Lyapunov caracterization of input-to-state stability for semilinear control systems over Banac spaces Andrii Mironcenko a, Fabian Wirt a a Faculty of Computer Science and Matematics, University of Passau, Innstraße 33, 943 Passau, Germany Abstract We prove tat input-to-state stability (ISS of nonlinear systems over Banac spaces is equivalent to existence of a coercive Lipscitz continuous ISS Lyapunov function for tis system. For linear infinite-dimensional systems, we sow tat ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions of coercive and non-coercive ISS Lyapunov functions for input-to-state stable linear systems. Keywords: nonlinear control systems, infinite-dimensional systems, input-to-state stability, Lyapunov metods Input-to-state stability (ISS was introduced by Sontag in is seminal paper [8] and as since become a backbone of robust nonlinear control teory. Applications of ISS include robust stabilization of nonlinear systems [7], design of nonlinear observers [], analysis of large-scale networks [, 6] and numerous oter brances of nonlinear control [6]. Te success of ISS teory of ordinary differential equations (ODEs and te need for proper tools for robust stability analysis of partial differential equations (PDEs motivated te development of ISS teory in te infinitedimensional setting [5,, 8,, 4,, 9]. Te two main lines of researc witin infinite-dimensional ISS teory are te development of a general ISS teory of evolution equations in Banac spaces and te application of ISS to stability analysis and control of particular important PDEs. Te results in te first area include for instance smallgain teorems for interconnected infinite-dimensional systems and teir applications to nonlinear interconnected parabolic PDEs over Sobolev spaces [5, ] and caracterizations of local and global ISS properties [9, 5]. Witin te second line of researc, constructions of ISS Lyapunov functions for nonlinear parabolic systems over L p -spaces [8], for linear time-variant systems of conservation laws [7], for nonlinear Kuramoto-Sivasinsky equation [] ave been investigated. Non-Lyapunov metods were successfully applied to linear parabolic systems wit boundary disturbances in [5]. In tis paper, we follow te first line of researc and prove converse Lyapunov teorems for ISS of linear and semilinear evolution equations in Banac spaces. For us te primary motivation comes from te papers [7, 9], Email addresses: andrii.mironcenko@uni-passau.de (Andrii Mironcenko, fabian.(lastname@uni-passau.de (Fabian Wirt in wic converse UGAS Lyapunov teorems ave been applied to prove, in te case of ODEs, te equivalence between ISS and te existence of a smoot ISS Lyapunov function. Tis result along wit furter restatements of ISS in terms of oter stability notions [9, 3] and smallgain teorems [, 6] is at te eart of ISS teory of systems of ordinary differential equations. In Section using te metod from [9] and converse Lyapunov teorems for global asymptotic stability of systems wit disturbances from [3] we prove tat ISS is equivalent to te existence of a coercive, Lipscitz continuous ISS Lyapunov function. Along te way, we sow tat ISS is equivalent to te existence of a globally stabilizing feedback wic is robust to multiplicative actuator disturbances of bounded magnitude (weak uniform robust stability, WURS. In Section we provide simpler constructions of coercive and non-coercive ISS Lyapunov functions for linear infinite-dimensional systems wit bounded input operators. In particular, we sow tat te existence of noncoercive ISS Lyapunov functions is already sufficient for ISS of linear systems wit bounded input operators. Weter te existence of a non-coercive ISS Lyapunov function is sufficient for ISS of infinite-dimensional nonlinear systems is not completely clear rigt now, altoug some positive results based on non-lyapunov caracterizations of ISS property ave been acieved in [5]. For systems witout disturbances, it was sown in [4] tat non-coercive Lyapunov functions ensure uniform global asymptotic stability of te system, provided certain additional mild conditions old. Extension of tese results to te systems wit inputs is a callenging question for future researc. In Section 3 we conclude te results of te paper. Some of te results of tis paper ave been presented at 54t IEEE Conference on Decision and Control (CDC 5 Preprint submitted to Systems & Control Letters July 9, 7
