS without this property. We conclude the present section with a few remarks. Remark IV.3: The general stability results concerning DDS reported in the current literature (e.g., [2], [3], [6], [12], and [13]) pertain essentially all to finite dimensional systems (“lumped parameter” systems) described by ordinary differential equations. These results are not applicable in the analysis of infinite-dimensional DDS which are capable of capturing the effects of distributed parameters (as in the case of partial differential equations), transportation and time delays (as in the case of differential-difference equations), hysteresis effects (as in the case of functional differential equations and Volterra integrodifferential equations), and the like. In contrast to this, the results of the present section are applicable in the stability analysis of finite-dimensional DDS as well as infinite dimensional DDS of this kind. Remark IV.4: The general stability results for DDS reported in the current literature (e.g., [2], [3], [6], [12]–[14], and [19]) are usually Lyapunov-type results, requiring the determination of appropriate Lyapunov functions, which is not necessarily an easy task. In contrast to this, Theorem IV.1 and Corollaries IV.1–IV.3 do not involve the existence of Lyapunov functions, simplifying their applications considerably. Instead of invoking the Lyapunov approach, we bring to bear the very extensive qualitative theory of semigroups in establishing these results. Remark IV.5: As pointed out in Remark IV.4, the Lyapunovtype results given in Theorem IV.2 are in general more difficult to apply than Theorem IV.1 and Corollaries IV.1, IV.2, and IV.3. However, the very ambiguity involved in the search for suitable Lyapunov functions offers flexibility in the application of Theorem IV.2 in efforts of reducing conservatism of results. In Section VI, we propose a systematic method of constructing Lyapunov functions for the classes of DDS considered herein. Remark IV.6: In the following comments, we compare the results of Theorem IV.2 with corresponding results for a class of impulsive systems described by ordinary differential equations reported in [2]. We emphasize that there are several other works whose results are in the same spirit as the ones reported in [2] (see., e.g., [3], [13], and [15]). The asymptotic stability results in [2, Ch. 16, pp. 185–191] involve the existence of Lyapunov functions which are strictly decreasing along the system’s motions over the in, and at the points of discontinuity, tervals , the Lyapunov functions are required to have “downward” jumps. In contrast to this, Theorem IV.2 requires that when evaluated along the system’s motions, the Lyapunov functions be strictly decreasing only at the points , and that over the intervals, of discontinuity, , the Lyapunov functions be only bounded in a certain way [refer to inequality (4.19)]. This allows us to consider systems which exhibit, e.g., unstable behavior over some , as long as there is sufficient or all of the intervals “attenuation” provided by the functions (given in (3.5) ). This is not possible in the at the points of discontinuity results given in [2] (or in other similar results given in [3], [13],
and [15]). Accordingly, even in the finite dimensional case, the results of Theorem IV.2 are less conservative than many of the existing results. Remark IV.7: It is possible to establish Lagrange stability results (i.e., uniform boundedness and uniform ultimate boundedness of motions) which are in the spirit of the results of this section. We did not include these, due to space limitations. V. APPLICATIONS We now apply the results of the preceding section in the stability analysis of three important classes of DDS: Systems described by nonlinear functional differential equations; systems described by linear functional differential equations; and an initial-value and initial-boundary value problem involving the heat equation. A. Functional Differential Equations In this section, we let Banach space with norm defined by
which is a
where denotes any norm on . Also, we let be for the function determined by and we let denote a neighborhood of the origin in . 1) DDS Determined by Nonlinear Semigroups: Now, consider the system of discontinuous retarded functional differential equations given by
where
and and
are given collections of mappings ( ) and is a given infinite unbounded discrete set. and We assume that for all (5.1)
for all that
, where and that
is a finite constant. Also, we assume satisfies the Lipschitz condition (5.2)
for all For every
. , the initial value problem (5.3)
possesses a unique solution for every iniwhich exists for all with tial condition . Therefore, it follows that for every possesses a unique solution which exists for all , given by
(5.4)
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
Note that
.
is continuous with respect and that at may be discontinuous. Furtheris an equilibrium of and that more, note that for all . Remark V.1: System merits perhaps some additional comments. First, we note that the state space for this system and that for , the state of this system, is , evolves according to the first equation in (5.4), starting at with the initial state (which is ). Next, , the state defined over the time interval at , is mapped by the function into the state , the initial state for the next interval [refer is defined over to the second equation in (5.4)]. Note that Also, note that to on
, where the interval , while is defined over the interval . Next, for the initial-value problem (5.3) we define . From the properties of the solutions of (5.3), it is is a nonlinear semigroup on . easily shown that is a quasicontractive semigroup, and In fact,
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of system (resp., Then the equilibrium ) is uniformly asymptotically stable and exponentially stable. c) In part a), replace iv) by hypothesis v) and assume that . Then, the conditions (5.1) and (5.2) hold for (resp., ) is uniformly equilibrium of system asymptotically stable in the large and exponentially stable in the large. Proof: In view of (5.5), we have, since (5.8) for all
, and , resp., . Setting , and , we can see that all hypotheses of Corollary IV.1 are satisfied. This completes the proof. 2) Dynamical Systems Determined by Linear Semi. We define the linear mapping groups: Now assume from to by the Stieltjes integral (5.9) to obtain the initial-value problem (see, e.g., [9]) (5.10)
(5.5) and all , where is given in (5.2) (see, for all e.g., [9]). The preceding allows us to characterize system as
Finally, it is clear that (resp., ) determines a discontinuous dynamical system which is a special case of the DDS . We will denote this nonlinear DDS by . Proposition V.1: (resp., ) assume the following. a) For system , the function satisfies the Lipsi) For each for chitz condition (5.2) with Lipschitz constant all , where is a neighborhood of the origin. , the function satisfies condition ii) For each for all . (5.1) with constant iii) For each , and . iv) For all (5.6)
b) v)
of system (resp., Then, the equilibrium ) is uniformly stable. In part a), replace iv) by the following hypothesis. For all (5.7)
is an matrix whose entries are In (5.9), assumed to be functions of bounded variation on . Then, is Lipschitz continuous on with Lipschitz constant less or equal to the variation of in (5.9). In this case, the semigroup is a -semigroup. The spectrum of its generator consists of all solutions of the equation (5.11) If in particular, all the solutions of (5.11) satisfy the relation , then it follows from Theorem II.2 that for any , there is a constant such that positive (5.12) When the previous assumptions do not hold, then in view of Theorem II.1, we still have the estimate (5.13) for some and . where is deNext, let fined similarly as in (5.9) by and where is assumed to be a let assumes the form linear operator. Then, system
It is clear that determines a DDS determined by linear . We will denote this semigroups which is a special case of . dynamical system by
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In the following, when all the solutions of the characteristic satisfy the condiequation , then given any , there is a tion such that constant
1) DDSs Determined by the Heat Equation: Now, consider the DDS determined by the equations
(5.14) [see (5.12)]. Otherwise, we still have the estimate (5.15) [see (5.13)]. for some When (5.14) applies, we let in the following
where family of mappings,
, are constants, is a given , and is a given infinite unand that there bounded discrete set. We assume that such that exists a constant (5.22)
(5.16) and when (5.15) applies, we let (5.17)
. for all Associated with we have a family of initial and boundary value problems determined by
Thus, in all cases, we have the estimate
(5.23) (5.18)
Proposition V.2: a) For system i) for each
, assume the following: , and
ii)
;
for each (5.19) and are given in (5.14)–(5.17). where of system Then, the equilibrium uniformly stable. In part a), replace (5.19) by
b)
It has been shown (e.g., [16]) that for each , the initial and boundary value problem (5.23) has a unique , such that solution for each fixed and is a continuously to with respect to the differentiable function from -norm given in (5.21). It now follows that for every possesses a unique which exists for all , given by solution
is (5.24)
(5.20) of is uniformly Then, the equilibrium asymptotically stable in the large and exponentially stable in the large. Proof: The proof follows directly from Corollary IV.3.
