DISCONTINUOUS dynamical systems (DDS) have motions

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER Stability Analysis of Discountinuous Dynamical Systems Determined by Semigroups Anthony N Michel, Life Fellow, IEEE, Ye Sun, Alexer P Molchanov, Senior Member, IEEE Abstract We present Lyapunov stability results for discontinuous dynamical systems (DDS) determined by linear nonlinear semigroups defined on Banach space DDS of the type considered herein arise in the modeling of a variety of finite- infinitedimensional systems, including certain classes of hybrid systems, discrete-event systems, switched systems, systems subjected to impulse effects, 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 DISCONTINUOUS 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, the like (see, eg, [2], [3], [6], [12] [15], [19], 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, more recently, with infinite-dimensional dynamical systems (defined on ) determined by functional differential equations [18] The results that were established in [14], [15], [19] are formulated for general dynamical systems in a metric space setting, 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 frequently requires further analysis [18] (This is usually also the case for continuous dynamical systems (see, eg, [8], [15], [20]) In this paper, we establish (Lyapunov) stability results for DDS determined by linear 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 USA ( anthonynmichel1@ndedu) Y Sun is with Credit Suisse First Boston, New York, NY USA ( yesun@csfbcom) A P Molchanov, deceased Digital Object Identifier /TAC are applicable to large classes of finite- infinite-dimensional DDS We demonstrate the applicability of our results in the analysis of DDS determined by linear nonlinear retarded functional differential equations a specific initial-value 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 nonlinear semigroups In Section III, we formulate the classes of DDS determined by linear nonlinear semigroups considered in this paper In Section IV, we establish stability results for the classes of DDS considered herein 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, let denote any one of the equivalent norms on For a real matrix (ie, ), let denote the norm of induced by the vector norm Let be Banach spaces let denote norms on Banach spaces Let be a linear operator defined on a domain with range in We call bounded if it maps each bounded set in into a bounded subset of,or equivalently, if it is continuous Given a bounded linear operator, its norm is defined by We let denote the identity transformation Finally, let Then, signifies that belongs to the set of continuous functions from into B -Semigroups We will require the following concepts (see, eg, [9] [11]) Definition II1: A one-parameter family of bounded linear operators, is said to be a -semigroup (or a linear semigroup) if i) ii) for any ; iii) for all -semigroups are generated by linear operators /$ IEEE

2 1278 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER 2005 Definition II2: The infinitesimal generator of a -semigroup is defined as the operator for where is the domain of given by The Hille Yoshida Phillips Theorem provides necessary 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 Theorem II1 [10]: For a -semigroup, there exists an an such that (21) for all The qualitative properties of a -semigroup in terms of the spectrum of its infinitesimal generator is given by the following result Definition II3: A -semigroup is called differentiable for if for each is continuously differentiable on Theorem II2 [17]: If is a -semigroup which is differentiable for, if is its infinitesimal generator, if for all, then given any positive, there is a constant such that for all (22) C Nonlinear Semigroups In the case of -semigroups, is a family of linear operators This restriction is removed in the following Definition II4 [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 II5: A (possibly multivalued) operator is said to generate a nonlinear semigroup on if for all The infinitesimal generator of a nonlinear semigroup is defined by for all such that this limit exists The operator the infinitesimal generator are generally different operators Of particular importance in applications are quasicontractive contraction semigroups A nonlinear semigroup is called a quasicontractive semigroup if there is a number such that (23) for all for all If in (23),, then 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 For a given -semigroup, we define the motion with initial state initial time by -semi- we define the dynamical system determined by a group as the family of motions (24) (25) We note that, in particular, when then for all We will call an equilibrium for the dynamical system we will call the corresponding motion the trivial motion For a given nonlinear semigroup,wedefine the motion with initial state initial time by (26) we define the dynamical system determined by a nonlinear semigroup as the family of motions (27) Henceforth, we will always assume that is in the interior of Once more we note that Throughout this paper, we will assume that for any nonlinear semigroup for all if We will call an equilibrium for the dynamical system we will call the corresponding motion,atrivial motion III DDS DETERMINED BY SEMIGROUPS To motivate the class of DDS which we will consider, to fix some of the ideas involved in our subsequent presentation, we first consider a specific case A An Example In Fig 1, we depict in block diagram form a configuration which is applicable to many classes of DDS, including hybrid

