1 Characterizations of input-to-state stability for infinite-dimensional systems Andrii Mironchenko and Fabian Wirth Abstract—WeprovecharacterizationsofISSforalargeclassof stability (GS/LS), and even local stability of the undisturbed infinite-dimensional control systems, including differential equa- system (0-LS): tions in Banach spaces, time-delay systems, ordinary differential equations, switched systems. These characterizations generalize AG∧GS ⇔ AG∧LS ⇔ AG∧0-LS ⇔ ISS, (1) 7 well-known criteria of ISS, proved by Sontag and Wang in [2] 1 forODEsystems.Forthespecialcaseofdifferentialequationsin see [17], [2]. 0 Banach spaces we prove even broader criteria for ISS property. These theorems greatly simplify the proofs of other funda- 2 Using an important technical result from [2] we show that mentalresults,suchassmall-gaintheorems[7],andareuseful n the characterizations obtained in [2] are a special case of our for analysis of other classes of systems, such as time-delay a results. We introduce the new notion of strong ISS which is systems in the Lyapunov-Razumikhin framework [18], [19], J equivalent to ISS in the ODE case, but which is strictly weaker 1 than ISS in the infinite-dimensional setting and prove several hybrid systems [20] to name a few examples. 3 criteria for sISS property. At the same time we show by means The significance of these characterizations of ISS makes it ofcounterexamples,thatmanycharacterizations,whicharevalid desirabletoextendtheresultstoinfinite-dimensionalsystems. ] in the ODE case, are not true for general infinite-dimensional In therecent paper [15] itwas shown thatuniform asymptotic C systems. stability at zero, local ISS and the existence of a LISS O Index Terms—input-to-state stability, nonlinear systems, Lyapunov function are equivalent properties for a system of h. infinite-dimensional systems. the form t a x˙(t)=Ax(t)+f(x(t),u(t)), x(t)∈X, u(t)∈U, (2) m I. INTRODUCTION provided the right hand side has some sort of uniform con- [ For ordinary differential equations, the concept of input-to- tinuity with respect to u. Here X is a Banach space, U is a 1 state stability (ISS) was introduced in [3]. The corresponding linear normed space, A is the generator of a C0-semigroup v theory is by now well developed with a firm theoretical basis. {T(t),t≥0} and f :X×U→X is sufficiently regular. It was 2 A variety of powerful tools for the investigation of ISS is alsodemonstratedbymeansofacounterexample,thatwithout 5 9 available and a multitude of applications have been developed this additional uniformity this characterization does not hold. 8 innonlinearcontroltheory,inparticular,torobuststabilization Inaddition,in[15]asystemoftheform(2)wasconstructed, 0 of nonlinear systems [4], design of nonlinear observers [5], whichislocallyISS(LISS),uniformlygloballyasymptotically 1. analysis of large-scale networks [6], [7], [8] etc. stable for a zero input (0-UGAS), globally stable (GS) and 0 ThesuccessofISStheoryofordinarydifferentialequations which has an asymptotic gain (AG), but which is not ISS, 7 and the need of proper tools for robust stability analysis of which strikingly contrasts to the ODE case, see (1). 1 partialdifferentialequationsmotivatedthedevelopmentofISS Thisnaturallyleadstoasetofchallengingquestions:which : v theoryininfinite-dimensionalsetting[9],[10],[11],[12],[13], combinations of properties considered in [2] are equivalent Xi [14], [15], [16]. to ISS for infinite-dimensional systems? Is it possible to generalize all characterizations of ISS from [2] to the general Characterizations of ISS in terms of other stability prop- r a erties [17], [2] are among the central theoretical results in infinite-dimensional setting, and under which conditions? Can one classify the properties, which are not equivalent to ISS in ISS theory of finite-dimensional systems. In [17] Sontag and anaturalway?IsitpossibletointroduceareasonableISS-like Wang have shown that ISS is equivalent to the existence of property which will be equivalent to ISS in finite dimensions, a smooth ISS Lyapunov function and in [2] the same authors but weaker than ISS for general systems (2)? proved a so-called ISS superposition theorem, saying that ISS In this paper we are going to answer these questions is equivalent to the combination of an asymptotic gain (AG) and obtain a broad picture of relationships between stability property of the system with inputs together with global/local properties for a large class of infinite-dimensional control systems,encompassingODEs,differentialequationsinBanach This research has been supported by the DFG under grant ”Input-to-state stability and stabilization of distributed parameter systems” (Wi1458/13-1). spaces, time-delay systems, switched systems etc. Thispaperisasubstantiallyrevisedandexpandedversionoftheconference In view of the examples in [15] we known that a ”naive” paperpresentedatthe22ndInternationalSymposiumonMathematicalTheory generalization of the equivalences (1) is not possible. These ofNetworksandSystems(MTNS2016)[1]. A. Mironchenko is with Faculty of Computer Science and preliminary studies reveal a lack of uniformity with respect Mathematics, University of Passau, 94030 Passau, Germany. Email: to the state in the definition of AG and other properties. [email protected]. These are frequently automatic in finite dimensions due to F.WirthiswithFacultyofComputerScienceandMathematics,University ofPassau,94030Passau,Germany.Email:[email protected]. local compactness of the state space. In infinite dimensions 2 this becomes a requirement. A further difficulty we encounter II. PRELIMINARIES in infinite-dimensional systems is that, in contrast to the ODE In this paper we consider abstract, axiomatically defined, case, forward completeness or global asymptotic stability do time-invariant, and forward complete systems. not guarantee the boundedness of reachability sets on finite time intervals. This is shown in the sequel by means of a Definition 1. Consider the triple Σ=(X,U,φ) consisting of counterexample. (i) A normed linear space (X,(cid:107)·(cid:107)X), called the state space, endowed with the norm (cid:107)·(cid:107) . Inordertoovercomethesedifficulties,weintroduceseveral