is continuous with , and that at may be discontinuous. Furthermore, is an equilibrium for and that for all is a trivial motion. , (5.23) can be cast as an initial-value problem For each in the space with respect to the -norm, letting with respect to on
. Notice that
B. A Class of Partial Differential Equations be a bounded domain with In this subsection, we let and we let denote smooth boundary where the Laplacian. Also we let and are Sobolev spaces (refer, e.g., to [15, , pp. 84–85] for the definition of Sobolev spaces). For any -norm by we define the (5.21) where
.
(5.25) and , denotes the where . Furthermore, it has solution of (5.25) with been shown (e.g., [16]) that (5.25) determines a -semigroup , where for any . Since , then is an equilibrium for (5.25) [respectively, (5.23)]. Also, it has been shown (e.g., [15, pp. 344–345]) that (5.26)
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
where into a cube of length . Letting can be characterized as
and
can be put
(i.e., (i.e.,
matrix matrix
) there exists a positive–definite ) such that
in (5.24), system (6.1)
Finally, it is clear that (respectively, ) determines a DDS which is a special case of the DDS . We will denote . this DDS by (respectively, ) asProposition V.3: For system sume that , . and a) If for each (5.27)
b)
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of system then the equilibrium stable with respect to the -norm. If for all
is uniformly
Further, all eigenvalues of have positive real parts if and only there exists a matrix if for every given matrix such that (6.1) holds. The preceding results are used in the stability analysis of , using the Lyapunov functions linear systems and . Now let and denote the smallest and largest eigenvalues of , respectively, and let and denote the smallest and largest eigenvalues of , respectively. Also, let when all eigenvalues of have negative real parts, and let when all eigenhave positive real parts. Assume that values of . , let denote the matrix norm inLet , and define duced by the Euclidean vector norm . , choose now the Lyapunov function For system
(5.28) is a constant, then the equilibrium where of system is uniformly asymptotically stable in the large and exponentially stable in the large, with respect -norm. to the Proof: Setting , it is clear that all hypotheses of Corollary IV.1 are satisfied. This proves the result. Remark V.2: We emphasize that in specific examples of the systems considered in the present section, all required parame, and in (5.7); , and in (5.20); ters [e.g., and , and in (5.27)] are either given, or can be computed, or can be estimated. VI. CONCLUDING REMARKS In this paper, we first formulated two important classes of and DDS determined by linear and nonlinear semigroups ( ) and we showed that under a reasonable assumption, the motions of these systems exhibit the time-invariance property. Next, we established sufficient conditions for various types of stability by invoking the properties of semigroups (Theorem IV.1 and Corollaries IV.1, IV.2, and IV.3) and the Lyapunov function approach (Theorem IV.2). We then applied some of these results in the analysis of three important classes of DDS. We conclude with some additional pertinent comments. Remark VI.1: There are no general rules for choosing Lyapunov functions in results such as Theorem IV.2. However, for cases where converse theorems are available for continuous dynamical systems, we propose a systematic procedure for constructing Lyapunov functions in the application of our results. We demonstrate this by considering a specific class of DDS. given in Remark IV.2. It is well We consider system , have negknown (e.g., [1]) that all eigenvalues of ative real parts if and only if for every given negative–definite
(6.2) . It is easily shown that for all (6.3) When satisfied with and with When is satisfied with
, hypothesis a) of Theorem IV.2 is , assuming that . Note that (4.22) is satisfied as for any . , hypothesis b) of Theorem IV.2 , where . Finally, in all cases (4.18) is satisfied with and , where and . All hypotheses of Theorem IV.2 are satisfied and we have the following result: Under the above assumptions, if for all , then the zero solutions of system is uniformly stable and if , then the zero solution of system is uniformly asymptotically stable in the large and exponentially stable in the large. Remark VI.2: We are not aware of any general stability results for DDS in the current literature which are applicable in the analysis of infinite dimensional systems of the type considered in this paper. However, results for many classes of finite dimensional DDS have been established. We compare in the following one of these with our results. Consider the linear system subject to impulsive effects given by (see [2])
where , and denotes the identity matrix. By applying Corollary IV.3 and Remark IV.2 (using a ), it is easily similar procedure as in the analysis of system
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shown that the equilibrium tially stable in the large if
of system
is exponen-
where . In ([2], p. 61) the same condition for exponential stability in the large is established, using a Lyapunov approach, for the case when . REFERENCES [1] P. J. Antsaklis and A. N. Michel, Linear Systems. New York: McGrawHill, 1997. [2] D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effects: Stability, Theory, and Applications. New York: Halsted, 1989. [3] 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. [4] M. G. Crandall, “Semigroups of nonlinear transformations in Banach spaces,” in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, Ed. New York: Academic, 1971. [5] M. G. Crandall and T. M. Liggett, “Generation of semigroups of nonlinear transformations on general Banach spaces,” Amer. J. Math., vol. 93, pp. 265–298, 1971. [6] 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. [7] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, 1975. [8] W. Hahn, Stability of Motion. Berlin, Germany: Springer-Verlag, 1967. [9] J. K. Hale, Functional Differential Equations. Berlin, Germany: Springer-Verlag, 1971. [10] E. Hille and R. S. Phillips, “Functional analysis and semigroups,” in Amer. Math. Soc. Colloquium Publ.. Providence, RI: Amer. Math. Soc., 1957, vol. 33. [11] S. G. Krein, “Linear differential equations in Banach spaces,” in Translation of Mathematical Monographs. Providence, RI: Amer. Math. Soc., 1970, vol. 29, ch. 1. [12] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999. [13] 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. [14] A. N. Michel and B. Hu, “Toward a stability theory of general hybrid dynamical systems,” Automatica, vol. 35, pp. 371–384, Apr. 1999. [15] A. N. Michel, K. Wang, and B. Hu, Qualitative Analysis of Dynamical Systems, 2nd ed. New York: Marcel Dekker, 2001. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983. [17] M. Slemrod, “Asymptotic behavior of C semigroups as determined by the spectrum of the generator,” Indiana Univ. Math. J., vol. 25, pp. 783–791, 1976.