3 MICHEL et al: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM 1279 Fig 1 DDS configuration systems 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, 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, eg, Petri nets, logic comms, various types of discrete event systems, 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 sample 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 (31) where denotes sampling instants, are real matrices of appropriate dimensions, are interface variables, Now, define, Then, at, at for all Let Letting where denotes the identity matrix, then system (31) can be described by the discontinuous differential equation (32) Next, for, let, denote the unique solution of the initial-value problem (33) define the mapping by Using the properties of the solutions of (33), it is easily shown that,isa -semigroup Given a set of initial conditions, the unique solution of (32) [, hence, of (31)] can now be expressed by the function (34) Consistent with the terminology used later, we refer to as a motion By varying over, we generate a family of motions,, the dynamical system determined by (34) We note that, in particular, when, then for all We call an equilibrium the trivial motion 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, 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 Nonlinear Semigroups In the following, we will require a given collection of linear or nonlinear semigroups defined on a Banach space, or on a set, respectively; a given collection of linear continuous operators, or of nonlinear continuous operators ; a given discrete, infinite, unbounded set We assume that when consists of linear semigroups, then consists of linear mappings The number of elements in may be finite or infinite

4 1280 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER 2005 We now consider dynamical systems whose motions with initial time initial state (respectively, ) are given by We define the DDS determined by semigroups as (35) (36) Note that every motion is unique, with, exists for all, is continuous with respect to on, that at may be discontinuous We call the set the set of discontinuities for the motion When in (36), consists of -semigroups consists of linear mappings, we speak of a DDS determined by linear semigroups, we denote this system by Similarly, when in (36), consists of nonlinear semigroups, we speak of a DDS determined by nonlinear semigroups we denote this system by When the types of the elements in are not specified, we simply speak of a DDS determined by semigroups we denote this system, as in (36), by Since in the case of, are linear operators, it follows that in particular for all We call the equilibrium for the DDS, the trivial motion In the case of, we assume that is in the interior of that for all if that if for all From this, it follows that for all if We call an equilibrium,atrivial motion for the DDS Finally, in the case of linear mappings, we use in (35) the notation We conclude the present section with a few remarks Remark III1: For different initial conditions, resulting in different motions, we allow the set of discontinuities, the set of semigroups, the set of functions to differ, accordingly, the notation, might be more appropriate However, since in all cases, all meaning will be clear from context, we will not use such superscripts Remark III2: To the best of our knowledge, the DDS models considered herein ( ) are original have not been considered previously These models are very general include large classes of finite dimensional dynamical systems determined by ordinary differential equations inequalities 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, more generally, differential equations 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, so forth) is confined to finite-dimensional systems determined by ordinary differential equations (eg, [2], [3], [6], [12], [13]) Remark III3: The dynamical system models are very flexible, 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) if for all, where denotes the identity transformation, then reduces to an autonomous, linear, continuous dynamical system 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, eg, [2])], one would choose for all while the 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, so in this case one would choose a finite family of semigroup (see, eg, [6] [12]); so forth Remark III4: Perhaps it needs pointing out that even though systems are determined by families of semigroups ( nonlinearities), by themselves they are not semigroups, since in general, they are time-varying do not satisfy the hypotheses i) iii) given in Definitions II1 II4 However, each individual semigroup, used in describing 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 nonlinear semigroups Before stating 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 by is defined on Recall also that the origin 0 is assumed to be in the interior of that is an equilibrium for both Since the following definitions pertain to both, we will refer to either one of them simply as Definition IV1: The equilibrium of is stable if for every every, there exists a such that for all of for all, whenever ( ) The equilibrium is uniformly stable if is independent of, ie, The equilibrium of is unstable if it is not stable