[] and at t IFAC Symposium on Nonlinear Control Systems (NOLCOS 6 [3]. Let R + := [,. For te formulation of stability properties te following classes of functions are useful: P := {γ : R + R + γ is continuous, γ(r = r = }, K := {γ P γ is strictly increasing}, K := {γ K γ is unbounded}, L := {γ : R + R + γ is continuous and strictly decreasing wit lim γ(t = }, t KL := {β : R + R + R + β is continuous,, β(, t K, β(r, L, t, r > }. For a normed space, we denote te closed ball of radius r around by B r or B r if we want to make te space clear. Given normed space, W, we call a function f : W locally Lipscitz continuous, if for all r > tere exists a constant L r suc tat f(x f(y W L r x y x, y B r. In te finite dimensional case, local Lipscitz continuity is sometimes defined using neigboroods of points, and in tis case, tis is of course equivalent. Note tat in te infinite-dimensional case it is necessary to go to a definition on bounded balls as tese are not compact. Te terminology we use ere is consistent wit [6, p. 85]. Tis concept is called Lipscitz continuity on bounded balls in [3]. Assumption. Let f : U be bi-lipscitz continuous on bounded subsets, wic means tat two following properties old:. C > L f (C >, suc tat x, y wit x C, y C and v U, it olds tat f(x, v f(y, v L f (C x y. (3. C > L f (C >, suc tat u, v U wit u U C, v U C and x, it olds tat f(x, u f(x, v L f (C u v U. (4 Due to standard arguments, Assumption implies tat mild solutions corresponding to any x( and any u U exist and are unique (actually, te second condition is too strong for mere existence and uniqueness, but we need it for te furter development. We call te system forward complete, if for all initial conditions x and all u U te solution exists on R +. We treat u as an external input, wic may ave significant influence on te dynamics of te system. For te stability analysis of suc systems a fundamental role is played by te concept of input-to-state stability, wic unifies external and internal stability concepts. Definition. System ( is called input-to-state stable (ISS, if it is forward complete and tere exist β KL and γ K suc tat x, u U and t te following inequality olds. Input-to-state stability and weak uniform robust stability In tis paper we consider infinite-dimensional systems of te form ẋ(t = Ax(t + f(x(t, u(t, x(t, u(t U, ( were A generates a strongly continuous semigroup of bounded linear operators, is a Banac space and U is a normed linear space of input values. As te space of admissible inputs, we consider te space U of globally bounded, piecewise continuous functions from R + to U. In tis paper we consider mild solutions of (, i.e. solutions of te integral equation x(t = T t x( + t T t s f(x(s, u(sds ( belonging to te class C([, τ], for certain τ >. Here {T t, t } is te C -semigroup over, generated by A. For te notions from te teory of C -semigroups and its applications to evolution equations we refer to [4, 3]. In te sequel, we will write φ(t, x, u to denote te solution corresponding to te initial condition φ(, x, u = x and te input u U. In te remainder of te paper we suppose tat te nonlinearity f satisfies te following assumption: φ(t, x, u β( x, t + γ( u U. (5 A key tool to study ISS is an ISS Lyapunov function. Definition. A continuous function V : R + is called a non-coercive ISS Lyapunov function, if V ( = and if tere exist ψ K, α P and χ K so tat < V (x ψ ( x x \ {}. (6 and so tat te Dini derivative of V along te trajectories of te system ( satisfies te implication x χ( u( U V u (x α( x (7 for all x and u U, were ( V u (x = lim V (φ(t, x, u V (x. (8 t + t If, in addition, tere exists ψ K suc tat ψ ( x V (x ψ ( x x, (9 ten V is called a coercive ISS Lyapunov function. In Definition we defined ISS Lyapunov function in te so-called implication form. For anoter (dissipative definition of ISS Lyapunov functions and for te relation between tese definitions please consult []. We ave te following result, see [5, Teorem ].