[18] Y. Sun, A. N. Michel, and G. Zhai, “Stability of discontinuous retarded functional differential equations with applications,” IEEE Trans. Autom. Control, vol. 50, no. 8, pp. 1090–1105, Aug. 2005. [19] 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. [20] V. I. Zubov, Methods of A. M. Lyapunov and Their Applications. Groningen, The Netherlands: Noordhoff, 1964.
Anthony N. Michel (S’55–M’59–SM’79–F’82 | acroread -toPostScript | lp–LF’95) 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 sixteen 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 the University of Notre Dame. He has also held visiting faculty positions at the Technical University of Vienna, Vienna, Austria, the Johannes Kepler University, Linz, Austria, and the 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.
Ye Sun received the B.S. degree in mathematics from the University of Science and Technology of China in 1999, and the M.S. and Ph.D. degrees from the University of Notre Dame, in 2002 and 2004, respectively. She is currently a Systems Analyst at Credit Suisse First Boston, New York, working on fixed-income models. Her research interests include systems modeling and qualitative analysis of discontinuous dynamical systems.
Alexander P. Molchanov, deceased.
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Stability Analysis of Discountinuous Dynamical Systems Determined by Semigroups Anthony N. Michel, Life Fellow, IEEE, Ye Sun, and Alexander P. Molchanov, Senior Member, IEEE
Abstract—We present Lyapunov stability results for discontinuous dynamical systems (DDS) determined by linear and nonlinear semigroups defined on Banach space. DDS of the type considered herein arise in the modeling of a variety of finite- and infinitedimensional systems, including certain classes of hybrid systems, discrete-event systems, switched systems, systems subjected to impulse effects, and the like. We apply our results in the analysis of several important specific classes of DDS. Index Terms—Asymptotic stability, 0 -semigroups, discontinuous dynamical systems (DDS), exponential stability, functional differential equations, heat equation, Lyapunov stability, nonlinear semigroups, partial differential equations.
I. INTRODUCTION
D
ISCONTINUOUS dynamical systems (DDS) have motions which are not continuous with respect to time. Such systems arise in the modeling process of a variety of systems, including hybrid dynamical systems, discrete event systems, switched systems, systems subjected to impulse effects, and the like (see, e.g., [2], [3], [6], [12]–[15], [19], and the references cited therein). The stability analysis of specific classes of such systems has thus far been concerned primarily with finite dimensional dynamical systems (defined on ) determined by ordinary differential equations, and more recently, ) with infinite-dimensional dynamical systems (defined on determined by functional differential equations [18]. The results that were established in [14], [15], and [19] are formulated for general dynamical systems in a metric space setting, and as such, are in principle applicable to finite dimensional dynamical systems as well as to infinite dimensional dynamical systems. However, in the latter case, the application of these results to specific classes of infinite dimensional systems is usually not straightforward and frequently requires further analysis [18]. (This is usually also the case for continuous dynamical systems (see, e.g., [8], [15], and [20]). In this paper, we establish (Lyapunov) stability results for DDS determined by linear and nonlinear semigroups defined on Banach spaces. We present general results which do not require determination of Lyapunov functions, as well as results which do involve Lyapunov functions. Our results are very general Manuscript received April 3, 2003; revised February 26, 2004. Recommended by Associate Editor S.-I. Niculescu. A. N. Michel is with the Department of Electrical Engineering, the University of Notre Dame, Notre Dame, IN 46554 USA (e-mail: [email protected]). Y. Sun is with Credit Suisse First Boston, New York, NY 10010 USA (e-mail: [email protected]). A. P. Molchanov, deceased. Digital Object Identifier 10.1109/TAC.2005.854582
and are applicable to large classes of finite- and infinite-dimensional DDS. We demonstrate the applicability of our results in the analysis of DDS determined by linear and nonlinear retarded functional differential equations and a specific initial-value and boundary-value problem governed by the heat equation. The remainder of this paper is organized as follows. In Section II, we provide essential background material on linear and nonlinear semigroups. In Section III, we formulate the classes of DDS determined by linear and nonlinear semigroups considered in this paper. In Section IV, we establish stability results for the classes of DDS considered herein and in Section V we apply these results in the analysis of several important specific classes of DDS. We conclude the paper in Section VI with appropriate comments. II. NOTATION AND BACKGROUND MATERIAL In this section, we provide essential background material concerning dynamical systems determined by semigroups. A. Notation , let Let denote real -space, and let denote any one of the . For a real matrix (i.e., equivalent norms on ) and , let denote the norm of induced by the vector norm . Let and be Banach spaces and let denote norms on Banach spaces. Let be a linear operator defined on a dowith range in . We call bounded if it main into a bounded subset of , or maps each bounded set in equivalently, if it is continuous. Given a bounded linear oper, its norm is defined by ator . We let denote the identity transformation. . Then, signifies that Finally, let belongs to the set of continuous functions from into . B.
-Semigroups
We will require the following concepts (see, e.g., [9]–[11]). Definition II.1: A one-parameter family of bounded linear operators , is said to be a -semigroup (or a linear semigroup) if i) ii) for any ; for all . iii) -semigroups are generated by linear operators.
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Definition II.2: The infinitesimal generator -semigroup is defined as the opof a for erator where is the domain of given by . The Hille–Yoshida–Phillips Theorem provides necessary and sufficient conditions for a linear operator to be the infinitesimal generator of some -semigroup. We refer the reader to [10] or to [15, p. 76] for a statement of this theorem. The following results which provide some of the basic properties of -semigroups will be of particular interest to us. , there exists an Theorem II.1 [10]: For a -semigroup and an such that (2.1) for all The qualitative properties of a -semigroup in terms of its infinitesimal generator is given of the spectrum by the following result. is called differentiable Definition II.3: A -semigroup if for each is continuously differentiable for . on Theorem II.2 [17]: If is a -semigroup which is differentiable for , if is its infinitesimal generator, and if for all , then given any positive , such that there is a constant (2.2) for all
.
C. Nonlinear Semigroups is a family of linear opIn the case of -semigroups, erators. This restriction is removed in the following. Definition II.4 [4], [5], [10]: Assume that is a subset of a Banach space . A family of one-parameter (nonlinear) operators , is said to be a nonlinear semigroup defined on if i) for ii) for ; iii) is continuous in on . As in the case of -semigroups, nonlinear semigroups are also generated by operators. Definition II.5: A (possibly multivalued) operator is said on if to generate a nonlinear semigroup
Of particular importance in applications are quasicontractive is and contraction semigroups. A nonlinear semigroup called a quasicontractive semigroup if there is a number such that (2.3) and for all . If in (2.3), , then for all is called a contraction semigroup. For a set of sufficient conditions under which an operator generates a quasi-contractive semigroup, refer to [10] or to [15, p. 78]. D. Continuous Dynamical Systems Determined by Semigroups -semigroup For a given with initial state by
, we define the motion and initial time (2.4)
and we define the dynamical system determined by a as the family of motions group
-semi-
(2.5) , and in particular, when We note that then for all . We will call an and we will call the equilibrium for the dynamical system the trivial motion. corresponding motion For a given nonlinear semigroup , we define the motion with initial state and initial time by (2.6) and we define the dynamical system determined by a nonlinear as the family of motions semigroup
(2.7) is in the interior Henceforth, we will always assume that of . . Throughout Once more we note that this paper, we will assume that for any nonlinear semigroup for all if . We will an equilibrium for the dynamcall ical system and we will call the corresponding motion , a trivial motion. III. DDS DETERMINED BY SEMIGROUPS
for all semigroup
. The infinitesimal generator is defined by
of a nonlinear
To motivate the class of DDS 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
for all such that this limit exists. The operator and the inare generally different operators. finitesimal generator
In Fig. 1, we depict in block diagram form a configuration which is applicable to many classes of DDS, including hybrid
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Fig. 1. DDS configuration.