X (ii) A set of input values U, which is a nonempty subset of novelstabilitynotions,whichnaturallyextendtheconceptsof a normed linear space S . limit property and of asymptotic gain. Namely: the uniform u (iii) A space of inputs U ⊂{f :R →U} endowed with a limit property (ULIM), the strong limit property (sLIM) as + well as strong asymptotic gain property (sAG). It turns out norm (cid:107)·(cid:107)U which satisfies the following two axioms: The axiom of shift invariance states that for all u ∈ that ULIM is the key to obtaining generalizations of the U and all τ ≥ 0 the time shift u(·+τ) ∈ U with characterizations of ISS. For the special case of systems on Banach spaces we obtain additional characterizations in (cid:107)u(cid:107)U ≥(cid:107)u(·+τ)(cid:107)U. Theaxiomofconcatenationisdefinedbytherequirement termsofULIMtogetherwithlocalstabilityoftheundisturbed thatforallu ,u ∈U andforallt>0theconcatenation system.UsingsLIMandsAGnotionwecancharacterizewhat 1 2 of u and u at time t we call strong ISS (sISS). For linear systems without inputs 1 2 this concept reduces to strong stability of the semigroup T, (cid:40) u (τ), if τ ∈[0,t], 1 whereas ISS for linear systems without inputs corresponds to u(τ):= (3) u (τ−t), otherwise, exponential stability of T. In order to characterize strong ISS 2 weintroducethestrongasymptoticgain(sAG)property,which belongs to U. is weaker than UAG, and prove that strong ISS is equivalent (iv) A transition map φ :R ×X×U →X. + to global stability together with sAG. The triple Σ is called a (forward complete) dynamical system, In the finite dimensional case, We show that the sLIM and if the following properties hold: ULIM properties are equivalent to the usual limit property (Σ1) Forward completeness: for every (x,u)∈X×U and for introduced in [2]. This proof relies in an essential manner on all t≥0 the value φ(t,x,u)∈X is well-defined. tools already developed in [2]. On the other hand ULIM is (Σ2) The identity property: for every (x,u)∈X×U it holds strictly stronger than sLIM or LIM already for linear infinite- that φ(0,x,u)=x. dimensional systems. In particular, we recover all character- (Σ3) Causality:forevery(t,x,u)∈R ×X×U,foreveryu˜∈ + izations of ISS for ODEs from [2] as a special case of our U,suchthatu(s)=u˜(s),s∈[0,t]itholdsthatφ(t,x,u)= results. φ(t,x,u˜). As argued above, for (2) ISS is no longer equivalent to (Σ4) Continuity:foreach(x,u)∈X×U themapt(cid:55)→φ(t,x,u) combinations of notions which are not fully uniform - like is continuous. AG ∧ GS or AG ∧ 0-UGAS. By means of counterexamples (Σ5) Thecocycleproperty:forallt,h≥0,forallx∈X,u∈U weshowthatthesecombinationsarenolongerequivalentwith we have φ(h,φ(t,x,u),u(t+·))=φ(t+h,x,u). each other. Instead, they can be classified into several groups, This class of systems encompasses control systems gen- according to the type and grade of uniformity. erated by ordinary differential equations (ODEs), switched The manuscript is structured as follows. In Section II we systems, time-delay systems, evolution partial differential introducethemainconceptswhichwillbeusedthroughoutthe equations (PDEs), abstract differential equations in Banach paper.InSectionIIIwemotivatethetopicofthepaperinmore spaces and many others. We single out two particular cases preciseterms,statethemainresultofthepaperandexplainthe which will be of interest. wayitisproved.Thesubsequentsectionscontaintheproofof The following class of semi-linear infinite-dimensional sys- themainresult.First,inSectionIVweprovecharacterizations tems has been studied in [15]. Let A be the generator of a of ISS for general infinite-dimensional systems in terms of strongly continuous semigroup T of bounded linear operators uniform limit and uniform attraction properties. In Section V on X and let f :X×U →X. Consider the system a concept of strong ISS is introduced and characterized in termsofstronglimitandstrongasymptoticgainproperties.In x˙(t)=Ax(t)+f(x(t),u(t)), x(t)∈X, u(t)∈U. (4) Section VI we construct four counterexamples, which clarify We study mild solutions of (4), i.e. solutions x:[0,τ]→X theinterrelationsbetweenthedifferentstabilitynotionsaswell of the integral equation as some of the difficulties and obstacles arising in infinite- (cid:90) t dimensional ISS theory. x(t)=T(t)x(0)+ T(t−s)f(x(s),u(s))ds, (5) Theresultsinthispaperarecloselyconnectedwithrecently 0 submitted papers on Lyapunov characterizations of ISS [21] belongingtothespaceofcontinuousfunctionsC([0,τ],X)for and on characterizations of UGAS for infinite-dimensional some τ >0. systemswithdisturbancesbymeansofnon-coerciveLyapunov In the sequel we assume that the state space X is a Banach functions[22].Someresultsofthesepapersaswellasof[15] space, the set of input valuesU is a normed linear space and are exploited in this paper. that the input functions belong to the space U :=PC(R ,U) + 3 ofgloballybounded,piecewisecontinuousfunctionsu:R → that + U, which are right continuous. The norm of u∈U is given by (cid:107)u(cid:107)U :=supt≥0(cid:107)u(t)(cid:107)U. sup{(cid:107)φ(t,x,u)(cid:107)X : (cid:107)x(cid:107)X ≤C, (cid:107)u(cid:107)U ≤C, t∈[0,τ]}<∞. Forsystem(4),weusethefollowingassumptionconcerning Notation: We use the following notation. The nonnegative the nonlinearity f. reals are R := [0,∞). The open ball of radius r around + 0 in X is denoted by B := {x ∈ X : (cid:107)x(cid:107) < r}. Similarly, Assumption 1. We assume that: r X Br,U :={u∈U :(cid:107)u(cid:107)U <r}. By lim we denote the superior (i) f :X×U→X isLipschitzcontinuousonboundedsubsets limit. of X, uniformly with respect to the second argument, i.e. For the formulation of stability properties the following for allC>0, there exists a L (C)>0, such that for all f classes of comparison functions are useful: x,y∈B and for all v∈U, it holds that C K :={γ :R →R | γ is continuous, strictly + + (cid:107)f(x,v)−f(y,v)(cid:107) ≤L (C)(cid:107)x−y(cid:107) . (6) increasing and γ(0)=0}, X f X K :={γ ∈K | γ is unbounded}, ∞ (ii) f(x,·) is continuous for all x∈X. L :={γ :R →R | γ is continuous and strictly + + (iii) f(0,0)=0. decreasing with limγ(t)=0}, t→∞ Since U = PC(R+,U), Assumption 1 ensures