5 MICHEL et al: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM 1281 Definition IV2: The equilibrium of is attractive if there exists an such that (41) for all of whenever ( ) We call the set of all such that (41) holds the domain of attraction of Definition IV3: The equilibrium of is asymptotically stable if it is stable attractive Definition IV4: The equilibrium of is uniformly attractive if for every every, there exists a, independent of, a, independent of, such that for all for all of, whenever ( ) Definition IV5: The equilibrium of is uniformly asymptotically stable if it is uniformly stable uniformly attractive Definition IV6: The equilibrium of is exponentially stable if there exists, for every every, there exists a such that for all for all of whenever ( ) 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 IV7: The equilibrium of is asymptotically stable in the large if i) it is stable, ii) for every of for all, (41) holds In this case, the domain of attraction of is all of Definition IV8: The equilibrium of is uniformly asymptotically stable in the large if i) it is uniformly stable, ii) it is uniformly attractive in the large, ie, for every every, for every, there exists a (independent of ), such that if, then for all of for all Definition IV9: The equilibrium of is exponentially stable in the large if there exists for every, there exists such that (42) for all of, for all, whenever Although system are determined by semigroups, they themselves are not semigroups 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 Assumption IV1: Assume that any two motions (of or ) with identical initial states but different initial times, say, are determined by identical sequences of semigroups operators Furthermore, assume that if for the motion, then for the motion,wehave, where By applying definitions, it is easily shown that under Assumption IV1, the motions of systems have the property that for all for every, ie, the motions possess the time-invariance property It has been shown (see, eg, [4], [5], [9] [11], [15]) that for dynamical systems whose motions have the time-invariance property, an equilibrium, say, is stable (respectively, asymptotically stable, asymptotically stable in the large) if 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 IV1: Under Assumption IV1, the following statements are true for systems : a) the equilibrium is stable if only if it is uniformly stable; b) the equilibrium is attractive if only if it is uniformly attractive; c) the equilibrium is asymptotically stable if only if it is uniformly asymptotically stable; d) the equilibrium is asymptotically stable in the large if only if it is uniformly asymptotically stable in the large We emphasize that in all subsequent results, Assumption IV1 is not required, the distinction between stability uniform stability (respectively, between asymptotic stability 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 these results that for each nonlinear semigroup there exist constants for each mapping there exists a constant such that (43) (44) for all We recall from Section 2-C (see (23)) that in particular, (43) is always satisfied for a quasicontractive semigroup for some computable parameters,, while for a contractive semigroup, inequality (43) is satisfied with We will require the following additional notation Let, for any given, we let, we let denote the finite products (45)

6 1282 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER 2005 Theorem IV1: a) For system, under (43) (44), assume that for any there exists a constant such that (46) for all, where is defined in (45) Then the equilibrium of is stable b) If in part a),, ie, in (46) can be chosen independent of, then the equilibrium of is uniformly stable c) If in part a), (46) is replaced by (47) for all, then the equilibrium of is asymptotically stable d) If the conditions of part b) are satisfied if in part c) relation (47) is satisfied uniformly with respect to (ie, for every every there exists a, independent of, such that for all ), then the equilibrium of is uniformly asymptotically stable e) Assume that in part a) (46) is replaced by where Assume also that (48) (49) where is a constant Then, the equilibrium of is exponentially stable f) If in parts c) e), respectively, conditions (43) (44) hold for all, then the equilibrium of is asymptotically stable in the large, uniformly asymptotically stable in the large 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 indexes To this end, let, where denotes the integer part of, let Then, we can relabel as as If, we have for Therefore, in view of (45) (410) is true It is clear that Similarly, for, if, then is true for, Therefore, by (45) (410), we have (411) For any, let From (46) (411), it now follows that, whenever Since since for all all we can equate, it follows that the equilibrium of is stable b) In proving part b), note that can be chosen independent of, consequently, can also be chosen independent of Therefore, the equilibrium of is uniformly stable c) From the assumption on it follows that Hence, we have as Since for any for some, then when Hence, it follows from (47) (411) that (41) holds for all of whenever Therefore, the equilibrium of is attractive its domain of attraction coincides with the entire set Since (46) follows from (47), then, as in part a), of is stable Hence, the equilibrium of is asymptotically stable d) Since the conditions of part b) are satisfied, the equilibrium of system is uniformly stable Therefore, we only need to prove that is uniformly attractive Choose in such a way that Since (47) is satisfied uniformly with respect to, then for every every there exists a (independent of ) such that for all Hence, from (411), we have for all for all Let Then, for all If we let, then we have that for all for all of, whenever Hence, the equilibrium of is uniformly attractive uniformly asymptotically stable e) To prove part e), note that as was shown in the proofs of parts a) c), for any any, there exists such that (411) holds Since in view of (49),, therefore, we have Hence, in view of (48), we have For any, let Then, for any with,wehave, where Therefore, the equilibrium of is exponentially stable