Proposition. If tere exists a coercive ISS Lyapunov function for (, ten ( is ISS. We intend to sow tat ISS of ( implies existence of a coercive, locally Lipscitz continuous Lyapunov function for (. On tis way we follow te metod developed in [9] for systems described by ODEs. In order to formalize te robust stability property of (, we consider te problem of global stabilization of ( by means of feedback laws wic are subject to multiplicative disturbances wit a magnitude bounded by. To tis end let ϕ : R + be locally Lipscitz continuous and consider inputs u(t := d(tϕ(x(t, t, ( were d D := {d : R + D, piecewise continuous}, D := {d U : d U }. Applying tis feedback law to ( we obtain te system ẋ(t = Ax(t + f(x(t, d(tϕ(x(t =: Ax(t + g(x(t, d(t. ( Let us denote te solution of ( at time t, starting at x and wit disturbance d D by φ ϕ (t, x, d. On its interval of existence, φ ϕ (t, x, d coincides wit te solution of ( for te input u(t = d(tϕ(x(t... Basic properties of te closed-loop system Te next lemma sows tat g in ( is Lipscitz continuous. Lemma. Let f be locally bi-lipscitz continuous. Ten g is Lipscitz continuous on bounded subsets of, uniformly wit respect to te second argument, i.e. C > L g (C >, suc tat x, y B C and d D, it olds tat g(x, d g(y, d L g (C x y. ( Proof. Pick an arbitrary C >, any x, y B C, and any d D. It olds g(x, d g(y, d = f(x, dϕ(x f(y, dϕ(y f(x, dϕ(x f(y, dϕ(x + f(y, dϕ(x f(y, dϕ(y. Since ϕ is Lipscitz continuous, it is bounded on B C by a bound R. According to Assumption and as d U, we can upper bound te first summand by L f (R x y and te second by L f (R ϕ(x ϕ(y. Te claim now follows from te local Lipscitz continuity of ϕ. Forward completeness of ( does not imply forward completeness of (. For example, consider ẋ = x + u, u(t = d x (t for d >. 3 In particular, Lemma sows tat te system ( is well-posed, i.e. its solution exists and is unique for any initial condition and any disturbance d. Remark. Lipscitz continuous feedbacks do not necessarily lead to Lipscitz continuous g if f is not Lipscitz wit respect to inputs. Consider e.g. ẋ(t = (u(t /3 and u(t := x(t. Definition 3. System ( is called robustly forward complete (RFC if for any C > and any τ > it olds tat sup φ ϕ (t, x, d <. x C, d D, t [,τ] Definition 4. We say tat te flow of ( is Lipscitz continuous on compact intervals, if for any τ > and any R > tere exists L > so tat for any x, y B R, for all t [, τ] and for all d D it olds tat φ ϕ (t, x, d φ ϕ (t, y, d L x y. (3 We will need te following result, see [4, Lemma 4.6], sowing te regularity properties of te system (. Lemma. Assume tat (i ( is robustly forward complete. (ii g is Lipscitz continuous on bounded subsets of, uniformly w.r.t. te second argument. Ten ( as a flow wic is Lipscitz continuous on compact intervals. Definition 5. System ( is called uniformly globally asymptotically stable (UGAS if tere exists a β KL suc tat d D, x, t φ ϕ (t, x, d β( x, t. (4 UGAS can be caracterized wit te elp of uniform global attractivity. Definition 6. System ( is called uniformly globally attractive (UGATT, if for any r, ε > tere exists τ = τ(r, ε so tat for all d D it olds tat x r, t τ(r, ε φ ϕ (t, x, d ε. (5 Definition 7. System ( is called uniformly globally stable (UGS, if tere exists σ K so tat d D, x, t φ ϕ (t, x, d σ( x. (6 Te following caracterization of UGAS follows easily from [3, Teorem.]. Proposition. System ( is UGAS if and only if ( is UGATT and UGS. Coercive Lyapunov functions corresponding to UGAS property are defined as follows:
Definition 8. A continuous function V : R + is called a Lyapunov function for (, if tere exist ψ, ψ K and α K suc tat ψ ( x V (x ψ ( x x (7 olds and Dini derivative of V along te trajectories of te system ( satisfies for all x, and all d D. V d (x α( x (8 Te following converse Lyapunov teorem will be crucial for our developments [3, Section 3.4]: Teorem 3. Let ( be UGAS and let its flow be Lipscitz continuous on compact intervals, ten ( admits a locally Lipscitz continuous Lyapunov function. We will need te following property, wic formalizes te robustness of ( wit respect to te feedback (. Remark. Te reader familiar wit te results in [9] will notice tat our assumptions on te dependence on u are stronger tan in te finite-dimensional case. For system ( we need to ensure existence of solutions if a feedback is applied. In te finite-dimensional case, it is sufficient to assume continuity by Peano s teorem. Tis guarantees existence but not uniqueness, but for te stability arguments, tis is not a major drawback. For system ( continuity is in general not sufficient for te existence of solutions [8, 9]. Lemma 4 ( is ISS Proposition ISS-LF for ( ( is WURS LF for ( Teorem 3 Figure : ISS Converse Lyapunov Teorem Lemma 5 Definition 9. System ( is called weakly uniformly robustly asymptotically stable (WURS, if tere exist a locally Lipscitz ϕ : R + and ψ K suc tat ϕ(x ψ( x and ( is uniformly globally asymptotically stable wit respect to D. Te next proposition sows ow te WURS property of system ( reflects te regularity of te solutions of (. Proposition 4. Consider a forward complete system (. Assume tat (i f is bi-lipscitz on bounded subsets of ; (ii ( is WURS. Ten for any ϕ satisfying te conditions of Definition 9, te closed-loop system ( as a flow, wic is Lipscitz continuous on compact intervals. Proof. Since ( is WURS and ϕ is a stabilizing feedback as required in Definition 9, system ( is forward complete and UGAS. Let β KL be a bound as in (4. Ten, for any C > and any τ > sup φ ϕ (t, x, d β(c, <. x C, d D, t [,τ] Assumption (i togeter wit Lemma imply tat g is locally Lipscitz continuous uniformly in te second argument. Tus, all assumptions of Lemma are satisfied, and te claim follows... Main result Te objective of tis paper is to prove tat for system ( (at least wit bi-lipscitz nonlinearities te notions depicted in Figure are equivalent. 4 First, we sow in Lemma 3 tat ISS implies WURS. Next, we apply Teorem 3 to prove tat WURS of ( implies te existence of a Lipscitz continuous coercive ISS Lyapunov function for (. Finally, te direct Lyapunov teorem (Proposition completes te proof. Lemma 3. If ( is ISS, ten ( is WURS. Proof. Te proof goes along te lines of [9, Lemma.]. Let ( be ISS. In order to prove tat ( is WURS we are going to use Proposition. Since ( is ISS, tere exist β KL and γ K so tat (5 olds for any t, x, u U. Define α(r := β(r,, for r R +. Substituting u and t = into (5 we see tat α(r r for all r R +. Pick any σ K so tat σ(r γ ( 4 α ( 3 r for all r. We may coose locally Lipscitz continuous maps ϕ : R + and ψ K suc tat ψ( x ϕ(x σ( x (just pick a locally Lipscitz continuous ψ K and set ϕ(x := ψ( x for all x, wic guarantees tat ϕ is locally Lipscitz continuous. We are going to sow tat for all x, all t and all d D it olds tat ( d(tϕ(φϕ γ (t, x, d U x. (9 First we sow tat (9 olds for all times t small enoug. Since α (r r for all r >, we ave ( γ d(tϕ(φ ϕ (t, x, d U γ ( σ( φ ϕ (t, x, d 4 α ( 3 φ ϕ(t, x, d 6 φ ϕ(t, x, d. For any d D and any x te latter expression can be made smaller tan x by coosing t small enoug, since φ ϕ is continuous in t.
Now pick any d D, x and define t = t (x, d by { ( t := inf t : γ d(t U ϕ(φ ϕ (t, x, d > x }. By te first step we know t >. Assume tat t < (oterwise our claim is true. Ten (9 olds for all t [, t. Tus, for all t [, t it olds tat φ ϕ (t, x, d β( x, t + x β( x, + α( x = 3 α( x. Using tis estimate we find out tat ( γ d(t ϕ(φϕ U (t, x, d 4 α ( 3 φ ϕ(t, x, d 4 α ( x 4 x. But tis contradicts te definition of t. Tus, t = +. Now we see tat for any x, any d D and all t we ave φ ϕ (t, x, d β( x, t + x, ( wic sows uniform global stability of (. Since β KL, tere exists a t = t ( x so tat β( x, t x 4 and consequently d D, x, t φ ϕ (t, x, d 3 4 x. By induction we obtain tat tere exists a strictly increasing sequence of times {t k } k=, wic depends on te norm of x but is independent of x and d so tat ( 3 k x φ ϕ (t, x, d, 4 for all x, any d D and all t t k. Tis means tat for all ε > and for all δ > tere exist a time τ = τ(δ so tat for all x wit x δ, for all d D and for all t τ we ave φ ϕ (t, x, d ε. Proof. Let ( be WURS, wic means tat ( is UGAS over D for suitable ϕ, ψ cosen in accordance wit Definition 9. Proposition 4 and Teorem 3 imply tat tere exists a