that mild K L :={β :R+×R+→R+| β is continuous, solutions of initial value problems of (4) exist and are unique, β(·,t)∈K , β(r,·)∈L, ∀t≥0, ∀r>0}. accordingtoavariationofaclassicalexistenceanduniqueness We continue with the list of stability notions, which will be theorem [23, Proposition 4.3.3]. For system (4) forward- used in the sequel. Several of these were already introduced completeness is a further assumption. The conditions (Σ2)- in [2] while others appear here for the first time as they only (Σ5) are satisfied by construction. become relevant in the infinite-dimensional case. In the finite- The second case of interest are finite-dimensional systems. dimensional case these new notions coincide with the classic LetX=Rn,U=Rm andU :=L (R ,U)(thespaceofglob- ∞ + ones. We discuss this issue in Section VIII. ally essentially bounded functions endowed with the essential supremum norm). For f :X×U →X consider the system A. Stability notions for undisturbed systems x˙= f(x,u). (7) We start with systems without inputs. We assume that f is continuous and locally Lipschitz con- Definition 5. System Σ=(X,U,φ) is called tinuous in x uniformly in u. With this assumption and • uniformlystableatzero(0-ULS),ifthereexistsaσ ∈K∞ the additional assumption of forward-completeness classical and r>0 so that Carathe´odory theory implies (Σ2)-(Σ5). We will sometimes (cid:107)φ(t,x,0)(cid:107) ≤σ((cid:107)x(cid:107) ) ∀x∈B ∀t≥0. (9) briefly speak of ODE systems, when referring to (7). X X r We start with some basic definitions. Without loss of • uniformly globally stable at zero (0-UGS), if there exists generality we restrict our analysis to fixed points of the form a σ ∈K so that ∞ (0,0)∈X×U, so that we tacitly assume that the zero input is an element of U. (cid:107)φ(t,x,0)(cid:107)X ≤σ((cid:107)x(cid:107)X) ∀x∈X ∀t≥0. (10) Definition 2. ConsiderasystemΣ=(X,U,φ).Wecall0∈X • globally attractive at zero (0-GATT), if anequilibriumpoint(oftheundisturbedsystem)ifφ(t,0,0)= lim(cid:107)φ(t,x,0)(cid:107) =0 ∀x∈X. (11) X 0 for all t≥0. t→∞ • a system with the limit property at zero (0-LIM), if Fordescribingthebehaviorofsolutionsneartheequilibrium the following notion is of importance inf(cid:107)φ(t,x,0)(cid:107)X =0 ∀x∈X. t≥0 Definition 3. Consider a system Σ=(X,U,φ) with equilib- • uniformlygloballyattractiveatzero(0-UGATT),ifforall riumpoint0∈X.Wesaythattheequilibriumpointisrobustif ε,δ >0 there is a τ =τ(ε,δ)<∞ such that foreveryε>0andforanyh>0thereexistsaδ=δ(ε,h)>0, so that t≥τ, x∈Bδ ⇒ (cid:107)φ(t,x,0)(cid:107)X ≤ε. (12) t∈[0,h], (cid:107)x(cid:107)X ≤δ, (cid:107)u(cid:107)U ≤δ ⇒ (cid:107)φ(t,x,u)(cid:107)X ≤ε. (8) • globally asymptotically stable at zero (0-GAS), if Σ is 0-ULS and 0-GATT. In this case we will also say that Σ has the REP property. • asymptoticallystableatzerouniformlywithrespecttothe state (0-UAS), if there exists a β ∈K L and r>0, such Even nonuniformly globally aymptotically stable systems that do not always have uniform bounds for their reachability sets on finite intervals (see Example 2). Systems exhibiting such (cid:107)φ(t,x,0)(cid:107) ≤β((cid:107)x(cid:107) ,t) ∀x∈B ∀t≥0. (13) X X r bounds deserve a special name. • globally asymptotically stable at zero uniformly with Definition 4. We say that Σ=(X,U,φ) has bounded reach- respect to the state (0-UGAS), if it is 0-UAS and (13) ability sets (BRS), if for any C>0 and any τ >0 it holds holds for all x∈X. 4 We stress the difference between the uniform notions 0- All three properties AG, sAG and UAG imply that all UGATT and 0-UGAS and the nonuniform notions 0-GATT trajectories converge to the ball of radius γ((cid:107)u(cid:107)U) around the and 0-GAS. For 0-GATT systems all trajectories converge to origin as t →∞. The difference between AG, sAG and UAG theorigin,buttheirspeedofconvergencemaydifferdrastically is in the kind of dependence of τ on the states and inputs. In forinitialvalueswiththesamenorm,incontrastto0-UGATT UAG systems this time depends (besides ε) only on the norm systems. The notions of 0-ULS and 0-UGS are uniform in of the state, in sAG systems it depends on the state x (and the sense that there exists an upper bound of the norm of may vary for different states with the same norm), but it does trajectorieswhichisequalforinitialstateswiththesamenorm. not depend on u. In AG systems τ depends both on x and on u. For systems without inputs, the AG and sAG properties Remark1. ForODEsystems0-GASisequivalentto0-UGAS, are reduced to 0-GATT, and the UAG property becomes 0- butitisweakerthan0-UGASintheinfinite-dimensionalcase. UGATT. For linear systems x˙ = Ax, where A generates a strongly Next we define properties, similar to AG, sAG and UAG, continuous semigroup, the Banach-Steinhaus theorem implies whichformalizereachabilityoftheε-neighborhoodoftheball that 0-GAS is equivalent to strong stability of the associated B by trajectories of Σ. semigroup T and implies the UGS property. γ((cid:107)u(cid:107)U) Definition 8. We say that Σ=(X,U,φ) has the Forsystemsthatare0-LIM,trajectoriesapproachtheorigin arbitrarily closely. Obviously, 0-GATT implies 0-LIM. (i) limit property (LIM) if there exists γ ∈K such that for all x∈X, u∈U and ε >0 there is a t=t(x,u,ε): B. Stability notions for systems with inputs (cid:107)φ(t,x,u)(cid:107)X ≤ε+γ((cid:107)u(cid:107)U). We now consider systems Σ=(X,U,φ) with inputs. (ii) stronglimitproperty(sLIM),ifthereexistsγ∈K sothat Definition 6. System Σ=(X,U,φ) is called foreveryε>0andforeveryx∈X thereexistsτ=τ(ε,x) • uniformly locally stable (ULS), if there exist σ,γ ∈K∞ such that for all u∈U there is a t≤τ: and r>0 such that for all x∈Br and all u∈Br,U: (cid:107)φ(t,x,u)(cid:107)X ≤ε+γ((cid:107)u(cid:107)U). (19) (cid:107)φ(t,x,u)(cid:107)X ≤σ((cid:107)x(cid:107)X)+γ((cid:107)u(cid:107)U) ∀t≥0. (14) (iii) uniform limit property (ULIM), if there exists γ ∈K so • uniformly globally stable (UGS), if there exist σ,γ∈K∞ that for every ε >0 and for every r>0 there exists a such that for all x∈X,u∈U the estimate (14) holds. τ=τ(ε,r)suchthatforallxwith(cid:107)x(cid:107) ≤randallu∈U X • practically uniformly globally stable (pUGS), if there there is a t≤τ such that exist σ,γ∈K and c>0 such that for all