7 MICHEL et al: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM 1283 f) We note that if the estimates (43) (44) hold for all, then inequality (411) is valid for all i) Repeating the reasoning in the proof of part c) for any any, we can conclude that in this case (41) holds for all of whenever Therefore, the equilibrium of is asymptotically stable in the large ii) Similarly as in the proof of part d), for every for every there exists a (independent of ), such that for all If we let, then we have that for all for all of, whenever Hence, the equilibrium of is uniformly asymptotically stable in the large iii) For every for every we have similarly as in the proof of part e) that for all, where Let It now follows that the equilibrium of is exponentially stable in the large This completes the proof Corollary IV1: a) For system, assume that the following statements are true i) Condition (43) holds (with parameters ) ii) Condition (44) holds (with parameter ): iii) for all ; iv) for all where are constants; v) for all (412) Then, the equilibrium of is stable uniformly stable b) If in part a), hypothesis v) is replaced by (413) for all, where, then the equilibrium of is asymptotically stable, uniformly asymptotically stable exponentially stable c) If in part a) it is assumed that inequalities (43) (44) hold for all inequality (412) is replaced by (413), then the equilibrium of is asymptotically stable in the large, uniformly asymptotically stable in the large, exponentially stable in the large Proof: a) It is easily shown that in part a) the estimate (46) is satisfied with, independent of Therefore, the conditions in part a) b) of Theorem IV1 are satisfied This proves part a) of the corollary b) In view of (413) the estimate (48) is true with Therefore, the limit relation (47) is satisfied uniformly with respect to This proves part b) of the corollary c) The conclusions of part c) of this corollary follow directly from part f) of Theorem IV1 From Theorem II1, we recall that for any -semigroup, there will exist such that (414) Furthermore, in accordance with Theorem II2, if is a -semigroup which is differentiable for,if is its infinitesimal generator, if for all, then given any positive, there is a constant such that (415) Similarly as in Theorem IV1, we will utilize in our next result the relation where, depending on the situation on h the constants are obtained from either (414) or (415) Similarly as in (45), we define in the case of DDS finite products (416) the (417) where, denotes the norm of the bounded linear operator used in defining the DDS in (35) By invoking Theorem IV1, we obtain in a straightforward manner the following result for system Corollary IV2: In Theorem IV1, replace (43) by (414) (416) (45) by (417) Then, all conclusions of Theorem IV1 are true for system Corollary IV3: In Corollary IV1, replace (412) (413) by, respectively, where denotes operator norm where are given in (416) Then, the conclusions of Corollary IV1 are true for system Remark IV1: Corollaries IV1 IV3 are more conservative than Theorem IV1 Corollary IV2 since in the case of the latter we put restrictions on partial products [see, eg, (46)] while in the case of the former, we put corresponding restrictions on the individual members of the partial products [see, eg, (412)] However, Corollaries IV1 IV3 are easier to apply than Theorem IV1 Corollary IV2 Remark IV2: For linear, finite-dimensional DDS given by

8 1284 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER 2005 where, Corollary IV3 can be improved somewhat by replacing the inequalities in that corollary, eg, by the inequalities for all, where assuming that, where denotes the measure of [see, eg, [7] for the definition of ], or by the inequalities We omit the proofs of the aforementioned assertions due to space limitations In the preceding results, we relied on fundamental definitions to establish various stability properties for systems It turns out that we can also obtain stability results for systems, making use of Lyapunov functions In doing so, we employ the following comparison functions Definition IV10: A function is said to belong to class, (ie, ), if if is strictly increasing on If if, we say that belongs to class (ie, ) Since the following results are applicable to both system, we will use the term system to indicate either system Theorem IV2: Assume that there exists a function functions such that (418) for all, where is a neighborhood of the origin a) Assume that there exists a neighborhood of the origin such that for every motion of, with for all is continuous for all except on a set Also, assume that is nonincreasing for all all, assume that there exists a function, independent of, such that (419) for all Then, the equilibrium of system is uniformly stable b) If in addition to the assumptions in part a), there exists a function such that (420) (421) then the equilibrium of system is uniformly asymptotically stable c) Assume that all assumptions in parts a) (b) are true with Then, the equilibrium of system is uniformly asymptotically stable in the large d) Assume that all assumptions in parts a) b) are true with,, where, are positive constants Furthermore, assume that the function in (419) satisfies as (422) 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: a) Since is continuous, then for any there exists such that as long as We assume that, Thus, for any, as long as the initial condition is satisfied, then for, since is nonincreasing Furthermore, for any, we can conclude that Therefore, by definition, the equilibrium of system is uniformly stable b) Letting, we obtain from the assumption of the theorem, that for If we let, then the previous inequality becomes Since is nonincreasing, it follows that for all We thus obtain that for all It follows that (423)

