Dashk vskiy Bj orn S uf fer abian R irth Abstract or tw classes of monotone maps on the dimensional positi orthant we sho that or discr ete dynamical system induced by map the origin of is globally asymptotically stable if and only if the map is suc ID: 25805 Download Pdf

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Dashk vskiy Bj orn S uf fer abian R irth Abstract or tw classes of monotone maps on the dimensional positi orthant we sho that or discr ete dynamical system induced by map the origin of is globally asymptotically stable if and only if the map is suc

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Discr ete time monotone systems: Criteria or global asymptotic stability and applications Ser ge N. Dashk vskiy Bj orn S. uf fer abian R. irth Abstract or tw classes of monotone maps on the -dimensional positi orthant we sho that or discr ete dynamical system induced by map the origin of is globally asymptotically stable, if and only if the map is such that or any point in the image-v ector is such that at least one component is strictly less than the corr esponding component of One class is the set of matrices of class functions; these induce monotone operators on Maps of the

other class satisfy some geometric pr operty or an in ariant set. eyw ords monotone maps, spectral radius, one compo- nent decr ease condition, global asymptotic stability consider monotone (that is order preserving) maps which map the nonne ati orthant of the dimensional Euclidean space into itself. Such map gi es rise to an associated dynamical system dened by 1) ( )) for (1) and (0) endo with the standard partial order dened by (2) If is linear the stability condition that the spectral radius of is less than is equi alent to the operator inequality there xists no

such that ( ); (3) note that the latter condition is also meaningful for nonlin- ear call monotone operators satisfying (3) nowher incr easing (on ). The concern of this paper are the relations between property (3) and global asymptotic stability of (1) for nite dimensional linear these are of course equi alent. Monotone maps satisfying (3) arise in the conte xt of lar ge-scale interconnections of input-to-state stable (ISS) subsystems. Here the map arises as matrix whose en- tries are strictly increasing functions (class -functions), which describe the interconnection ains

between ISS subsystems. Recently the authors pro ed in [4 that if there xists `rob ustness' operator such is no where increasing, then the lar ge-scale interconnection system is also ISS. S. Dashk vskiy and B. uf fer are with the Zentrum ur echnomath- ematik, Uni ersit at Bremen, German dsn@math.uni-bremen.de rueffer@math.uni-bremen.de irth is with the Hamilton Institute, NUI Maynooth, Ireland, fabian.wirth@nuim.ie The rst small-g ain theorem for the feedback intercon- nection of tw ISS systems as gi en by Jiang, eel and Praly in [8]. Since then man more ISS small-g ain type theorems

follo wed, for references and discussion see [4]. Monotone maps induced by matrices of class func- tions and their dynamics are of course ery special case. The theory of more general monotone maps and induced dynamics is still an acti eld of research, see for xample [11 or [12 ]. In [6] Hirsch and Smith gi state of the art ervie on discrete-time monotone dynamics. ork related to this article is [1], where Angeli and Sontag present results on monotone systems that were inspired by questions in molecular biology modeling. In [2] the introduce signed graphs for monotone maps, which in the

case of matrices with entries in agree with the graphs that we associate to such maps. are going to in estig ate the relations between prop- erty (3) and global asymptotic stability of the origin (0- GAS) of (1) for general monotone maps Using only monotonicity we establish (3) as consequence of 0-GAS of (1) in Proposition 5.2, while the con erse implication is true for matrices with entries in if we add `rob ustness term' in (3), see Theorem 7.10. In tw dimensions we can sw ap the matrix structure for geometrical properties of an in ariant set to obtain similar result (Proposition 6.3).

rigorous problem statement will be gi en in Section II. Notational foundations and some remarks on class functions follo in Sections III and IV. Results requiring only monotonicity continuity and/or condition (3) are stated in Section V, while Section VI deals with the tw dimensional case. The theory for matrices with entries in is presented in Section VII. In this case the special structure allo ws for stronger statements. Con- sequences for lar ge-scale `small-g ain' interconnections of ISS systems are gi en in SectionVIII. or monotone map such that (0) we are interested in the

relations between the follo wing tw properties: 1) id, that is, by denition for all we ha ( and 2) the discrete system dened by (0) 1) := ( )) (4) is globally asymptotically stable in (0-GAS), i.e.,