systems and switched systems. There is a block which contains continuous-time dynamics, a block which contains phenomena that evolve at discrete points in time (discrete-time dynamics) or at discrete events, and a block which contains interface elements for the above two system components. The block which contains the continuous-time dynamics is usually characterized in the existing current literature by ordinary differential equations while the block on the right in Fig. 1 is usually characterized by difference equations, or it may involve other types of discrete characterizations, e.g., Petri nets, logic commands, various types of discrete event systems, and the like. The block labeled interface elements may vary from the very simple to the very complicated. At the simplest level, this block involves samplers and sample and hold elements. The sampling process may involve only one uniform rate, or it may be nonuniform (variable rate sampling), or there may be several different (uniform or nonuniform) sampling rates occurring simultaneously (multi-rate sampling). Perhaps the simplest specific example of the above class of systems are sampled-data control systems described by the equations
(3.1) where sampling instants, priate dimensions,
denotes are real matrices of approare interface variables, and
and .
Now, define at
and
, and . Then, for all
at . Let
,
.
, let Next, for solution of the initial-value problem
, denote the unique
(3.3) by . Using the properties of the so, is a lutions of (3.3), it is easily shown that -semigroup. Given a set of initial conditions , the unique solution of (3.2) [and, hence, of (3.1)] can now be expressed by the function and define the mapping
(3.4) Consistent with the terminology used later, we refer to as a motion. By varying over , we generate a family of motions, , the dynamical system , and in determined by (3.4). We note that , then for all . We particular, when an equilibrium and the trivial motion. call We conclude the present example with an observation. As noted earlier, in the current literature, the continuous-time dynamics in Fig. 1 are almost always assumed to be determined by finite dimensional systems (lumped parameter systems) described by ordinary differential equations. However, in many applications, the continuous dynamics may more appropriately be determined by infinite dimensional systems, to capture the effects of time delays, hysteresis phenomena, distributed parameters, and the like. We will present in the present section system models which allow the continuous-time dynamics in Fig. 1 to be represented by finite-dimensional as well as infinite-dimensional systems. B. DDS Determined by Linear and Nonlinear Semigroups
Letting
where denotes the identity matrix, then system (3.1) can be described by the discontinuous differential equation
(3.2)
In the following, we will require a given collection of linear or defined on a Banach space nonlinear semigroups , or on a set , respectively; and a given collection of linear and continuous operators , or of nonlinear and continuous operators ; and a given discrete, infinite, and unbounded set . We assume that when consists of linear semigroups, then consists of linear mappings. The number of elements in and may be finite or infinite.
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We now consider dynamical systems whose motions with initial time and initial state (respectively, ) are given by
(3.5) We define the DDS determined by semigroups as
(3.6) is unique, with Note that every motion , exists for all , and is continuous with respect to on , and that at may be discontinuous. We call the set the set of discontinuities for the motion . consists of -semigroups and conWhen in (3.6), sists of linear mappings, we speak of a DDS determined by . Simlinear semigroups, and we denote this system by consists of nonlinear semigroups, we ilarly, when in (3.6), speak of a DDS determined by nonlinear semigroups and we . When the types of the elements in denote this system by and are not specified, we simply speak of a DDS determined by semigroups and we denote this system, as in (3.6), by . , and are linear Since in the case of for all operators, it follows that in particular . We call the equilibrium for the DDS and , the trivial motion. , we assume that is in the interior In the case of for all if and that of and that if for all . From this, it follows that for all if . We call an , a trivial motion for the equilibrium and . DDS Finally, in the case of linear mappings, we use in (3.5) the and notation . We conclude the present section with a few remarks. , reRemark III.1: For different initial conditions , we allow the set of sulting in different motions , the set of semigroups discontinuities , and the set of functions to differ, and accordingly, the notation , and might be more appropriate. However, since in all cases, all meaning will be clear from context, we will not use such superscripts. Remark III.2: To the best of our knowledge, the DDS models and ) are original and have not considered herein ( been considered previously. These models are very general and include large classes of finite dimensional dynamical systems determined by ordinary differential equations and inequalities and by large classes of infinite dimensional dynamical systems
determined by differential-difference equations, functional differential equations, Volterra integrodifferential equations, certain classes of partial differential equations, and more generally, differential equations and inclusions defined on Banach space. In contrast, most of the existing literature concerning stability of DDS (including hybrid systems, switched systems, systems subjected to impulse disturbances, and so forth) is confined to finite-dimensional systems determined by ordinary differential equations (e.g., [2], [3], [6], [12], and [13]). Remark III.3: The dynamical system models and are very flexible, and include as special cases, many of the DDS considered thus far in the literature, as well as general autonomous continuous dynamical systems: a) If for all ( has only one element) and if for all , where denotes the identity transformation, then reduces to an autonomous, linear, continuous dynamical system and to an autonomous, nonlinear, continuous dynamical system; b) in the case of dynamical systems subjected to impulse effects [considered in the literature for finite dimensional systems (see, for all while the e.g., [2])], one would choose impulse effects are captured by an infinite family of functions ; c) in the case of switched systems, frequently only a finite number of systems that are being switched is required, and so in this case one would choose a finite family of (see, e.g., [6] and [12]); and so forth. semigroup Remark III.4: Perhaps it needs pointing out that even though and are determined by families of semisystems groups (and nonlinearities), by themselves they are not semigroups, since in general, they are time-varying and do not satisfy the hypotheses i)–iii) given in Definitions II.1 and II.4. How, used in describing ever, each individual semigroup or , does possess the semigroup properties, albeit, only . over a finite interval IV. STABILITY RESULTS FOR DDS DETERMINED BY SEMIGROUPS In this section, we establish several stability results for discontinuous dynamical systems determined by linear and nonlinear semigroups. Before stating and proving these results, we give the definitions of the various stability concepts that we will employ. A. Qualitative Characterization of DDS Recall that the DDS determined by linear semigroups, is defined on a Banach space while the nonlinear DDS given is defined on . Recall also that the origin 0 is by assumed to be in the interior of and that is an equiand . Since the following definitions librium for both and , we will refer to either one of pertain to both them simply as . Definition IV.1: The equilibrium of is stable if for and every , there exists a every such that for all of for all , (and ). The equilibrium is whenever uniformly stable if is independent of , i.e., . The of is unstable if it is not stable. equilibrium
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
Definition IV.2: The equilibrium such that there exists an
of
is attractive if