locally Lipscitz continuous Lyapunov function V : R +, satisfying (7 for certain ψ, ψ K and wose Lie derivative along te solutions of ( for all x and for all d D satisfies te estimate V d (x α(v (x. ( Tis is equivalent to te fact tat V u (x α(v (x. ( olds for all x and all u U satisfying u U ϕ(x. Tis automatically implies tat ( olds for all x and all u U wit u U ψ( x. In oter words, V is an ISS Lyapunov function for ( in an implication form wit Lyapunov gain χ := ψ. We conclude our investigation wit te following caracterization of ISS property: Teorem 5. Let Assumption be fulfilled. Ten te following statements are equivalent:. ( is ISS.. ( is WURS. 3. Tere exists a coercive ISS Lyapunov function for ( wic is locally Lipscitz continuous. Proof. Te claim follows from Proposition and Lemmas 3 and 4. Teorem 5 sows tat ISS is equivalent to te existence of a Lipscitz continuous coercive ISS Lyapunov function. At te same time, te question weter te existence of a non-coercive ISS Lyapunov function is sufficient for ISS of ( remains open. Tis question is essentially infinitedimensional, since in te ODE case non-coercive Lyapunov functions are automatically coercive, at least locally. In contrast to ODEs, for linear infinite-dimensional systems, non-coercive ISS Lyapunov functions naturally arise wen one constructs Lyapunov functions by solving Lyapunov operator equation, see [4, Teorem 5..3 ]. Hence it is of great interest to study criteria of ISS in terms of noncoercive ISS Lyapunov functions. In te next section, we sow some preliminary results in tis direction. An extensive treatment of tis topic for nonlinear systems witout inputs as been performed in [4]. Tis sows uniform global attractivity of (. Now we are ready to apply Proposition, wic sows tat ( is UGAS and tus ( is WURS. Lemma 4. If ( is WURS and Assumption is satisfied ten tere exists a locally Lipscitz continuous ISS Lyapunov function for (. 5. Linear systems In tis section, we derive a converse Lyapunov teorem for linear systems wit a bounded input operator B of te form ẋ = Ax + Bu. (3 Te assumptions on A are as before. definition. We start wit a
Definition. System ( is globally asymptotically stable at zero uniformly wit respect to te state (-UGAS, if tere exists a β KL, suc tat x, t φ(t, x, β( x, t. (4 Now we proceed wit a tecnical lemma; its proof is straigtforward and is omitted. Lemma 5. Let B L(U, and let T be a C -semigroup. Ten for any u U it olds tat lim + T s Bu(sds = Bu(. (5 Te main tecnical result of tis section is as follows: Proposition 6. If (3 is -UGAS, ten V : R +, defined as V (x = T t x dt (6 is a non-coercive ISS Lyapunov function for (3 wic is locally Lipscitz continuous. Moreover, x, u U and ε > it olds tat V u (x x + εm were M, λ > are so tat λ x + M λε B u( U, (7 T t Me λt. (8 Proof. Let (3 be -UGAS and pick u. Ten (4 implies T t x β(, t for all t and for all x wit x =. Since β KL, tere exists a t suc tat T t x < for all x, x =. Tus, T t < and consequently T is an exponentially stable semigroup [4, Teorem..6], i.e. tere exist M, λ > suc tat (8 olds. Consider V : R + as defined in (6. We ave V (x T t x dt M λ x. (9 Let V (x =. Ten T t x a.e. on [,. Strong continuity of T implies tat x =, and tus (6 olds. Next we estimate te Dini derivative of V : V u (x = lim (V (φ(, x, u V (x + ( = lim T t φ(, x, u + dt T t x dt ( = lim T t (T x + T s Bu(sds dt + T t x dt = lim + ( T t+ x + T t T s Bu(sds dt T t x dt 6 ( ( Tt+ lim x + dt + T t T s Bu(sds T t x dt =I + I, were ( I := lim + and I := lim + Let us compute I : ( I = lim + T t+ x dt T t x dt ( Tt T t+ x T s Bu(sds + T t T s Bu(sds = lim + = x. Now we proceed wit I : I = lim + T t x dt T t x dt dt. T t x dt Tt T t+ x T s Bu(sds dt + lim T t T s Bu(sds + dt. Te limit of te second term equals zero since lim + T t T s Bu(sds dt lim + M 4 e λt B u U dt =. To compute te limit of te first term, note tat Tt T t+ x T s Bu(sds M x T t M B sup u(r U r [,] M 3 x B u U e λt. Tus, we can apply te dominated convergence teorem. Togeter wit Lemma 5 and Young s inequality tis leads to I = T t x T t Bu( dt ε T t x + ε T tbu( dt ε T t dt x + ε εm λ x + M λε B u( U, T t Bu( dt