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a) (Stability) or ery there xists such that whene er (0) then for all b) (Attracti vity) as Note that so ar we did not mak an continuity assump- tions whatsoe er will pro vide the follo wing answers: Proposition 5.2 states that 2b) implies 1. Lemma 6.3 gi es the con erse implication and en stability for for continuous if in addition for diagonal rob

ustness operator we ha id and some constraints on the geometry of the set which is to be dened in equation (6), are satised. If is matrix with entries in the same is pro ed without the restriction on in Theorem 7.10. Finally in the conte xt of lar ge-scale netw orks of ISS systems this gi es suf cient condition for the input-to-state stability (ISS) of the interconnected system, see Theorem 8.1. A. Number s, or dering By we denote the positi inte gers, by the union by the real numbers, by the nonne ati real numbers, and by the nonne ati orthant of On the latter we ha

partial ordering, which is induced by the componentwise ordering on or x; the relation is dened by for all While the denition of agrees with the one in (2) it is important to note that the meaning of does not coincide with the usual meaning, and Consequently the notation means that there xists an inde such that or maps A; we dene as point wise relation with the xception of the origin; that is, there xists no such that The map is monotone if for x; such that we ha monotone map is nowher incr easing if id. B. Comparison functions Recall that function is of class or if (0)

is strictly increasing and continuous. If, in addition, is unbounded, then we say is of class (or ). Sometimes we enrich these spaces by the function mapping erything to which we denote by or function is said to be of class if it is of class in the rst ar gument and, whene er the rst ar gument is x ed, it is decreasing in the second ar gument with limit zero. C. Matrices of -functions, or dering for monotone maps By we denote the set of -matrices with elements in Gi en matrix and ector we dene the ector ( by ( := =1 ij for n: Note, that if all ij

are linear then this denition is compatible with the usual matrix ector multiplication from linear algebra. D. Pr ojections On for an inde set we denote by the projection of the coordinates in corresponding to the indices in onto The corresponding injection is mapping 7! where we write and denote by =1 ;:::;n the standard basis of or an inde set and ector denote by the ector in with elements if and otherwise. E. Gr aphs, adjacency matrix ith an matrix with entries in we associate (directed) graph with ertices and edges which consists of all ordered pairs i; such that ij 6 The adjacency

matrix of graph with and is the matrix := ij with ij if i; and else. Gi en we dene matrix := diag id := ij id )) where ij denotes the Kroneck er symbol, ij if else. nonne ati matrix is irr educible if for ery i; there is such that the i; -th entry of is positi e, which is denoted by ij This can be stated equi alently as that the graph of is strongly connected, i.e., from an erte there is path to an other erte x. If the number can be chosen independently of i; then is primitive If is not irreducible, then it is educible In Section VII we will need the graph concept also for po wers of

this end let and consider for some Clearly the map is monotone, continuous and satises (0) ith

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we associate the graph ( with ertices and edges i; 7! (( )) is unbounded i; 7! (( )) is of class or notational simplicity we just say that map has some of the properties as (ir -) reducibility/primiti vity if the adjacenc matrix of the graph () does ha this property In the sequel we sometimes refer to as the adjacenc matrix of the graph of In [4], as well as in the follo wing sections, the notion is used, being diagonal matrix with entries in monotone

map. Proposition 4.3 states equi alent formulations of this notion that will be useful in the proof of Lemma 6.1, ut may be of independent interest. be gin with tw technical observ ations. Lemma 4.1: or an there xist such that id id id Pr oof: Choose, e.g., and id Then id id id id id id id id id id This immediately xtends to diagonal operators. Lemma 4.2: or ery diag id there xist diag id and diag id such that Note, as consequence, for as in Lemma 4.2, there xists diag id such that This class function is dened by min for So in Proposition 4.3 without loss of generality we may en assume