9 MICHEL et al: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM 1285 Now, consider a fixed For any given,we can choose a such that Let We obtain from that (424) since For any with 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 (423) that, which implies that which yields (427) where If is true for some, then for all all Thus, for all In the following, we assume that for all Since, it follows from (427) that (425) Hence (426) In the case when, it follows from (425) that, noticing that (419) holds In the case when, we can conclude from (426) that This proves that the equilibrium of system is uniformly asymptotically stable c) From part a), the equilibrium of system is uniformly stable We need to show that the equilibrium is uniformly attractive in the large For any fixed, we can choose a a such that Let For any with any, we can show that whenever This is true since for any, we can find some such that Therefore,, which implies that is true for all that have made use of the fact that easily seen that It now follows from In the last step, we From (422), it is Let Then, for all It follows from relations (419) (422) that for all, it is true that The last inequality follows since Thus, For any any such that, let Thus, we have when In the case when, we can conclude from above that This proves that the equilibrium of system is uniformly asymptotically stable in the large d) We only prove the case of global exponential stability The case of local exponential stability can be proved similarly Then for all of Therefore, the equilibrium of system is exponentially stable in the large

10 1286 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER 2005 We close by noting that in the previous proofs, we did not require the semigroup property ii) in Definitions II1 II4, 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 IV3: The general stability results concerning DDS reported in the current literature (eg, [2], [3], [6], [12], [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 time delays (as in the case of differential-difference equations), hysteresis effects (as in the case of functional differential equations Volterra integrodifferential equations), 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 IV4: The general stability results for DDS reported in the current literature (eg, [2], [3], [6], [12] [14], [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 IV1 Corollaries IV1 IV3 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 IV5: As pointed out in Remark IV4, the Lyapunovtype results given in Theorem IV2 are in general more difficult to apply than Theorem IV1 Corollaries IV1, IV2, IV3 However, the very ambiguity involved in the search for suitable Lyapunov functions offers flexibility in the application of Theorem IV2 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 IV6: In the following comments, we compare the results of Theorem IV2 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, eg, [3], [13], [15]) The asymptotic stability results in [2, Ch 16, pp ] involve the existence of Lyapunov functions which are strictly decreasing along the system s motions over the intervals, at the points of discontinuity,, the Lyapunov functions are required to have downward jumps In contrast to this, Theorem IV2 requires that when evaluated along the system s motions, the Lyapunov functions be strictly decreasing only at the points of discontinuity,, that over the intervals,, the Lyapunov functions be only bounded in a certain way [refer to inequality (419)] This allows us to consider systems which exhibit, eg, unstable behavior over some or all of the intervals, as long as there is sufficient attenuation provided by the functions (given in (35) at the points of discontinuity ) This is not possible in the results given in [2] (or in other similar results given in [3], [13], [15]) Accordingly, even in the finite dimensional case, the results of Theorem IV2 are less conservative than many of the existing results Remark IV7: It is possible to establish Lagrange stability results (ie, uniform boundedness 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; an initial-value 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 the function determined by for 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 are given collections of mappings ( ) is a given infinite unbounded discrete set We assume that for all (51) for all, where is a finite constant Also, we assume that that satisfies the Lipschitz condition for all For every, the initial value problem (52) (53) possesses a unique solution for every initial condition which exists for all with Therefore, it follows that for every possesses a unique solution which exists for all, given by (54)