x∈X, and all ∞ u∈U it holds that (cid:107)φ(t,x,u)(cid:107)X ≤ε+γ((cid:107)u(cid:107)U). (20) (cid:107)φ(t,x,u)(cid:107)X ≤σ((cid:107)x(cid:107)X)+γ((cid:107)u(cid:107)U)+c ∀t≥0. (15) Rexeismteanrcke o3f. aItγi∈s Keasyfotrowsheeichthat AG is equivalent to the ∞ Remark 2. It is easy to see that the notion of pUGS is equivalent to the boundedness property (BND), as defined in lim(cid:107)φ(t,x,u)(cid:107)X ≤γ((cid:107)u(cid:107)U) t→∞ [2,p.1285].AlsopUGSimpliesBRS,buttheconversefailsin and LIM is equivalent to existence of a γ ∈K so that general as BRS is only a property of the finite-time reachable ∞ sets. inf(cid:107)φ(t,x,u)(cid:107)X ≤γ((cid:107)u(cid:107)U), t≥0 C. Attractivity properties for systems with inputs where in both cases the conditions hold for all x∈X,u∈U. Wedefinetheattractivitypropertiesforsystemswithinputs. Inparticular,AGimpliesLIMandontheotherhanditiseasy to see that LIM and UGS together imply AG. Definition 7. System Σ=(X,U,φ) has the • asymptoticgain(AG)property,ifthereisaγ∈K∞∪{0} Remark 4. For systems without inputs the notions of sLIM such that for all ε >0, for all x∈X and for all u∈U and LIM coincide and are strictly weaker than the ULIM, even for linear infinite-dimensional systems generated by C - there exists a τ =τ(ε,x,u)<∞ such that 0 semigroups, see [22]. t≥τ ⇒ (cid:107)φ(t,x,u)(cid:107)X ≤ε+γ((cid:107)u(cid:107)U). (16) • strong asymptotic gain (sAG) property, if there is a γ ∈ D. Input-to-state stability K ∪{0} such that for all x∈X and for all ε >0 there ∞ Now we proceed to the main concept of this paper: exists a τ =τ(ε,x)<∞ such that for all u∈U Definition 9. System Σ = (X,U,φ) is called (uniformly) t≥τ ⇒ (cid:107)φ(t,x,u)(cid:107)X ≤ε+γ((cid:107)u(cid:107)U). (17) input-to-state stable (ISS), if there exist β ∈K L and γ∈K • uniform asymptotic gain (UAG) property, if there exists such that for all x∈X, u∈U and t≥0 it holds that a γ ∈K ∪{0} such that for all ε,r>0 there is a τ = τ(ε,r)<∞∞ such that for all u∈U and all x∈B (cid:107)φ(t,x,u)(cid:107)X ≤β((cid:107)x(cid:107)X,t)+γ((cid:107)u(cid:107)U). (21) r The local counterpart of the ISS property is t≥τ ⇒ (cid:107)φ(t,x,u)(cid:107)X ≤ε+γ((cid:107)u(cid:107)U). (18) 5 UAG ISS ∃ ISS-LF Definition 10. System Σ = (X,U,φ) is called (uniformly) locally input-to-state stable (LISS), if there exist β ∈K L, AG∧0-UGAS AG∧UGS γ ∈K and r>0 such that the inequality (21) holds for all x∈Br, u∈Br,U and t≥0. AG∧LISS LIM∧UGS Lyapunovfunctionsareapowerfultoolfortheinvestigation of ISS and local ISS. For the class of semilinear systems (4) AG∧ULS LIM∧0-ULS LIM∧ULS they are defined as follows. Let x∈X andV be a real-valued function defined in a neighborhood of x. The Dini derivative Fig.1:CharacterizationsofISSinfinitedimensions of V at x corresponding to the input u along the trajectories of Σ is defined by ♦ there exist σ ∈K and r>0 so that for all v∈B and r,U 1 V˙u(x)= lim (V(φ(t,x,u))−V(x)). (22) all x∈Br we have t→+0t (cid:107)f(x,v)−f(x,0)(cid:107) ≤σ((cid:107)v(cid:107) ). (25) Definition 11. Let D⊂X be open with 0∈D. A continuous X U function V :D→R+ is called a LISS Lyapunov function for Then the following statements are equivalent: a system Σ=(X,φ,U), if there exist r >0, ψ ,ψ ∈K , 1 2 ∞ (i) (4) is 0-UAS. α ∈K and σ ∈K such that B ⊂D, ∞ r (ii) (4)hasaLipschitzcontinuous0-UASLyapunovfunction. ψ ((cid:107)x(cid:107) )≤V(x)≤ψ ((cid:107)x(cid:107) ), ∀x∈B (23) (iii) (4) has a Lipschitz continuous LISS Lyapunov function. 1 X 2 X r (iv) (4) is LISS. andtheDiniderivativeofV alongthetrajectoriesofΣsatisfies This result is reminiscent of a classical result on the V˙u(x)≤−α((cid:107)x(cid:107)X)+σ((cid:107)u(cid:107)U) (24) robustness of the 0-UAS property [24, Corollary 4.2.3]. As an easy consequence we have that for system (4) 0-UGAS for all x∈Br and u∈Br,U. implies LISS, which has already been shown in [2, Lemma (i) A function V : X → R is called an ISS Lyapunov + I.1] for ODE systems. function, if (24) holds for all x∈X,u∈U. In the ODE case the assumption ”♦” in Proposition 2 is (ii) V :D→R is called a (0-UAS) Lyapunov function, if + automatically fulfilled. However, this assumption cannot be (23) is satisfied and if (24) holds for u≡0. dropped for infinite-dimensional systems (4) as demonstrated Remark 5. We point out that on the right hand side of the by a counterexample in [15, Section 4]. We recall another dissipation inequality (24) the growth bound is given in terms example from [15, Section 5]: of (cid:107)u(cid:107)U instead of the more familiar (cid:107)u(cid:107)U for u∈U. For Example 1. Consider a system Σ with state space X =l := some input spaces this is a necessity, but for the input space 1 of bounded piecewise continuous functions, as well as for {(xk)∞k=1:∑∞k=1|xk|<∞} and input space U :=PC(R+,R). Let the dynamics of the k-th mode of Σ be given by L∞(R ,U) it is equivalent to require the condition + 1 V˙u(x)≤−α((cid:107)x(cid:107)X)+σ((cid:107)u(cid:107)U) x˙k(t)=−1+|u(t)|kxk(t). (26) for all x∈X,u∈U. This may be shown similarly to the proof (cid:3) for ”implication form” Lyapunov functions provided in, [9, According to the analysis in [15] system (26) is 0-UGAS, Proposition 5]. sAG, AG with zero gain, UGS with zero gain, and LISS with zerogain,butitisnotISS(fromthemainresultofthepresent III. MOTIVATIONANDMAINRESULT paper it follows that (26) is not ULIM). This means that all Theprimarymotivationforthismanuscriptisthefollowing characterizations of ISS in terms of AG or LIM together with fundamental result due to Sontag and Wang [17], [2], which GS or 0-UGAS, depicted in Figure 1 are no longer valid for we informally described in the introduction. infinite-dimensional systems. This makes the characterization of ISS in infinite dimensions a challenging problem. Proposition 1. For a forward complete, finite dimensional In order to reflect the essential distinctions occurring in system (7), the equivalences depicted in Figure 1 hold. thesestabilityproperties,andtoobtainapropergeneralization Inparticular,ISSisnotonlyequivalenttotheuniformprop- of the criteria for ISS, developed by Sontag and Wang in erties (UAG, existence of a smooth ISS Lyapunov function), Proposition 1, we introduce several new