that (using monotonicity of ): Pr oposition 4.3: or the follo wing are equi alent: (i) diag id id, (ii) diag id id, (iii) diag id diag id id. The easy equi alence transformations are omitted. In this section we in estig ate the problem stated in Section II for maps that are only monotone. or some results we also need continuity of rst general result is Proposition 5.2 Lemma 5.1: Let be monotone, (0) Assume id. Then for an inde set such that we ha id Pr oof: The easy proof is left to the reader One of the highlights of this section, though not dif cult to pro e, is the

follo wing statement, that terms of the problem of Section II for general already property 2b implies property 1: Pr oposition 5.2: Let be monotone, (0) Then attracti vity of for the associated discrete dynamical system implies id. Pr oof: ar gue indirectly: Suppose there xists an such that ( By monotonicity of we ha ( )( and inducti ely Hence 6! as contradicting 2b, i.e., lim !1 The follo wing result will pro as po werful tool in the indirect proofs of se eral results to follo Lemma 5.3: Let be monotone, (0) Then id implies id for all Pr oof: ar gue indirectly:

Suppose there xists an and such that Dene := max =0 ;:::;k Since the case is included we ha and by monotonicity obtain ( max =1 ;:::;k max =0 ;:::;k max =0 ;:::;k which in turn contradicts id. Hence the lemma is pro ed. If in addition is continuous then already boundedness of trajectory of system (1) implies that this trajectory con er ges to zero; this act will be used frequently in the follo wing. Lemma 5.4: Let be monotone and con- tinuous, (0) or x ed let =0 ;::: be bounded and let id. Then as Pr oof: Consider the -limit set of := subsequence =1 ;::: s.t. as

1g Since an bounded sequence in contains con er gent subsequence is not empty Note that by continuity of the set is in ariant under or an the image ( is also in and there xists preimage such that ( dene := sup which is nite. or ery we ha and hence ( ( By in ariance

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this yields ( sup ( But this contradicts id if Hence In the follo wing we will occasionally mak use of properties of the sets dened belo Suppose an operator is gi en. Then dene := (( )) (5) := =1 ( (6) := (( ))

(7) := =1 ( (8) If there is no ambiguity re arding the operator then we will omit the north-east inde Ob viously we ha Lemma 5.5: Let be monotone and continuous, (0) Then the follo wing are equi alent: (i) id (ii) =1 Pr oof: This is easily seen, so the proof is left to the reader The ne xt result has interesting consequences: Under mild assumptions there are points arbitrarily ar ay from the origin, such that for ery initial alue with for system (1) the corresponding trajectory is attracted to Pr oposition 5.6: Let be monotone and continuous, (0) Then id implies for all

where denotes the sphere around the origin in of radius with respect to the 1-norm, =1 or the proof of this proposition we need amous result, that we state here for the con enience of the reader: Theor em 5.7 (Knaster -K ur atowski-Mazurkie wicz, 1929): Let denote unit -simple x, and for ace of let (0) denote the set of ertices of If amily (0) of subsets of is such that all the sets are closed or all are open, and each ace of is contained in the corresponding union (0) then there is point common to all the sets. Pr oof: The original proof for closed sets as gi en in [9], while the formulation

abo is tak en from [7] and as pro ed in [10]. Pr oof: [Proof of Proposition 5.6] Note that for is simple with ertices Each (nonempty) ace spanned by fullls the assumptions of the Knaster urato wski-Mazurk wicz theorem, i.e., it is contained in the union ( Then the KKM-theorem implies that ( Lemma 5.8: Let be monotone, (0) id. Then each trajectory of gi en by (4) starting in is bounded. Pr oof: This follo ws easily by monotonicity of since implies ( and iterated application of gi es ( for all As consequence we ha the follo wing intermediate result, that is also