(4.1) for all of whenever (and ). We call the set of all such that (4.1) holds the domain . of attraction of of is asymptotiDefinition IV.3: The equilibrium cally stable if it is stable and attractive. Definition IV.4: The equilibrium of is uniformly and every , there exists a attractive if for every , independent of and , and a , independent for all and for all of , such that of , whenever (and ). Definition IV.5: The equilibrium of is uniformly asymptotically stable if it is uniformly stable and uniformly attractive. Definition IV.6: The equilibrium of is exponentially , and for every and every stable if there exists , there exists a such that for all and for all of whenever (and ). The preceding concern local characterizations of an equilibrium. In the following, we address global characterizations. In . this case, we will find it convenient to let Definition IV.7: The equilibrium of is asymptotically stable in the large if i) it is stable, and of and for all ii) for every , (4.1) holds. In this case, the domain of attraction of is all of . of is uniformly Definition IV.8: The equilibrium asymptotically stable in the large if i) it is uniformly stable, and ii) it is uniformly attractive in the large, i.e., for every and every , and for every , there exists (independent of ), such that if a , then for all of for all . Definition IV.9: The equilibrium stable in the large if there exists such that there exists
of is exponentially and for every ,
(4.2) of , for all , whenever . for all Although system and are determined by semigroups, they themselves are not semigroups and in general, they are time-varying systems. However, in the following we identify an assumption under which the motions of these systems exhibit the time-invariance property. or Assumption IV.1: Assume that any two motions (of ) with identical initial states but different initial times, say and , are determined by identical and operators . sequences of semigroups
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Furthermore, assume that if for the motion , then for the motion , we have , where . By applying definitions, it is easily shown that under Assumpand have the proption IV.1, the motions of systems for all and erty that , i.e., the motions possess the time-invariance for every property. It has been shown (see, e.g., [4], [5], [9]–[11], and [15]) that for dynamical systems whose motions have the time-invariance , is stable (respectively, property, an equilibrium, say asymptotically stable, asymptotically stable in the large) if and only if it is uniformly stable (respectively, uniformly asymptotically stable, uniformly asymptotically stable in the large). From the above observations, the following result follows readily. Proposition IV.1: Under Assumption IV.1, the following and : statements are true for systems is stable if and only if it is a) the equilibrium uniformly stable; is attractive if and only if it is b) the equilibrium uniformly attractive; is asymptotically stable if and c) the equilibrium only if it is uniformly asymptotically stable; is asymptotically stable in the d) the equilibrium large if and only if it is uniformly asymptotically stable in the large. . We emphasize that in all subsequent results, Assumption IV.1 is not required, and the distinction between stability and uniform stability (respectively, between asymptotic stability and uniform asymptotic stability) is required. B. Principal Stability Results In our first results, we establish sufficient conditions for various stability properties for system . We will assume in there exist these results that for each nonlinear semigroup and and for each mapping constants there exists a constant such that (4.3) and (4.4) for all . We recall from Section 2-C (see (2.3)) that in particular, (4.3) is always satisfied for a quafor some computable paramesicontractive semigroup , and , while for a contractive ters , inequality (4.3) is satisfied with and semigroup . We will require the following additional notation. Let , for any given and , we let , and we let and denote the finite products (4.5)
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Theorem IV.1: a) For system , under (4.3) and (4.4), assume that for there exists a constant such that any
is true for . and Therefore, by (4.5) and (4.10), we have
(4.6)
b)
c)
d)
e)
(4.11)
, where is defined in (4.5). Then the for all equilibrium of is stable. , i.e., in (4.6) can be chosen If in part a), , then the equilibrium of independent of is uniformly stable. If in part a), (4.6) is replaced by (4.7) for all , then the equilibrium of is asymptotically stable. If the conditions of part b) are satisfied and if in part c) relation (4.7) is satisfied uniformly with respect to (i.e., for every and every there , independent of , such that exists a for all ), then the equilibrium of is uniformly asymptotically stable. Assume that in part a) (4.6) is replaced by
b)
c)
(4.8) where
and
. Assume also that (4.9)
is a constant. Then, the equilibrium where of is exponentially stable. f) If in parts c)–e), respectively, conditions (4.3) and (4.4) , then the equilibrium of hold for all is asymptotically stable in the large, uniformly asymptotically stable in the large and exponentially stable in the large, respectively. Proof: a) For system , with , we associate each interval with the index . We will find it convenient to employ a relabeling of , where indexes. To this end, let denotes the integer part of , and let . Then, we can relabel as and as . and , If we have for . Therefore, in view of (4.5) (4.10) is
true.
It
is
clear
that
. Similarly, for , if
, then
,
d)
e)
For any and , let . From (4.6) and (4.11), it now follows that , whenever and . Since and since for all and we can equate , all it follows that the equilibrium of is stable. can In proving part b), note that , and consequently, be chosen independent of can also be chosen independent of . Therefore, the equiof is uniformly stable. librium it From the assumption on follows that . Hence, as . Since for any we have for some , then when . Hence, it follows from (4.7) and (4.11) that (4.1) holds for all of whenever . of is attractive Therefore, the equilibrium and its domain of attraction coincides with the entire . Since (4.6) follows from (4.7), then, as in set part a), of is stable. Hence, the equilibrium of is asymptotically stable. Since the conditions of part b) are satisfied, the equilibof system is uniformly stable. Thererium fore, we only need to prove that is uniformly attractive. in such a way that Choose . Since (4.7) is satisfied uniformly with respect to , then for every and every there exists a (independent of ) for all . Hence, from such that for all (4.11), we have and for all . Let . and for Then, . If we let , then all for all and we have that for all of , whenever . Hence, of is uniformly attractive the equilibrium and uniformly asymptotically stable. To prove part e), note that as was shown in the proofs of and any , there parts a) and c), for any and such that and exists (4.11) holds. Since and in view of (4.9), , and therefore, we have . Hence, in . view of (4.8), we have , let . Then, for any For any with , we have , where . Therefore, the equilibrium of is exponentially stable.
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
f)
We note that if the estimates (4.3) and (4.4) hold for all , then inequality (4.11) is valid for all . i) Repeating the reasoning in the proof of part c) for any and any , we can conclude that in this of whenever case (4.1) holds for all and . Therefore, the equilibrium of is asymptotically stable in the large. ii) Similarly as in the proof of part d), for every and for every there exists a (independent of ), such that for all . If we let , then we have that for all and for all of , whenever . of is uniformly Hence, the equilibrium asymptotically stable in the large. and for every we have iii) For every similarly as in the proof of part e) that for , where . Let all . It now follows that the equilibrium of is exponentially stable in the large. This completes the proof. Corollary IV.1: a) For system , assume that the following statements are true. ). i) Condition (4.3) holds (with parameters ii) Condition (4.4) holds (with parameter ): ; iii) for all and iv) for all where and are constants; v) for all
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b)
c)
In view of (4.13) the estimate (4.8) is true with and . Therefore, the limit relation (4.7) is satisfied uniformly with respect to . This proves part b) of the corollary. The conclusions of part c) of this corollary follow directly from part f) of Theorem IV.1.