11 MICHEL et al: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM 1287 Note that Also, note that is continuous with respect to on that at may be discontinuous Furthermore, note that is an equilibrium of that for all Remark V1: System merits perhaps some additional comments First, we note that the state space for this system is that for, the state of this system,, evolves according to the first equation in (54), starting at with the initial state (which is defined over the time interval ) Next,, the state at, is mapped by the function into the state, the initial state for the next interval [refer to the second equation in (54)] Note that is defined over the interval, where, while is defined over the interval Next, for the initial-value problem (53) we define From the properties of the solutions of (53), it is easily shown that is a nonlinear semigroup on In fact, is a quasicontractive semigroup, (55) for all all, where is given in (52) (see, eg, [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 V1: a) For system (resp, ) assume the following i) For each, the function satisfies the Lipschitz condition (52) with Lipschitz constant for all, where is a neighborhood of the origin ii) For each, the function satisfies condition (51) with constant for all iii) For each, iv) For all (56) Then, the equilibrium of system (resp, )isuniformly stable b) In part a), replace iv) by the following hypothesis v) For all (57) Then the equilibrium of system (resp, )isuniformly asymptotically stable exponentially stable c) In part a), replace iv) by hypothesis v) assume that conditions (51) (52) hold for Then, the equilibrium of system (resp, )isuniformly asymptotically stable in the large exponentially stable in the large Proof: In view of (55), we have, since (58) for all,, resp, Setting,, we can see that all hypotheses of Corollary IV1 are satisfied This completes the proof 2) Dynamical Systems Determined by Linear Semigroups: Now assume Wedefine the linear mapping from to by the Stieltjes integral to obtain the initial-value problem (see, eg, [9]) (59) (510) In (59), is an matrix whose entries are assumed to be functions of bounded variation on Then, is Lipschitz continuous on with Lipschitz constant less or equal to the variation of in (59) In this case, the semigroup is a -semigroup The spectrum of its generator consists of all solutions of the equation (511) If in particular, all the solutions of (511) satisfy the relation, then it follows from Theorem II2 that for any positive, there is a constant such that (512) When the previous assumptions do not hold, then in view of Theorem II1, we still have the estimate (513) for some Next, let where is defined similarly as in (59) by let where is assumed to be a linear operator Then, system assumes the form It is clear that determines a DDS determined by linear semigroups which is a special case of We will denote this dynamical system by

12 1288 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER 2005 In the following, when all the solutions of the characteristic equation satisfy the condition, then given any, there is a constant such that 1) DDSs Determined by the Heat Equation: Now, consider the DDS determined by the equations [see (512)] Otherwise, we still have the estimate for some [see (513)] When (514) applies, we let in the following when (515) applies, we let Thus, in all cases, we have the estimate Proposition V2: a) For system, assume the following: i) for each, ; ii) for each (514) (515) (516) (517) (518) (519) where, are constants, is a given family of mappings,, is a given infinite unbounded discrete set We assume that that there exists a constant such that (522) for all Associated with we have a family of initial boundary value problems determined by (523) It has been shown (eg, [16]) that for each, the initial boundary value problem (523) has a unique solution, such that for each fixed is a continuously differentiable function from to with respect to the -norm given in (521) It now follows that for every possesses a unique solution which exists for all, given by where are given in (514) (517) Then, the equilibrium of system is uniformly stable b) In part a), replace (519) by (520) Then, the equilibrium of is uniformly asymptotically stable in the large exponentially stable in the large Proof: The proof follows directly from Corollary IV3 B A Class of Partial Differential Equations In this subsection, we let be a bounded domain with smooth boundary we let denote the Laplacian Also we let where are Sobolev spaces (refer, eg, to [15, pp 84 85] for the definition of Sobolev spaces) For any, we define the -norm by where (521) (524) with Notice that is continuous with respect to on, that at may be discontinuous Furthermore, is an equilibrium for that for all is a trivial motion For each, (523) can be cast as an initial-value problem in the space with respect to the -norm, letting (525) where, denotes the solution of (525) with Furthermore, it has been shown (eg, [16]) that (525) determines a -semigroup, where for any Since, then is an equilibrium for (525) [respectively, (523)] Also, it has been shown (eg, [15, pp ]) that (526)