used in the companion paper [5]. Pr oposition 5.9: Let be monotone and continuous, (0) id. Then ery trajectory of system (1) starting in is attracted to Pr oof: This follo ws from Lemma 5.8 and Lemma 5.4. So ar we only considered attracti vity of the origin of ut ne glected stability As it turns out, the latter is consequence of the rst: Lemma 5.10: Let be monotone and continuous, (0) and id. If ery trajectory of system (4) is attracted to the origin, then the origin is also stable for system (4) Pr oof: By Proposition 5.6 and gi en we may choose =1 where denotes the

sphere of radius in with respect to the 1-norm. Dene by := sup (0) where (0) denotes the open ball of radius with respect to the 1-norm around the origin. Clearly for we ha By the choice of we ha ( and hence ( The same applies for hence for all which pro es stability As consequence of Proposition 5.9 we obtain: Cor ollary 5.11: Let be monotone and continuous, (0) id. If for initial alue of system (1) there xists and some such that then the corresponding trajectory con er ges to Another easy ut important consequence is the ne xt statement, that relies on the tw

preceding results: If the set is unbounded in ery component, then (1) is 0-GAS. Pr oposition 5.12: Let be monotone and continuous, (0) id. If for ery there xists such that then the origin of is GAS for system (1). Pr oof: or ery initial alue of (1), there xists such that By monotonicity of we ha for all The remainder follo ws by application of Proposition 5.9 and Lemma 5.10. or completeness, we state the by no ob vious result: If is bounded, then we immediately get 0-GAS of (1). Lemma 5.13: Let be monotone, contin- uous, and bounded, such that id. Then system (1) is 0-GAS. Pr oof: Note

that (0) is consequence of id and continuity No apply Lemma 5.4 and Lemma 5.10.

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or matrix with entries in it has been sho wn in [4] that id for some diag id (9) where is suf cient to deduce that (1) is 0-GAS. In the case we pro vide dichotomy for more general such that if (9) holds, then for subsequence at least one component of each trajectory of (1) con er ges to Lemma 6.1: Let be monotone and continuous, (0) Suppose there xists an operator diag id such that id. Suppose further that for some the orbit =0 ;::: is unbounded. Then there xists subsequence

such that as xactly one of the follo wing holds: (i) ( )) and ( )) (ii) ( )) and ( )) Pr oof: First note, that by assumption and Lemma 4.2 there xist operators diag id such that and therefore id. By Lemma 5.3 we en ha id for all Since is unbounded, necessarily one compo- nent must be unbounded, and without loss of general- ity that is the rst component. So we ha to sho that ( )) as may pick subsequence =0 such that the rst component of )) := := is strictly monotone and unbounded. The sequence must be strictly decreasing, since otherwise we ould ha +1 ut we kno +1 It remains to

pro that decreases to zero. So suppose there xists constant such that Then for ]0 )[ there xist and such that No for we ha )) for some 00 for some 00 00 ))) )) where 00 The second component of the last line can be estimated ))] id )( 00 and for the rst component we nd ))] ogether this gi es point such that )) in contradiction to id Hence we must ha This pro es the lemma. In the follo wing xample all assumptions of Lemma 6.1 are satised, ut nonetheless only the second component of trajectories with certain initial alues con er ge to while the rst component di er

ges. Example 6.2: Fix some real constants ]0 1[ and Let be gi en by s s s for all 22 The map fullls (0) is continuous, monotone, and for (1 id where = it satises id, as can easily be seen. (Just consider the cases and separately .) If is such that s then the trajectory of (4) starting in is unbounded in the rst component: The condition s )) with implies 1) )( s )) = )) s 1)) and clearly 1) By induction we obtain trajectory that con er ges to in the second component and di er ges in the rst one.