From Theorem II.1, we recall that for any -semigroup , there will exist and such that (4.14) is a Furthermore, in accordance with Theorem II.2, if -semigroup which is differentiable for , if is its infor all , finitesimal generator, and if , there is a constant then given any positive such that (4.15) Similarly as in Theorem IV.1, we will utilize in our next result the relation (4.16) where, depending on the situation on hand the constants are obtained from either (4.14) or (4.15). Similarly as in (4.5), we define in the case of DDS finite products
and the
(4.17) (4.12)
b)
of is stable and Then, the equilibrium uniformly stable. If in part a), hypothesis v) is replaced by (4.13)
c)
a)
, where , then the equilibrium for all of is asymptotically stable, uniformly asymptotically stable and exponentially stable. If in part a) it is assumed that inequalities (4.3) and and inequality (4.12) is re(4.4) hold for all of placed by (4.13), then the equilibrium is asymptotically stable in the large, uniformly asymptotically stable in the large, and exponentially stable in the large. Proof: It is easily shown that in part a) the estimate (4.6) is , indepensatisfied with . Therefore, the conditions in part a) and dent of b) of Theorem IV.1 are satisfied. This proves part a) of the corollary.
, denotes the norm of the bounded linear where used in defining the DDS in (3.5). operator By invoking Theorem IV.1, we obtain in a straightforward . manner the following result for system Corollary IV.2: In Theorem IV.1, replace (4.3) by (4.14)–(4.16) and (4.5) by (4.17). Then, all conclusions of . Theorem IV.1 are true for system Corollary IV.3: In Corollary IV.1, replace (4.12) and (4.13) and , respecby denotes operator norm and where and tively, where are given in (4.16). Then, the conclusions of Corollary IV.1 are . true for system Remark IV.1: Corollaries IV.1 and IV.3 are more conservative than Theorem IV.1 and Corollary IV.2 since in the case of the latter we put restrictions on partial products [see, e.g., (4.6)] while in the case of the former, we put corresponding restrictions on the individual members of the partial products [see, e.g., (4.12)]. However, Corollaries IV.1 and IV.3 are easier to apply than Theorem IV.1 and Corollary IV.2. Remark IV.2: For linear, finite-dimensional DDS given by
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where and , Corollary IV.3 can be improved somewhat by replacing the inequalities in that corollary, e.g., by the inequalities
for all
, where
(4.21)
and
assuming that , where [see, e.g., [7] for the definition of the measure of by the inequalities
denotes ], or
c)
d)
We omit the proofs of the aforementioned assertions due to space limitations. In the preceding results, we relied on fundamental definitions and to establish various stability properties for systems . It turns out that we can also obtain stability results for and , making use of Lyapunov functions. In systems doing so, we employ the following comparison functions. is said to beDefinition IV.10: A function ), if and if is strictly long to class , (i.e., . If and if , we say increasing on that belongs to class (i.e., ). Since the following results are applicable to both system and , we will use the term “system ” to indicate either system. Theorem IV.2: Assume that there exists a function and functions such that
then the equilibrium of system is uniformly asymptotically stable. Assume that all assumptions in parts a) and (b) are true and . Then, the equiwith of system is uniformly asymptotically librium stable in the large. Assume that all assumptions in parts a) and b) are true , and , with , and are positive constants. Furtherwhere more, assume that the function in (4.19) satisfies as
(4.22)
where is some positive constant. Then, the equilibrium of system is exponentially stable. e) Assume that all assumptions in part d) are true with . Then, the equilibrium of system is exponentially stable in the large. Proof: , then for any a) Since is continuous and there exists such that as long as . We assume that , and . Thus, for any , is satas long as the initial condition isfied, then
(4.18) and , where is a neighborhood of for all the origin . a) Assume that there exists a neighborhood of such that for every motion the origin of , with for all and is continuous for all except on a set . Also, assume that is nonincreasing for all and all , and assume that there exists a function , independent of , such that
and since more, for any
for , is nonincreasing. Further, we can conclude that
and
b)
Therefore, by definition, the equilibrium of system is uniformly stable. , we obtain from the Letting assumption of the theorem, that for
(4.19)
b)
. for all of system is uniThen, the equilibrium formly stable. If in addition to the assumptions in part a), there exists such that a function (4.20)
If we let , then inequality becomes Since is nonincreasing and thus obtain that . It follows that all
and the previous . , it follows that for all . We for
(4.23)
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
Now, consider a fixed . For any given such that can choose a
, we
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and
Let We obtain from
. and that
(4.24) since and . For any with and any , we are now able to show that whenever . The aforementioned statement is true because for any must belong to some interval for some . Therefore, we know that . It follows from (4.23) that , which implies that
(4.25)
which yields (4.27) . If where , then and and all . Thus, following, we assume that Since
is true for some for all for all
. In the for all . , it follows from (4.27) that
Hence
and
(4.26) , it follows from (4.25) that , noticing that (4.19) holds. In the , we can conclude from (4.26) . This proves that the equilibof system is uniformly asymptotically
In the case when
c)
case when that rium stable. of system is From part a), the equilibrium uniformly stable. We need to show that the equilibrium is uniformly attractive in the large. For any fixed and , we can choose and a a such that . Let and . For any with and any , we can show that whenever . This is true , we can find some since for any such that . Therefore, and , which implies that
is true for all that
. It now follows from . In the last step, we . From (4.22), it is . Let
have made use of the fact that easily seen that
Then, for all relations (4.19) and (4.22) that for all that
. It follows from , it is true
The last inequality follows since . Thus, . For any
and any
such that
, let
and Thus, we have
d)
and when . , we can conclude In the case when . This proves that from above that the equilibrium of system is uniformly asymptotically stable in the large. We only prove the case of global exponential stability. The case of local exponential stability can be proved similarly.
Then
for all of system
of and . Therefore, the equilibrium is exponentially stable in the large.