concepts. These are but also to the combination of the limit property with local the uniform and the strong limit property (ULIM and sLIM), stability of the system. strong input-to-state stability (sISS) as well as the strong In [15] characterizations of LISS for nonlinear infinite- asymptotic gain property (sAG). These notions naturally ex- dimensional systems of the form (4) have been studied and tend the notions of LIM, AG and UAG introduced in [2]. the following result [15, Theorem 4] has been obtained: The first positive result in characterizations of ISS is the following Lyapunov characterization of ISS, shown in [21]. Theorem 2. Let Assumption 1 hold and assume furthermore 6 Theorem 3. Assume that f :X×U →X is bi-Lipschitz con- (i) ⇒ (ii). It is immediate that ISS implies REP and BRS. tinuous on bounded subsets, which means that two following The remaining claim is shown in Lemma 4. properties hold: (ii) ⇒ (iii). Evidently, UAG implies ULIM. By Lemma 5 1) ∀C > 0 ∃L1(C) > 0, such that ∀x,y ∈ X : (cid:107)x(cid:107) ≤ UAG∧REP implies ULS. f X C, (cid:107)y(cid:107) ≤C and ∀v∈U, it holds that (iii) ⇒ (iv). Let Σ be ULIM and ULS. By Proposition 7, X ULIM together with BRS implies pUGS. By Lemma 3, UGS (cid:107)f(x,v)−f(y,v)(cid:107) ≤L1(C)(cid:107)x−y(cid:107) . X f X is equivalent to pUGS∧ULS. 2) ∀C > 0 ∃L2(C) > 0, such that ∀u,v ∈ U : (cid:107)u(cid:107) ≤ (iv) ⇒ (ii). It is clear that UGS implies REP and BRS. The f U claim follows using Lemma 6. C, (cid:107)v(cid:107) ≤C and ∀x∈X, it holds that U (ii) ⇒ (i). Follows from the equivalence (ii) ⇔ (iv) and (cid:107)f(x,u)−f(x,v)(cid:107)X ≤L2f(C)(cid:107)u−v(cid:107)U. Lemma 7. The next step in the outline of the proof of Theorem 4 Then (4) is ISS if and only if there exists a Lipschitz is as follows. In Section V we introduce the new concept continuous ISS Lyapunov function for (4). of strong input-to-state stability (sISS). For nonlinear ODE systems this is equivalent to ISS, see Proposition 9, and A. Main result and structure of the paper for linear infinite-dimensional systems without inputs sISS is The central result in this paper is the following theorem: equivalent to strong stability of the associated semigroup T (which justifies the name ”strong” for this notion). We show Theorem 4. Let Σ=(X,U,φ) be a forward complete system in Theorem 8 that satisfying the BRS and the REP property. Then the relations depicted in Figure 2 hold. sISS ⇔ sAG ∧ UGS ⇔ sLIM ∧ GS. Blackarrowsshowimplicationsorequivalenceswhichhold On the other hand, ISS implies the combination AG ∧ 0- for the class of infinite dimensional systems defined in Defi- UGAS,whichisverydifferentfromsISS:sISSdoesnotimply nition 1; blue arrows are valid for semi-linear systems of the theexistenceofauniformconvergenceratefortheundisturbed class (4) under additional assumptions; the red arrows (with system (and thus, it does not imply 0-UGAS). At the same the negation sign) are implications which do not hold, due time AG ∧ 0-UGAS does not ensure the existence of uniform to the counterexamples presented in this paper; the question globalboundsforasystemwithinputs,i.e.GSisnotimplied. marks indicate that it is not known to us, whether or not the BelowthelevelofsISSandAG∧0-UGAStherearefurther converse implication holds. levels with even weaker properties. The counterexamples, Proof:FollowsfromTheorems5,8.Thecounterexamples discussing delicate properties of infinite-dimensional systems for the red arrows are discussed in Section VI, see Remark 7. and giving the necessary counterexamples for Figure 2 are discussed in detail in Section VI. For ODEs all the combinations in Figure 2 are equivalent Finally, we specialize ourselves to the important subclass as for system (7) we have that AG∧0-GAS is equivalent to of infinite-dimensional systems described by (4) and discuss ISS by to Proposition 1. In contrast, for infinite-dimensional the proof of the blue (dashed) implications. The equivalences systems these notions are divided into several groups, which labeled by Theorems 2 and 3 are cited from the literature and are not equivalent to each other. includedtoprovideabroaderpicture.WeshowinSectionVII The proof of Theorem 4 will be given in several steps. thatforsuchsystemsstandardassumptionsonthenonlinearity The upper level in Figure 2 consists of notions, which are f together with the BRS property imply robustness of the equivalent to ISS. As in the ODE case, ISS is equivalent to trivial equilibrium. This helps to make our main result more the uniform asymptotic gain property, and to the existence of precise for systems of the form (4). aLipschitzcontinuousLyapunovfunction.ByExample1,ISS Theorem 6. Let (4) satisfy Assumption 1 and property ”♦” isnot equivalentto combinationsof AGorLIM togetherwith fromTheorem2.Thenthefollowingstatementsareequivalent. LS or 0-UGAS. But it turns out, that ISS is equivalent to the (i) (4) is ISS. combination of ULIM and LS. This shows that uniformity of (ii) (4) is UAG and BRS. attractivity/reachability plays a much more important role in (iii) (4) is ULIM, ULS and BRS. the infinite-dimensional setting than it does in the ODE case. (iv) (4) is ULIM and UGS. The main result in this respect is: (v) (4) is ULIM, 0-ULS, and BRS. Theorem 5. LetΣ=(X,U,φ)beaforwardcompletecontrol Proof: By Lemma 10, it holds for system (4) that As- system. The following statements are equivalent: sumption 1 together with BRS implies REP. In conjunction (i) Σ is ISS. with Theorem 5 this shows equivalence of (i) – (iv). (ii) Σ is UAG, REP, and, BRS. Clearly, (iii) implies (v). Assume that (4) is ULIM, 0-ULS, (iii) Σ is ULIM, ULS, and BRS. and BRS. By the BRS property the value (iv) Σ is ULIM and UGS. β˜(r,t):=sup{(cid:107)φ(t,x,0)(cid:107) : x∈B } Proof: The proof of this result is divided into several r lemmas, which will be shown in Section IV. Here we show is finite for all (r,t)∈R2. The function β˜ is increasing in r. + how the result follows from them. Alsoforfixedr≥0weclaimthatlim β˜(r,t)=0byULIM t→∞ 7 Theorem5 Theorem3 ULIM∧0-ULS ULIM∧ULS ULIM∧UGS UAG ISS ∃ ISS-LF Theorem6 / (ix) (iii)/ Theorem8 / (iv) sLIM∧UGS sAG∧UGS sISS AG∧0-UGAS / (v) ??? /(ii) Remark3 / (vi) Theorem2 