Hence the monotone system induced by is not 0-GAS. Geometrically we ha s s (1 s The picture in Figure is dra wn for and 16 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 s Fig. 1. Attracting and repelling sets in Example 6.2 In the ne xt result the assumptions on LL and (acron yms of lo wer -left and upper -right) roughly state that the set has entually to stay ay from the coordinate ax es.

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Pr oposition 6.3: Let be monotone and continuous, (0) Suppose there xists an operator diag id such that id. Suppose that the set

satises the conditions (i) LL := inf for some is nondecreasing in and there xists an such that LL (ii) := inf for some is nondecreasing in and there xists an such that Then system (4) is 0-GAS. Pr oof: By our assumptions, there xists max such that := min LL or each initial alue of (4) by Lemma 6.1 we nd an inde such that one component of is less that while the other is greater than So without loss of generality assume and Hence we nd point such that By Corollary 5.11 the proof is complete. This section is de oted to the special case that is matrix with entries in Of

course, if ery entry of is bounded, then by Lemma 5.13 it follo ws that (4) is 0-GAS. So here we consider which will lead to the satisfying result that if is irreducible, then is no where increasing if and only if system (1) is 0-GAS (Theorem 7.7). If is reducible we reformulate the problem in terms of diagonal rob ustness operator There xists diagonal rob ustness operator such that is no where increasing if and only if there xists diagonal rob ustness operator such that denes 0-GAS system (Theorem 7.10). In later proofs in this section we will rely on the follo wing tw acts: Lemma

7.1: Let Let be the adjacenc matrix of for Then ij ij Lemma 7.2: Let Let for and dene diag Let be permutation matrix and let Then the graphs of and coincide and the graph of is the graph of the former maps with renumbered ertices. The proofs are not dif cult (just write do wn the respec- ti maps xplicitly) and left to the reader A. The irr educible case Lemma 7.3: Let be primiti and id. Then for an the sequence is bounded. Pr oof: Suppose there xists an such that lim sup !1 and id. Let =1 ;::: ;n denote the standard basis of Since is primiti there is such that the graph ( is

fully connected (i.e., an i; ( for i; ), see Lemma 7.1. Hence there xists such that for all and some x ed we ha (recall, this means ( )) ). Since lim sup !1 there xists and inde such that ( )) By monotonicity of we ha max ( )) ( )) x: This contradicts id for all which is asserted by Lemma 5.3. The proof is complete. Remark 7.4 (coor dinate hang e): By change of coor dinates, using permutation matrix it is possible to rearrange the adjacenc matrix of There are tw main cases: Either is irreducible, or it is not. In the latter case, is similar to an upper triangular block matrix of the

form AP 11 22 where all diagonal blocks are irreducible or zero. No let be irreducible. By well kno wn results (see, e.g., [3, Chapter 2: Theorems 2.20, 2.30, 2.33],[4 ]) from graph theory either is primiti e, or there xists an inte ger such that is similar to block diagonal matrix with primiti blocks i.e., there xists permutation matrix such that diag (10) Lemma 7.5: trajectory dened by (4) is unbounded if and only if for an the sequence dened by 1) := )) (0) (0) (11) is unbounded. The easy proof is omitted. Lemma 7.5 allo ws us to consider instead of when we in estig ate

boundedness of orbits of (4) No we are able to establish consequence of Lemma 7.3: Lemma 7.6: Let be irreducible and no where increasing. Then for an the sequence is bounded. Pr oof: Let be the adjacenc matrix of and assume is similar to the right hand side form of (10) for some Let denote the indices