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We close by noting that in the previous proofs, we did not require the semigroup property ii) in Definitions II.1 and II.4 and, therefore, the present results will also be applicable to appropriately formulated DDS without this property. We conclude the present section with a few remarks. Remark IV.3: The general stability results concerning DDS reported in the current literature (e.g., [2], [3], [6], [12], and [13]) pertain essentially all to finite dimensional systems (“lumped parameter” systems) described by ordinary differential equations. These results are not applicable in the analysis of infinite-dimensional DDS which are capable of capturing the effects of distributed parameters (as in the case of partial differential equations), transportation and time delays (as in the case of differential-difference equations), hysteresis effects (as in the case of functional differential equations and Volterra integrodifferential equations), and the like. In contrast to this, the results of the present section are applicable in the stability analysis of finite-dimensional DDS as well as infinite dimensional DDS of this kind. Remark IV.4: The general stability results for DDS reported in the current literature (e.g., [2], [3], [6], [12]–[14], and [19]) are usually Lyapunov-type results, requiring the determination of appropriate Lyapunov functions, which is not necessarily an easy task. In contrast to this, Theorem IV.1 and Corollaries IV.1–IV.3 do not involve the existence of Lyapunov functions, simplifying their applications considerably. Instead of invoking the Lyapunov approach, we bring to bear the very extensive qualitative theory of semigroups in establishing these results. Remark IV.5: As pointed out in Remark IV.4, the Lyapunovtype results given in Theorem IV.2 are in general more difficult to apply than Theorem IV.1 and Corollaries IV.1, IV.2, and IV.3. However, the very ambiguity involved in the search for suitable Lyapunov functions offers flexibility in the application of Theorem IV.2 in efforts of reducing conservatism of results. In Section VI, we propose a systematic method of constructing Lyapunov functions for the classes of DDS considered herein. Remark IV.6: In the following comments, we compare the results of Theorem IV.2 with corresponding results for a class of impulsive systems described by ordinary differential equations reported in [2]. We emphasize that there are several other works whose results are in the same spirit as the ones reported in [2] (see., e.g., [3], [13], and [15]). The asymptotic stability results in [2, Ch. 16, pp. 185–191] involve the existence of Lyapunov functions which are strictly decreasing along the system’s motions over the in, and at the points of discontinuity, tervals , the Lyapunov functions are required to have “downward” jumps. In contrast to this, Theorem IV.2 requires that when evaluated along the system’s motions, the Lyapunov functions be strictly decreasing only at the points , and that over the intervals, of discontinuity, , the Lyapunov functions be only bounded in a certain way [refer to inequality (4.19)]. This allows us to consider systems which exhibit, e.g., unstable behavior over some , as long as there is sufficient or all of the intervals “attenuation” provided by the functions (given in (3.5) ). This is not possible in the at the points of discontinuity results given in [2] (or in other similar results given in [3], [13],
and [15]). Accordingly, even in the finite dimensional case, the results of Theorem IV.2 are less conservative than many of the existing results. Remark IV.7: It is possible to establish Lagrange stability results (i.e., uniform boundedness and uniform ultimate boundedness of motions) which are in the spirit of the results of this section. We did not include these, due to space limitations. V. APPLICATIONS We now apply the results of the preceding section in the stability analysis of three important classes of DDS: Systems described by nonlinear functional differential equations; systems described by linear functional differential equations; and an initial-value and initial-boundary value problem involving the heat equation. A. Functional Differential Equations In this section, we let Banach space with norm defined by
which is a
where denotes any norm on . Also, we let be for the function determined by and we let denote a neighborhood of the origin in . 1) DDS Determined by Nonlinear Semigroups: Now, consider the system of discontinuous retarded functional differential equations given by
where
and and
are given collections of mappings ( ) and is a given infinite unbounded discrete set. and We assume that for all (5.1)
for all that
, where and that
is a finite constant. Also, we assume satisfies the Lipschitz condition (5.2)
for all For every
. , the initial value problem (5.3)
possesses a unique solution for every iniwhich exists for all with tial condition . Therefore, it follows that for every possesses a unique solution which exists for all , given by
(5.4)
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
Note that
.
is continuous with respect and that at may be discontinuous. Furtheris an equilibrium of and that more, note that for all . Remark V.1: System merits perhaps some additional comments. First, we note that the state space for this system and that for , the state of this system, is , evolves according to the first equation in (5.4), starting at with the initial state (which is ). Next, , the state defined over the time interval at , is mapped by the function into the state , the initial state for the next interval [refer is defined over to the second equation in (5.4)]. Note that Also, note that to on
, where the interval , while is defined over the interval . Next, for the initial-value problem (5.3) we define . From the properties of the solutions of (5.3), it is is a nonlinear semigroup on . easily shown that is a quasicontractive semigroup, and In fact,
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of system (resp., Then the equilibrium ) is uniformly asymptotically stable and exponentially stable. c) In part a), replace iv) by hypothesis v) and assume that . Then, the conditions (5.1) and (5.2) hold for (resp., ) is uniformly equilibrium of system asymptotically stable in the large and exponentially stable in the large. Proof: In view of (5.5), we have, since (5.8) for all
, and , resp., . Setting , and , we can see that all hypotheses of Corollary IV.1 are satisfied. This completes the proof. 2) Dynamical Systems Determined by Linear Semi. We define the linear mapping groups: Now assume from to by the Stieltjes integral (5.9) to obtain the initial-value problem (see, e.g., [9]) (5.10)
(5.5) and all , where is given in (5.2) (see, for all e.g., [9]). The preceding allows us to characterize system as
Finally, it is clear that (resp., ) determines a discontinuous dynamical system which is a special case of the DDS . We will denote this nonlinear DDS by . Proposition V.1: (resp., ) assume the following. a) For system , the function satisfies the Lipsi) For each for chitz condition (5.2) with Lipschitz constant all , where is a neighborhood of the origin. , the function satisfies condition ii) For each for all . (5.1) with constant iii) For each , and . iv) For all (5.6)
b) v)
of system (resp., Then, the equilibrium ) is uniformly stable. In part a), replace iv) by the following hypothesis. For all (5.7)
is an matrix whose entries are In (5.9), assumed to be functions of bounded variation on . Then, is Lipschitz continuous on with Lipschitz constant less or equal to the variation of in (5.9). In this case, the semigroup is a -semigroup. The spectrum of its generator consists of all solutions of the equation (5.11) If in particular, all the solutions of (5.11) satisfy the relation , then it follows from Theorem II.2 that for any , there is a constant such that positive (5.12) When the previous assumptions do not hold, then in view of Theorem II.1, we still have the estimate (5.13) for some and . where is deNext, let fined similarly as in (5.9) by and where is assumed to be a let assumes the form linear operator. Then, system
It is clear that determines a DDS determined by linear . We will denote this semigroups which is a special case of . dynamical system by
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In the following, when all the solutions of the characteristic satisfy the condiequation , then given any , there is a tion such that constant
1) DDSs Determined by the Heat Equation: Now, consider the DDS determined by the equations
(5.14) [see (5.12)]. Otherwise, we still have the estimate (5.15) [see (5.13)]. for some When (5.14) applies, we let in the following
where family of mappings,
, are constants, is a given , and is a given infinite unand that there bounded discrete set. We assume that such that exists a constant (5.22)
(5.16) and when (5.15) applies, we let (5.17)
. for all Associated with we have a family of initial and boundary value problems determined by
Thus, in all cases, we have the estimate
(5.23) (5.18)
Proposition V.2: a) For system i) for each
, assume the following: , and
ii)
;
for each (5.19) and are given in (5.14)–(5.17). where of system Then, the equilibrium uniformly stable. In part a), replace (5.19) by
b)
It has been shown (e.g., [16]) that for each , the initial and boundary value problem (5.23) has a unique , such that solution for each fixed and is a continuously to with respect to the differentiable function from -norm given in (5.21). It now follows that for every possesses a unique which exists for all , given by solution
is (5.24)
(5.20) of is uniformly Then, the equilibrium asymptotically stable in the large and exponentially stable in the large. Proof: The proof follows directly from Corollary IV.3.
is continuous with , and that at may be discontinuous. Furthermore, is an equilibrium for and that for all is a trivial motion. , (5.23) can be cast as an initial-value problem For each in the space with respect to the -norm, letting with respect to on
. Notice that
B. A Class of Partial Differential Equations be a bounded domain with In this subsection, we let and we let denote smooth boundary where the Laplacian. Also we let and are Sobolev spaces (refer, e.g., to [15, , pp. 84–85] for the definition of Sobolev spaces). For any -norm by we define the (5.21) where
.