LIM∧UGS AG∧UGS AG∧0-UAS AG∧LISS / (vii) /(viii) /(i) ??? AG∧ULS AG∧0-ULS AG∧0-GAS Fig.2:Relationsbetweenstabilitypropertiesofinfinite-dimensionalsystems,whichhavearobustequilibriumpointandboundedreachabilitysets: • Blackarrowsshowimplicationsorequivalenceswhichholdforgeneralcontrolsystemsininfinitedimensions. • Redarrows(withthenegationsign)areimplicationswhichdonothold,duetothecounterexamplespresentedinthispaper(seeRemark7). • Bluedashedequivalencesareprovedonlyforsystemsoftheform(4)andundercertainadditionalconditions. • Questionmarksmeanthatitisnotknownrightnow(asfarastheauthorsareconcerned),whethertheconverseimplicationsholdornot. and 0-ULS. To see this let σ be the function characterizing Lemma1canbeinterpretedasfollows:thesystemΣisULS 0-ULS. Given ε >0 we may by ULIM choose a τ >0 such if and only if 0 is a robust equilibrium of Σ and the function that for all x∈B there is a t≤τ with δ in Definition 3 is independent of the time h. r It is useful to have a restatement of the BRS property in a (cid:107)φ(t,x,0)(cid:107)≤σ−1(ε). comparison-functions-like manner. By 0-ULS and the cocycle property it follows that We call a function h:R3 →R increasing, if (r ,r ,r )≤ + + 1 2 3 (cid:107)φ(t,x,0)(cid:107) ≤ ε for all t ≥ τ so that we have the desired (R1,R2,R3) implies that h(r1,r2,r3) ≤ h(R1,R2,R3), where convergence. We now have that for all x∈X and all t≥0 we use the component-wise partial order on R3. We call h + strictlyincreasingif(r ,r ,r )≤(R ,R ,R )and(r ,r ,r )(cid:54)= (cid:107)φ(t,x,0)(cid:107)≤max{β˜((cid:107)x(cid:107),t+s) : s≥0}+(cid:107)x(cid:107)e−t. 1 2 3 1 2 3 1 2 3 (R ,R ,R ) imply h(r ,r ,r )<h(R ,R ,R ). 1 2 3 1 2 3 1 2 3 This upper bound is well-defined, strictly increasing in (cid:107)x(cid:107) We have the following lemma: and strictly decreasing in t. If necessary, we can upper bound Lemma 2. Consider a forward complete system Σ. The it by a continuous β ∈K L. It follows that (4) is 0-UGAS. following statements are equivalent: Since we suppose that the assumption ”♦” of Theorem 2 (i) Σ has bounded reachability sets. holds, Theorem 2 implies LISS (and in particular, ULS) of (ii) there exists a continuous, increasing function µ :R3 → (4). Hence, (v) implies (iii). + R , such that for all x∈X,u∈U and allt≥0 we have This completes the proof of Theorem 4. + In Section VIII we show that our results contain Propo- (cid:107)φ(t,x,u)(cid:107)X ≤µ((cid:107)x(cid:107)X,(cid:107)u(cid:107)U,t). (28) sition 1 as a special case. This is done by proving that the notions of LIM and ULIM coincide for ODE systems. (iii) thereexistsacontinuousfunction µ:R3 →R suchthat + + Finally, we specialize our results to the system (4) without for all x∈X,u∈U and for all t≥0 the inequality (28) inputs to obtain characterizations of 0-UGAS. holds. Theproofisanalogoustothetheproofof[22,Lemma2.12] IV. CHARACTERIZATIONSOFISS and is omitted. The technical results needed to prove Theorem 5 will Finally, we recall [2, Lemma I.2], which was shown for be divided into 2 parts. First in Section IV-A we recall ODEs, but is proved in the same way for Σ. restatementsofBRSandULS.NextinSectionIV-Bweprove ourmaintechnicallemmas.Weassumethroughoutthissection Lemma 3. Consider a forward complete system Σ. Then Σ is that Σ is a forward complete system. ULS and pUGS if and only if it is UGS. A. Stability and boundedness of reachable sets B. Proof of Theorem 5 We start with a standard reformulation of the ε-δ formu- We start with a simple lemma: lations of stability in terms of K -functions (the proof is Lemma 4. If Σ is ISS, then it is UAG. straightforward and thus omitted). Lemma 1. System Σ is ULS if and only if for all ε>0 there Proof: Let Σ be ISS with the corresponding β ∈K L exists a δ >0 such that and γ ∈K∞. Take arbitrary ε,r >0. Define τ =τ(ε,r) as the solution of the equation β(r,τ)=ε (if it exists, then it is (cid:107)x(cid:107)X ≤δ, (cid:107)u(cid:107)U ≤δ, t≥0 ⇒ (cid:107)φ(t,x,u)(cid:107)X ≤ε. (27) unique,becauseofmonotonicityofβ inthesecondargument, 8 if it does not exist, we set τ(ε,r):=0). Then for allt≥τ, all and u(·+t )∈U, since Σ satisfies the axiom of shift invari- m x∈X with (cid:107)x(cid:107) ≤r and all u∈U we have ance. Assume thatt−t ≤τ(r). Since (cid:107)φ(t ,x,u)(cid:107) ≤r, (32) X m m X implies that (cid:107)φ(t,x,u)(cid:107) ≤σ˜(r) for all t ∈[t ,t]. Otherwise, X m (cid:107)φ(t,x,u)(cid:107)X ≤ β((cid:107)x(cid:107)X,t)+γ((cid:107)u(cid:107)U) ift−t >τ(r), then due to (31) there existst∗<τ(r), so that m ≤ β((cid:107)x(cid:107)X,τ)+γ((cid:107)u(cid:107)U) ≤ ε+γ((cid:107)u(cid:107)U), (cid:13)(cid:13)φ(cid:0)t∗,φ(tm,x,u),u(·+tm)(cid:1)(cid:13)(cid:13)X =(cid:107)φ(t∗+tm,x,d)(cid:107)X ≤ 34r, and the estimate (18) holds. which contradicts the definition oft , sincet +t∗<t. Hence m m Lemma 5. If Σ is UAG and REP, then it is ULS. x∈Br,(cid:107)u(cid:107)U ≤γ−1(4r),t≥0 ⇒ (cid:107)φ(t,x,u)(cid:107)X ≤σ˜(r). (33) Proof: We will show that (27) holds so that the claim Denote σ(r):=σ˜(r)−σ˜(0), for any r≥0. Clearly, σ ∈K . ∞ follows from Lemma 1. Let τ and γ be the functions given For each x ∈ X and u ∈ U we can pick r := by (18). Let ε>0 and τ:=τ(ε/2,1). Pick any δ >0 so that γ(δ )<ε/2. Then for all x∈X with (cid:107)x(cid:107) ≤1 an1d all u∈U max{(cid:107)x(cid:107)X,4γ((cid:107)u(cid:107)U)} and (33) immediately shows for all 1 X x∈X, u∈U, t≥0 that with (cid:107)u(cid:107)U ≤δ1 we have (cid:0) (cid:1) ε (cid:107)φ(t,x,u)(cid:107)X ≤ σ max{(cid:107)x(cid:107)X,4γ((cid:107)u(cid:107)U)} +σ˜(0) sup(cid:107)φ(t,x,u)(cid:107)X ≤ +γ((cid:107)u(cid:107)U)<ε. (29) (cid:0) (cid:1) t≥τ 2 ≤ σ((cid:107)x(cid:107)X)+σ 4γ((cid:107)u(cid:107)U) +σ˜(0), Since Σ is REP, there is some δ2=δ2(ε,τ)>0 so that which shows pUGS of Σ. (cid:107)η(cid:107)X ≤δ2 ∧ (cid:107)u(cid:107)U ≤δ2 ⇒ sup (cid:107)φ(t,η,u)(cid:107)X ≤ε. Lemma 6. If Σ is ULIM and UGS, then Σ is UAG. t∈[0,τ] Proof: Without loss of generality assume that γ in the Together with (29), this proves (27) with δ :=min{1,δ ,δ }. 