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corresponding to Since we ha ( )) ( )) and each ( j )) satises the premises of Lemma 7.3, by Lemma 7.5 the problem reduces to parallel applications of Lemma 7.3. Theor em 7.7: Let be irreducible. Then id if and only if system (4) is globally asymptotically stable

in Pr oof: If is 0-GAS, then in particular each trajectory is attracted to zero, hence Proposition 5.2 estab- lishes id. Zero is an equilibrium point and each trajectory of the system is globally attracted to zero by Lemma 7.6 and Lemma 5.4. Stability follo ws from Lemma 5.10. B. The educible case Let be reducible. ithout loss of generality by Remark 7.4, we may assume that the adjacenc matrix of () is 11 12 22 (12) where 11 is irreducible, or equi alently stated 11 12 22 (13) with 11 an irreducible matrix with entries of class and 12 and 22 some and, respecti ely matrices with entries

in Note that 11 and 22 for all or x ed (0) we recursi ely dene sequence by 1) := ( )) (14) also consider the projected sequences := )) for (15) This moti ates the follo wing statement. Lemma 7.8: Let be reducible and of the form (13). Suppose that there xists diag id such that id. Suppose there are blocks on the diagonal, all irreducible and the th block of dimension such that we ha or we dene the sets for +1 ::: by := := ::: !) Then we ha 1) for an +1 ::: it holds that 2) for an +1 ::: we ha and 3) Pr oof: The proof is straightforw ard and thus omitted.

Dene subsystem of system (4) as the projected dynamical system on 1) 1)) 22 )) (16) Lemma 7.9: Let be reducible. Let diag id for some Assume that id. Let satisfy equation (13), such that 11 is irreducible. If system (4) is such that system (16) is globally attracted to zero, then for system (4) the origin of is GAS. Pr oof: First note, that by our assumptions the system +1 11 )( with initial alue is 0-GAS, since 11 id satises the premises of Theorem 7.7. Fix an initial alue (0) (0) (0) Fix some By of Lemma 7.8 there xists such that 6 Further there xists such that for all Hence,

for and 6 we ha 1) 11 )) Inducti ely as long as 6 we get 11 )) so at some point we arri at an such that (since 11 is assumed to by 0-GAS). Let (( Since 11 is 0-GAS, there xists bounded set 11 ithout loss of generality is such that implies for all since ( implies ( ( ( No if then 1) There remain tw cases: 1) 1) Then 2) 2) 1) 62 Hence 1) and therefore we nd 1) Hence 2) 11 1)) It follo ws, that is bounded. No we apply Lemma 5.4 which gi es us !1 Since we already kno that !1 we can deduce stability from Lemma 5.10. This completes the proof. The main

result of this section is the follo wing: Theor em 7.10: Let Then the follo wing are equi alent: 1) There xists such that for diag id we ha id. 2) There xists such that for diag id the discrete dynamical system dened by (0) 1) := ( )) (17)

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is globally asymptotically stable in Pr oof: 2: First note, that by Lemma 4.2 we can decompose then apply Lemma 7.9 inducti ely 1: This follo ws by an application of Proposition 5.2 with replaced by In [4] an xample as presented, that id alone does not imply that system (1) is 0-GAS. In [4] the authors sho wed that xistence

of such that for diag id the condition id (18) implies input-to-state stability (ISS) of lar ge scale dimensional dynamical system. This lar ge-scale system is gi en by x; (19) and can be decomposed into smaller -dimensional systems with gi en by n; (20) where Suppose each system satises the standard assumptions for xistence and uniqueness of solutions and is forw ard-complete. Assume that each fullls the ISS condition: There xists function and functions ij with ii such that for all (0) and the estimate (0) =1 ij [0 ;t [0 ;t (21) holds. Here is the nonlinear gain matrix simply

dened by ij The functions ij and are called ains in this conte xt, hence the name. No by putting the results of the pre vious sections together we can state the main result of this section: Theor em 8.1: Let Then the follo wing are equi alent: 1) There xists such that for diag id we ha id. 2) There xists such that for diag id the discrete dynamical system dened by (0) 1) := ( )) (22) is globally asymptotically stable in Both imply ISS of the corresponding lar ge-scale dynamical system dened by (19). Pr oof: The equi alence has already been pro ed in Theorem