(5.25) and , denotes the where . Furthermore, it has solution of (5.25) with been shown (e.g., [16]) that (5.25) determines a -semigroup , where for any . Since , then is an equilibrium for (5.25) [respectively, (5.23)]. Also, it has been shown (e.g., [15, pp. 344–345]) that (5.26)
MICHEL et al.: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM
where into a cube of length . Letting can be characterized as
and
can be put
(i.e., (i.e.,
matrix matrix
) there exists a positive–definite ) such that
in (5.24), system (6.1)
Finally, it is clear that (respectively, ) determines a DDS which is a special case of the DDS . We will denote . this DDS by (respectively, ) asProposition V.3: For system sume that , . and a) If for each (5.27)
b)
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of system then the equilibrium stable with respect to the -norm. If for all
is uniformly
Further, all eigenvalues of have positive real parts if and only there exists a matrix if for every given matrix such that (6.1) holds. The preceding results are used in the stability analysis of , using the Lyapunov functions linear systems and . Now let and denote the smallest and largest eigenvalues of , respectively, and let and denote the smallest and largest eigenvalues of , respectively. Also, let when all eigenvalues of have negative real parts, and let when all eigenhave positive real parts. Assume that values of . , let denote the matrix norm inLet , and define duced by the Euclidean vector norm . , choose now the Lyapunov function For system
(5.28) is a constant, then the equilibrium where of system is uniformly asymptotically stable in the large and exponentially stable in the large, with respect -norm. to the Proof: Setting , it is clear that all hypotheses of Corollary IV.1 are satisfied. This proves the result. Remark V.2: We emphasize that in specific examples of the systems considered in the present section, all required parame, and in (5.7); , and in (5.20); ters [e.g., and , and in (5.27)] are either given, or can be computed, or can be estimated. VI. CONCLUDING REMARKS In this paper, we first formulated two important classes of and DDS determined by linear and nonlinear semigroups ( ) and we showed that under a reasonable assumption, the motions of these systems exhibit the time-invariance property. Next, we established sufficient conditions for various types of stability by invoking the properties of semigroups (Theorem IV.1 and Corollaries IV.1, IV.2, and IV.3) and the Lyapunov function approach (Theorem IV.2). We then applied some of these results in the analysis of three important classes of DDS. We conclude with some additional pertinent comments. Remark VI.1: There are no general rules for choosing Lyapunov functions in results such as Theorem IV.2. However, for cases where converse theorems are available for continuous dynamical systems, we propose a systematic procedure for constructing Lyapunov functions in the application of our results. We demonstrate this by considering a specific class of DDS. given in Remark IV.2. It is well We consider system , have negknown (e.g., [1]) that all eigenvalues of ative real parts if and only if for every given negative–definite
(6.2) . It is easily shown that for all (6.3) When satisfied with and with When is satisfied with
, hypothesis a) of Theorem IV.2 is , assuming that . Note that (4.22) is satisfied as for any . , hypothesis b) of Theorem IV.2 , where . Finally, in all cases (4.18) is satisfied with and , where and . All hypotheses of Theorem IV.2 are satisfied and we have the following result: Under the above assumptions, if for all , then the zero solutions of system is uniformly stable and if , then the zero solution of system is uniformly asymptotically stable in the large and exponentially stable in the large. Remark VI.2: We are not aware of any general stability results for DDS in the current literature which are applicable in the analysis of infinite dimensional systems of the type considered in this paper. However, results for many classes of finite dimensional DDS have been established. We compare in the following one of these with our results. Consider the linear system subject to impulsive effects given by (see [2])
where , and denotes the identity matrix. By applying Corollary IV.3 and Remark IV.2 (using a ), it is easily similar procedure as in the analysis of system
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shown that the equilibrium tially stable in the large if
of system
is exponen-
where . In ([2], p. 61) the same condition for exponential stability in the large is established, using a Lyapunov approach, for the case when . REFERENCES [1] P. J. Antsaklis and A. N. Michel, Linear Systems. New York: McGrawHill, 1997. [2] D. D. Bainov and P. S. Simeonov, Systems with Impulsive Effects: Stability, Theory, and Applications. New York: Halsted, 1989. [3] 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. [4] M. G. Crandall, “Semigroups of nonlinear transformations in Banach spaces,” in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, Ed. New York: Academic, 1971. [5] M. G. Crandall and T. M. Liggett, “Generation of semigroups of nonlinear transformations on general Banach spaces,” Amer. J. Math., vol. 93, pp. 265–298, 1971. [6] 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. [7] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, 1975. [8] W. Hahn, Stability of Motion. Berlin, Germany: Springer-Verlag, 1967. [9] J. K. Hale, Functional Differential Equations. Berlin, Germany: Springer-Verlag, 1971. [10] E. Hille and R. S. Phillips, “Functional analysis and semigroups,” in Amer. Math. Soc. Colloquium Publ.. Providence, RI: Amer. Math. Soc., 1957, vol. 33. [11] S. G. Krein, “Linear differential equations in Banach spaces,” in Translation of Mathematical Monographs. Providence, RI: Amer. Math. Soc., 1970, vol. 29, ch. 1. [12] D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999. [13] 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. [14] A. N. Michel and B. Hu, “Toward a stability theory of general hybrid dynamical systems,” Automatica, vol. 35, pp. 371–384, Apr. 1999. [15] A. N. Michel, K. Wang, and B. Hu, Qualitative Analysis of Dynamical Systems, 2nd ed. New York: Marcel Dekker, 2001. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983. [17] M. Slemrod, “Asymptotic behavior of C semigroups as determined by the spectrum of the generator,” Indiana Univ. Math. J., vol. 25, pp. 783–791, 1976.
[18] Y. Sun, A. N. Michel, and G. Zhai, “Stability of discontinuous retarded functional differential equations with applications,” IEEE Trans. Autom. Control, vol. 50, no. 8, pp. 1090–1105, Aug. 2005. [19] 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. [20] V. I. Zubov, Methods of A. M. Lyapunov and Their Applications. Groningen, The Netherlands: Noordhoff, 1964.
Anthony N. Michel (S’55–M’59–SM’79–F’82 | acroread -toPostScript | lp–LF’95) 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 sixteen 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 the University of Notre Dame. He has also held visiting faculty positions at the Technical University of Vienna, Vienna, Austria, the Johannes Kepler University, Linz, Austria, and the 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.
Ye Sun received the B.S. degree in mathematics from the University of Science and Technology of China in 1999, and the M.S. and Ph.D. degrees from the University of Notre Dame, in 2002 and 2004, respectively. She is currently a Systems Analyst at Credit Suisse First Boston, New York, working on fixed-income models. Her research interests include systems modeling and qualitative analysis of discontinuous dynamical systems.
Alexander P. Molchanov, deceased.