1 2 definitions of ULIM and UGS is the same (otherwise take the maximum of the two). We proceed with Pickanyε>0andanyr>0.Bytheuniformlimitproperty, Proposition 7. Assume that Σ is BRS and has the uniform there exists γ ∈K∞, independent of ε and r, and τ =τ(ε,r) limit property. Then Σ is pUGS. so that for any x∈Br, u∈Br,U there exists a t ≤τ so that (cid:107)φ(t,x,u)(cid:107)X ≤ε+γ((cid:107)u(cid:107)U). Proof: Pick any r>0 and set ε := r. Since Σ has the 2 In view of the cocycle property, we have from the UGS uniform limit property, there exists a τ =τ(r) so that property that for the above x,u,t and any s≥0 r x∈Br,u∈U ⇒ ∃t≤τ : (cid:107)φ(t,x,u)(cid:107)X ≤ 2+γ((cid:107)u(cid:107)U). (30) (cid:107)φ(s+t,x,u)(cid:107)X = (cid:107)φ(s,φ(t,x,u),u(s+·))(cid:107)X Inparticular,ifx∈Br,(cid:107)u(cid:107)U ≤γ−1(4r)thenthereexistsat≤τ ≤ σ((cid:107)φ(t,x,u)(cid:107)X)+γ((cid:107)u(cid:107)U) such that ≤ σ(ε+γ((cid:107)u(cid:107)U))+γ((cid:107)u(cid:107)U). (cid:107)φ(t,x,u)(cid:107) ≤ 3r. (31) Now let ε˜ :=σ(2ε)>0. Using the evident inequality σ(a+ X 4 b)≤σ(2a)+σ(2b),whichholdsforanya,b≥0,weproceed Without loss of generality we can assume that τ is increasing to in r, in particular, it is locally integrable. Defining τ¯(r):= 1(cid:82)2rτ(s)ds we see that τ¯(r)≥τ(r) and τ¯ is continuous. (cid:107)φ(s+t,x,u)(cid:107)X ≤ε˜+γ˜((cid:107)u(cid:107)U), r r Since Σ is BRS, Lemma 2 implies that there exists a where γ˜(r)=σ(2γ(r))+γ(r). continuous, increasing function µ :R3 →R , such that for + + Overall, for any ε˜ > 0 and any r > 0 there exists τ = all x∈X,u∈U and all t ≥0 the estimate (28) holds. This τ(ε,r)=τ(1σ−1(ε˜),r), so that for t≥τ we have implies that 2 x∈Br,(cid:107)u(cid:107)U ≤γ−1(4r),t≤τ(r) ⇒ (cid:107)φ(t,x,u)(cid:107)X ≤σ˜(r), (32) (cid:107)φ(t,x,u)(cid:107)X ≤ε˜+γ˜((cid:107)u(cid:107)U). whereσ˜ :r(cid:55)→µ(cid:0)r,γ−1(r),τ(r)(cid:1)iscontinuousandincreasing, This shows that Σ is UAG. 4 The final technical lemma of this section is: since µ,γ,τ are continuous increasing functions. Also from (31) and (32) it is clear that σ˜(r)≥ 3r for any r>0. Lemma 7. If Σ is UAG and UGS, then Σ is ISS. 4 Assumethatthereexistx∈Br,u∈U with(cid:107)u(cid:107)U ≤γ−1(4r) Proof: Fix arbitrary r∈R . We are going to construct a and t≥0 so that (cid:107)φ(t,x,u)(cid:107) >σ˜(r). Define + X function β ∈K L so that (21) holds. tm:=sup{s∈[0,t]:(cid:107)φ(s,x,u)(cid:107)X ≤r}≥0. From global stability it follows that there exist γ,σ ∈K∞ such that for all t≥0, all x∈B and all u∈U we have r Thequantityt iswell-defined,since(cid:107)φ(0,x,u)(cid:107) =(cid:107)x(cid:107) ≤r m X X due to the identity property (Σ2). (cid:107)φ(t,x,u)(cid:107)X ≤σ(r)+γ((cid:107)u(cid:107)U). (34) In view of the cocycle property Σ5, it holds that Define ε :=2−nσ(r), for n∈N. The UAG property implies n (cid:0) (cid:1) φ(t,x,u)=φ t−t ,φ(t ,x,u),u(·+t ) , that there exists a sequence of times τ :=τ(ε ,r), which we m m m n n 9 maywithoutlossofgeneralityassumetobestrictlyincreasing, (ii)⇒(i).AssumethatΣisUGSandsAG.Fixanarbitrary such that for all x∈B and all u∈U δ ∈R and any x∈X with (cid:107)x(cid:107) ≤δ. We are going to con- r + X structβ:X×R →R withthepropertiesasinDefinition12, + + (cid:107)φ(t,x,u)(cid:107)X ≤εn+γ((cid:107)u(cid:107)U) ∀t≥τn. so that (35) holds. Uniform global stability of Σ implies that there exist σ,γ∈ From (34) we see that we may set τ :=0. Define ω(r,τ ):= ε , for n∈N, n(cid:54)=0. 0 n K∞ so that for all t≥0 and for all u∈U it holds that n−1 Now extend the definition of ω to a function ω(r,·)∈L. (cid:107)φ(t,x,u)(cid:107)X ≤σ(δ)+γ((cid:107)u(cid:107)U). We obtain for t ∈ (τ ,τ ), n = 0,1,... and x ∈ B that n n+1 r (cid:107)φ(t,x,u)(cid:107)X ≤εn+γ((cid:107)u(cid:107)U)<ω(r,t)+γ((cid:107)u(cid:107)U). Doing this Define εn:=2−nσ(δ), for n∈N. Due to sAG there exists a for all r∈R+ we obtain the definition of the function ω. sequence of times τn :=τ(εn,x), which we assume without Now define βˆ(r,t):=sup ω(s,t)≥ω(r,t) for (r,t)∈ loss of generality to be strictly increasing in n, such that 0≤s≤r iRs2+in.cFrreoamsintghisindrefiannidtioβˆn(ri,t·)foilslodwescrtehaasti,nfgorineatchfotr≥ea0c,hβˆr(>·,t0) (cid:107)φ(t,x,u)(cid:107)X ≤εn+γ((cid:107)u(cid:107)U) ∀u∈U, ∀t≥τn. as every ω(r,·)∈L. Thus, βˆ+|r|e−t may be upper bounded Define β(x,τn):=εn−1, for n∈N, n(cid:54)=0. by a continuous β ∈K L and the estimate (21) is satisfied Now extend the function β(x,·) for t∈R+\{τn,n∈N} so with such a β. that β(x,·)∈L and β(x,t)≤2σ((cid:107)x(cid:107)X) for allt≥0 (this can be done by choosing the values of β(x,t) sufficiently small for t∈[0,τ )). V. STRONGISS 1 Thefunctionβ satisfiestheestimate(35),becauseforallt∈ As has been shown in Lemma 8, the combination of the (τn,τn+1)itholdsthat(cid:107)φ(t,x,u)(cid:107)X≤εn+γ((cid:107)u(cid:107)U)<β(x,t)+ properties AG and UGS is weaker than ISS. Therefore it γ((cid:107)u(cid:107)U). Performing this procedure for all x∈X we obtain is natural to ask for a weaker property than ISS which is the definition of the function β. This shows Σ is sISS. equivalent to the combination AG ∧ UGS. In this section we prove a partial result of this kind. VI. COUNTEREXAMPLES Definition12. SystemΣiscalledstronglyinput-to-statestable Beforeweproceedtoourmainexamples,letustakeaquick (sISS), if there exist γ ∈K , σ ∈K and β :X×R →R , ∞ + + look at linear systems. so that 1) β(x,·)∈L for all x∈X, x(cid:54)=0 Lemma8. Forlinearinfinite-dimensionalsystemsoftheform (4), i.e. systems with f ≡0, sAG ∧ UGS does not imply LISS. 2) β(x,t)≤σ((cid:107)x(cid:107) ) for all x∈X and all t≥0 X 3) for all x∈X, all u∈U and all t≥0 it holds that Proof: Consider the linear system x˙=Ax, where A is the generator of a C -semigroup T(·). For this system it is (cid:107)φ(t,x,u)(cid:107)X ≤β(x,t)+γ((cid:107)u(cid:107)U). (35) observedin[25]that0ISSisequivalentto0-UGASwhichisin Remark 6. Clearly, ISS implies sISS, but not vice versa. turn equivalent to exponential stability of the semigroup T(·). By linearity and as there is no input these properties are also Strong ISS can be characterized as follows. equivalent to LISS. Theorem 8. LetΣ=(X,U,φ)beaforwardcompletecontrol Also, as there is no input sAG is equivalent to AG. On the system. The following statements are equivalent. other hand, using linearity we have the equivalences AG ∧ UGS⇔AG∧ULS⇔0-GATT∧0-ULS⇔0-GAS⇔strong (i) Σ is sISS. stabilityofT(·)(forthelastequivalenceseeRemark1).Since (ii) Σ is sAG and