for any ε >. Overall, we obtain tat x, u U and for all ε > te inequality (7 olds. Considering ε < λ M tis sows tat V is a non-coercive ISS Lyapunov function (in dissipative form for (3. It can be brougt into implication form (as in (7 by coosing te Lyapunov gain χ(s := Rs for all s R + and for R large enoug. It remains to sow te local Lipscitz continuity of V.Pick arbitrary r > and any x, y B r. It olds tat + V (x V (y = T t x T t y dt + ( T t x T t y Tt x + T t y dt + + T t x T t y ( Tt x + T t y dt Me λt x y Me λt ( x + y dt M r λ x y, wic sows te Lipscitz continuity of V. Remark 3. Te ISS Lyapunov function V defined in (6 is not coercive in general. Noncoercivity of V defined by (6 implies tat te system ẋ = Ax, y = x is not exactly observable on [, + (even toug we can measure te full state!, see [4, Corollary 4..4]. Te reason for tis is tat for any given exponential decay rate tere are states tat decay faster tan tis given rate, and tus we lose a part of te information about te state infinitely fast. Remark 4. Note tat according to [5, Section III.B], te existence of a non-coercive Lyapunov function satisfying (7 ensures ISS of (3. Below we provide anoter construction of ISS Lyapunov functions for te system (3 wit bounded input operators. It is based on a standard construction in te analysis of C -semigroups, see e.g. [6, Eq. (5.4]. For exponentially stable C -semigroup T tere exist M, λ > suc tat te estimate (8 olds. Coose γ > suc tat γ λ <. Ten V γ (x := max s eγs T s x (3 defines an equivalent norm on, for wic we ave V γ (T t x = max s eγs T s T t x = e γt max s eγ(s+t T s+t x e γt V γ (x. (3 Based on tis inequality we obtain te following statement for ISS Lyapunov functions. 7 Proposition 7. Let (3 be -UGAS. Let M, λ > be suc tat (8 olds and let < γ < λ. Ten V γ : R +, defined by (3 is a coercive ISS Lyapunov function for (3. In particular, for any u U, x, we ave te dissipation inequality V γ u (x γ V γ (x + V γ (Bu(. (3 Proof. In order to obtain te infinitesimal estimate, we compute, using te triangle inequality (V γ is a norm, te estimate (3, and Lemma 5, V u γ (x = lim + = lim + lim + lim + (V γ (φ(, x, u V γ (x ( V γ( T x + T s Bu(sds (V γ( T x + V γ( ((e γ V γ (x + V γ( γ V γ (x + V γ (Bu(. V γ (x T s Bu(sds V γ (x T s Bu(sds Tis sows V γ is an ISS-Lyapunov function (in te dissipative form and tat (3 olds. Coosing a suitable Lyapunov gain χ K, one can sow tat (7 olds and tus V γ is an ISS Lyapunov function in implication form. Coercivity is evident by construction. It remains to sow Lipscitz continuity of V γ. Pick any x, y and assume tat V γ (x > V γ (y. Ten V γ (x V γ (y = max s eγs T s x max s eγs T s y ( max e γs T s x e γs T s y s max e γs T s x e γs T s y s max s eγs T s (x y M x y, wic sows tat V γ is globally Lipscitz continuous. Te case V γ (y > V γ (x can be treated analogously. Finally, we can state te main result of tis section: Teorem 8. Let B L(U,. Te following statements are equivalent: (i (3 is ISS. (ii (3 is -UGAS. (iii {T t } t is an exponentially stable semigroup. (iv V defined in (6 is a (not necessarily coercive locally Lipscitz continuous ISS Lyapunov function for (3. (v V γ defined in (3 is a coercive globally Lipscitz continuous ISS Lyapunov function for (3.
Proof. Equivalence between items (i and (ii can be easily derived from te variation of constants formula. Te implications (ii (iii (iv follow from Proposition 6. Item (iv implies (iii due to Datko s Lemma, see [4, Lemma 5.., Teorem 5..3, p. 5]. Implication (iii (ii is clear. (ii implies (v due to Proposition 7 and (v implies (i by Proposition. 3. Conclusions We ave sown tat input-to-state stability of a nonlinear infinite-dimensional system is equivalent to te existence of a coercive Lipscitz continuous ISS Lyapunov function. For linear systems, we ave proposed simpler direct constructions of coercive as well as non-coercive Lipscitz continuous ISS Lyapunov functions. Weter te existence of a non-coercive ISS Lyapunov function is sufficient for ISS of nonlinear infinite-dimensional systems, remains an open question. 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