13 MICHEL et al: STABILITY ANALYSIS OF DISCOUNTINUOUS DYNAMICAL SYSTEM 1289 where can be put into a cube of length Letting in (524), system can be characterized as Finally, it is clear that (respectively, ) determines a DDS which is a special case of the DDS We will denote this DDS by Proposition V3: For system (respectively, ) assume that, a) If for each (527) then the equilibrium of system is uniformly stable with respect to the -norm b) If for all (528) where is a constant, then the equilibrium of system is uniformly asymptotically stable in the large exponentially stable in the large, with respect to the -norm Proof: Setting, it is clear that all hypotheses of Corollary IV1 are satisfied This proves the result Remark V2: We emphasize that in specific examples of the systems considered in the present section, all required parameters [eg,, in (57);, in (520);, in (527)] are either given, or can be computed, or can be estimated VI CONCLUDING REMARKS In this paper, we first formulated two important classes of DDS determined by linear nonlinear semigroups ( ) 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 IV1 Corollaries IV1, IV2, IV3) the Lyapunov function approach (Theorem IV2) We then applied some of these results in the analysis of three important classes of DDS We conclude with some additional pertinent comments Remark VI1: There are no general rules for choosing Lyapunov functions in results such as Theorem IV2 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 We consider system given in Remark IV2 It is well known (eg, [1]) that all eigenvalues of, have negative real parts if only if for every given negative definite matrix (ie, ) there exists a positive definite matrix (ie, ) such that (61) Further, all eigenvalues of have positive real parts if only if for every given matrix there exists a matrix such that (61) holds The preceding results are used in the stability analysis of linear systems, using the Lyapunov functions Now let denote the smallest largest eigenvalues of, respectively, let denote the smallest largest eigenvalues of, respectively Also, let when all eigenvalues of have negative real parts, let when all eigenvalues of have positive real parts Assume that Let, let denote the matrix norm induced by the Euclidean vector norm, define For system, choose now the Lyapunov function It is easily shown that for all (62) (63) When, hypothesis a) of Theorem IV2 is satisfied with, assuming that Note that (422) is satisfied with as for any When, hypothesis b) of Theorem IV2 is satisfied with, where Finally, in all cases (418) is satisfied with, where All hypotheses of Theorem IV2 are satisfied we have the following result: Under the above assumptions, if for all, then the zero solutions of system is uniformly stable if, then the zero solution of system is uniformly asymptotically stable in the large exponentially stable in the large Remark VI2: 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, denotes the identity matrix By applying Corollary IV3 Remark IV2 (using a similar procedure as in the analysis of system ), it is easily

14 1290 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 9, SEPTEMBER 2005 shown that the equilibrium of system is exponentially stable in the large if 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 A N Michel, Linear Systems New York: McGraw- Hill, 1997 [2] D D Bainov P S Simeonov, Systems with Impulsive Effects: Stability, Theory, Applications New York: Halsted, 1989 [3] M S Branicky, Multiple Lyapunov functions other analysis tools for switched hybrid systems, IEEE Trans Autom Control, vol 43, no 4, pp , Apr 1998 [4] M G Crall, Semigroups of nonlinear transformations in Banach spaces, in Contributions to Nonlinear Functional Analysis, E H Zarantonello, Ed New York: Academic, 1971 [5] M G Crall T M Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer J Math, vol 93, pp , 1971 [6] R DeCarlo, M Branicky, S Pettersson, B Lennartson, Perspectives results on the stability stabilizability of hybrid systems, Proc IEEE, vol 88, no 7, pp , Jul 2000 [7] C A Desoer 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 R S Phillips, Functional analysis 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 A S Morse, Basic problems in stability 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 , Jan 1999 [14] A N Michel B Hu, Toward a stability theory of general hybrid dynamical systems, Automatica, vol 35, pp , Apr 1999 [15] A N Michel, K Wang, B Hu, Qualitative Analysis of Dynamical Systems, 2nd ed New York: Marcel Dekker, 2001 [16] A Pazy, Semigroups of Linear Operators 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 , 1976 [18] Y Sun, A N Michel, G Zhai, Stability of discontinuous retarded functional differential equations with applications, IEEE Trans Autom Control, vol 50, no 8, pp , Aug 2005 [19] H Ye, A N Michel, L Hou, Stability theory for hybrid dynamical systems, IEEE Trans Autom Control, vol 43, no 4, pp , Apr 1998 [20] V I Zubov, Methods of A M Lyapunov Their Applications Groningen, The Netherls: Noordhoff, 1964 Anthony N Michel (S 55 M 59 SM 79 F 82 acroread -topostscript lp LF 95) received the PhD degree in electrical engineering from Marquette University, Milwaukee, WI, the DSc degree in applied mathematics from the Technical University of Graz, Graz, Austria He has seven years of industrial experience 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 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, the Ruhr University, Bochum, Germany He is the author or coauthor of eight books numerous archival publications His more recent work is concerned with stability analysis of finite- infinite-dimensional dynamical systems qualitative analysis synthesis of recurrent neural networks Dr Michel has rendered substantial service to several professional organizations, especially the IEEE Circuits Systems Society the IEEE Control Systems Society He has been honored by a number of prestigious awards for his work as an educator scholar Alexer P Molchanov, deceased Ye Sun received the BS degree in mathematics from the University of Science Technology of China in 1999, the MS PhD degrees from the University of Notre Dame, in , 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 qualitative analysis of discontinuous dynamical systems