UGS. strong stability of a semigroup does not imply exponential (iii) Σ is sLIM and UGS. stability in general, the claim of the lemma follows. Proof: (i) ⇒ (ii). Let Σ be sISS with the corresponding In this section we construct two nonlinear systems without β :X×R →R and σ ∈K and γ ∈K . By definition, Σ inputs Σ1, Σ3, and two nonlinear systems with inputs Σ2, + + ∞ is UGS characterized by σ and γ. Σ4 providing counterexamples which show that the following Fix any x∈X and any ε >0. Define τ =τ(ε,x) as the implications are false: solutionoftheequationβ(x,τ)=ε (ifthissolutionexists,then Σ1: (Σ1)∧0-GAS∧0-UAS (cid:54)⇒ BRS. it is unique, because of the monotonicity of β in the second Σ2: (Σ1)∧0-UGAS∧AG∧LISS (cid:54)⇒ BRS. argument, if it does not exist, we set τ(ε,x)=0). Then for all Σ3: (Σ1)∧BRS∧0-GAS∧0-UAS (cid:54)⇒ 0-UGS. t≥τ and all u∈U Σ4: (Σ1)∧BRS∧0-UGAS∧AG∧LISS (cid:54)⇒ UGS. (cid:107)φ(t,x,u)(cid:107)X ≤ β(x,t)+γ((cid:107)u(cid:107)U) Remark 7. Before we proceed to the constructions of the ≤ β(x,τ)+γ((cid:107)u(cid:107)U) systems Σi, let show how they justify the negated implications depicted in Figure 2. ≤ ε+γ((cid:107)u(cid:107)U), (i) Follows from Lemma 8. and the estimate (17) holds. Thus, sISS implies sAG. (ii) Follows by construction of Σ3. (ii) ⇒ (iii). This is clear. (iii) Follows by construction of Σ4. (iii) ⇒ (ii). Can be proved along the lines of Lemma 6. (iv) Follows by construction of Σ4. 10 (v) Follows from Lemma 8. According to the previous arguments, ∑Nk=−11|zk(t)|2 → 0 as (vi) Follows by construction of Σ3 and/or Σ4. t → 0 since all Σ1 are GAS for k = 1,...,N −1. Also k (vii) Follows from Lemma 8. ∑∞k=N|zk(t)|2 decays monotonically and exponentially to 0 as (viii) Follows by construction of Σ4. t→∞. Overall, (cid:107)z(t)(cid:107) →0 ast→∞ which shows that Σ1 is l2 (ix) Follows from Lemma 8. forward complete, 0-GAS and 0-UAS. Inaddition,werecallthatExample1,whichisfullydiscussed Finally we show that Σ1 is not BRS. To prove this, it is in [15], shows that 0-UGAS ∧ sAG ∧ AG with zero gain ∧ enough to show that there exists an r>0 and τ >0 so that UGS with zero gain ∧ LISS with zero gain do not imply ISS for any M>0 there exist z∈l andt∈[0,τ] so that (cid:107)z(cid:107) =r 2 l2 (and even do not imply ULIM). Hence, the properties of the and (cid:107)φ(t,z,0)(cid:107) >M. l2 ”second”level(sISSandAG∧0-UGAS)notonlyaredifferent Let us consider Σ1. For y ≥1 and for x ∈[0,k] it holds k k k from each other (in the sense that they do not imply each that other), but also even taken together they do not imply ISS. Finally,systemsΣ1andΣ2showthatthesystemswithglobal x˙k≥−2xk+xk2. (39) nonuniform attractivity properties together with very strong properties near the equilibrium may not even be BRS. Pick an initial state xk(0)=c>0 (which is independent of k) so that the solution of x˙ =−2x +x2 blows up to infinity in k k k Example 2 ((Σ1)∧0-GAS∧0-UAS (cid:54)⇒ BRS). According timet∗=1. Now pick y (0)=e (Euler’s constant) for all k= k to Remark 1 for linear infinite-dimensional systems 0-GAS 1,2,.... For this initial condition we obtain y (t)=e1−t ≥1 k implies 0-UGS. Now we show that for nonlinear systems 0- for t∈[0,1]. And consequently for z (0)=(c,e)T there exists k GAS does not even imply BRS of the undisturbed system. a time τ ∈(0,1) such that x (τ )=k for the solution of Σ1. Consider the nonlinear infinite-dimensional system Σ1 defined Now ckonsider an initial stkatek z(0) for Σ1, where z (0)k= k by (c,e)T and z (0)=(0,0)T for j(cid:54)=k. For this initial state we j  (cid:26) x˙ =−x +x2y − 1x3, have that (cid:107)z(t)(cid:107) =|z (t)| and  Σ1: k k k k k2 k l2 k Σ1: k y˙ =−y . (36) k k  k=1,2,..., sup(cid:107)z(t)(cid:107)l2 =sup|zk(t)|≥|xk(τk)|≥k. t≥0 t≥0 with the state space (cid:40) (cid:41) As k∈N was arbitrary, this shows that the system Σ1 is not ∞ X :=l = (z )∞ : ∑|z |2<∞, z =(x ,y )∈R2 . (37) BRS. 2 k k=1 k k k k k=1 Example 3 ((Σ1)∧0-UGAS∧AG∧LISS (cid:54)⇒ BRS). In this We show that Σ1 is forward complete, 0-GAS and 0-UAS but modification of Example 2 it is demonstrated that 0-UGAS ∧ does not have bounded reachability sets. AG ∧ LISS does not imply BRS. Let Σ2 be defined by First we argue that Σ1 is 0-UAS. Indeed, for r < 1 the Lyapunov function V(z)=(cid:107)z(cid:107)2 =∑∞ (x2+y2) satisfies for  (cid:26) x˙ =−x +x2y |u|− 1x3, all zk with |xk|≤r and |yk|≤rl2, k∈Nk=,1thek estikmate Σ2: Σ2k : y˙kk=−ykk. k k k2 k ∞ 1  k=1,2,..., V˙(z)=2∑(−x2+x3y − x4−y2) k k k k2 k k k=1 And let the state space of Σ2 be l (see (37)) and its input 2 ≤2∑∞ (−x2+|x |3|y |− 1 x4−y2) space be U :=PC(R+,R). k k k k2 k k (38) Evidently, this system is 0-UGAS. Also it is clear that Σ2 k=1 ∞ is not BRS, since for u≡1 we obtain exactly the system from ≤2∑((r2−1)x2−y2) k k Example 2, which is not BRS. The proof that this system is k=1 forward complete, LISS and AG with zero gain mimics the ≤2(r2−1)V(z). argument we exploited to show 0-GATT of Example 2 and WeseethatV isanexponentiallocalLyapunovfunctionforthe thus we omit it. system (36) and thus (36) is locally uniformly asymptotically Example 4 ((Σ1)∧BRS∧0-GAS∧0-UAS (cid:54)⇒ 0-UGS). We stable. Indeed it it not hard to show that the domain of construct a counterexample in 3 steps. attraction contains {z∈l :|x |<r,|y |<r,∀k}. 2 k k To show forward completeness and global attractivity of Σ1 Step 1. Let us revisit Example 2 and find useful estimates we first point out that every Σ1 is 0-GAS. This follows from from above for the dynamics of the subsystems Σ1k. the fact that any Σ1 is a cascakde interconnection of an ISS We first note that for initial conditions z0k =(xk0,y0k) with xk-system (with yk aks an input) and a globally asymptotically xk0y0k ≤ 0 we have for the solutions of Σk1 (see (36)) that |z (t)|≤|z0| for all t≥0. stable yk-system, see [3]. k k Furthermore, for any z(0)∈l2 there exists a finite N >0 It is easy to check that for each k∈N and each zk(0)= such that |zk(0)|≤ 21 for all k≥N. Decompose the norm of (xk(0),yk(0))∈R2 with yk(0)xk(0)>0 the solution of Σ1k for z(t) as follows the initial condition zk(0) (see (36)) can be estimated in norm by N−1 ∞ (cid:107)z(t)(cid:107) = ∑ |z (t)|2+ ∑ |z (t)|2. l2 k k |x (t)|≤|xˆ (t)| k=1 k=N k k