7.10. In [4] the authors pro ed that 1) implies that is ISS. This claries and establishes the role of the discrete dynamical system associated with as suf cient stability criterion for the lar ge-scale system A. The equivalence of 0-GAS and id for mor ener al monotone maps In this section the graph structure associated to matrices with entries in played an important role. It is possible to dene such graph also for more general maps. or xample, Angeli and Sontag dene the signed incidence graph of monotone map in [2]. or matrices with entries in this

denition agrees with our denition of Section III. natural question to ask is, if the equi alence stated in Theorem 7.10 does hold for more general monotone maps which also possess an embedded graph structure, lik the incidence graph of Angeli and Sontag. The follo wing xample sho ws, that there are monotone maps possessing an incidence graph, such that at least assertion of Lemma 7.8 does not hold: Example 8.2: Let be gi en by ( s s ]0 1[ The set of edges of the incidence graph is (1 1) (2 1) (2 2) or = and (1 id we clearly ha id. No look at the set as

dened in Lemma 7.8 for nd that (1 is less or equal to ) 0) (1 s if and only if (1 (1 (23) Thus (1 if and only if (23) holds. This clearly violates property of Lemma 7.8, which is an important ingredient in the proof in Lemma 7.9. or monotone maps on the positi orthant of it has been sho wn that if the induced discrete dynamical system is 0-GAS, the inequality id must hold. or the con erse implication we ha to mak stronger assumptions on or matrices with entries in an equi alence relation as obtained. Some questions remain open. It is unclear if for there are

-dimensional xtensions of Lemma 6.1. Also, is it possible to restate the condition on in Proposition 6.3 in ay that is easier to check? This research as supported by the German Research oundation (DFG) as part of the Collaborati Research Centre 637 Autonomous Cooperating Logistic Processes”. abian irth as supported by the Science oundation Ireland grants 04-IN3-I460 and 00/PI.1/C067.

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[1] Da vid Angeli and Eduardo Sontag. Interconnections of monotone systems with steady-state characteristics. In Optimal contr ol, stabilization and nonsmooth analysis olume 301 of Lectur Notes in

Contr ol and Inform. Sci. pages 135–154. Springer Berlin, 2004. [2] Da vid Angeli and Eduardo D. Sontag. Multi-stability in monotone input/output systems. Systems Contr ol Lett. 51(3-4):185–202, 2004. [3] A. Berman and R. J. Plemmons. Nonne gative matrices in the mathematical sciences Academic Press [Harcourt Brace Jo ano vich Publishers], Ne ork, 1979. [4] S. Dashk vskiy B. uf fer and irth. An ISS small-g ain theorem for general netw orks. Berichte aus der echnomathematik 05-05, Zentrum ur echnomathematik, Bremen, German 2005. Online ailable at http://www

.math.uni-bremen.de/zetem/reports/reports- liste.html, ISSN 1435-7968. [5] S. Dashk vskiy B. uf fer and irth. An ISS yapuno function for netw orks of ISS systems. In Pr oceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, yoto uly 24-28, apan 2006. [6] M. Hirsch and Hal Smith. Monotone maps: re vie Dif fer ence Equ. Appl. 11(4-5):379–398, 2005. [7] C. D. Horv ath and M. Lassonde. Intersection of sets with connected unions. Pr oc. Am. Math. Soc. 125(4):1209–1214, 1997. [8] Z.-P Jiang, A. R. eel, and L. Praly Small-g ain theorem for ISS systems and

applications. Math. Contr ol Signals Systems 7(2):95 120, 1994. [9] B. Knaster C. urato wski, and S. Mazurkie wicz. Ein Be weis des Fixpunktsatzes ur -dimensionale Simple e. Fundamenta 14:132 137, 1929. [10] Marc Lassonde. Sur le principe KKM. (On the KKM principle). 1990. [11] H. L. Smith. Complicated dynamics for lo w-dimensional strongly monotone maps. Nonlinear Anal. 30(3):1911–1914, 1997. [12] Peter ak c. Domains of attraction of generic -limit sets for strongly monotone discrete-time semigroups. Reine Ang Math. 423:101–173, 1992.

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