Fellow, IEEE David Angeli

Senior Member, IEEE Denis Efimov

Characterizations of Input-to-State Stability for systems with multiple invariant sets

We generalize the theory of Input-to-State Stability (ISS) and of its characterizations by means of Lyapunov dissipation inequalities to the study of systems admitting invariant sets, which are not necessarily stable in the sense of Lyapunov but admit a suitable hierarchical decomposition. It is the latter which allows to greatly extend the class of systems to which ISS theory can be applied, allowing in a unified treatement to deal with oscillators in Euclidean coordinates, almost globally asymptotically stable systems on manifolds, systems with multiple equilibria in R n just to name a few.

I. INTRODUCTION AND MOTIVATIONS

A well established technique for the study of stability and robustness of nonlinear systems, which are described by a set of differential equations globally defined in Euclidean space, is the Input-to-State Stability approach, (see [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF] and references therein). The classical definition potentially allows to formulate and characterize stability properties with respect to arbitrary compact invariant sets (and not simply equilibria). The implicit requirement that these sets should be simultaneously Lyapunov stable and globally attractive, however, makes the basic theory not applicable for a global analysis of many dynamical behaviours of interest, such as multistability or periodic oscillations, just to name a few, and only local analysis remains possible [START_REF] Chaves | Bistable biological systems: A characterization through local compact input-to-state stability[END_REF]. In fact, it is well-known that such systems, when defined in Euclidean space, normally admit invariant sets (such as additional equilibria) that fail to be Lyapunov stable.

As an attempt to overcome such limitations for the case of nonlinear autonomous systems, the almost global stability property was introduced, [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF], and short afterwards, almost Input-to-State Stability, [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF], for systems admitting exogenous disturbances. In particular, for the case of almost ISS, sufficient criteria based on a combination of dual Lyapunov techniques [START_REF] Rantzer | A dual to Lyapunov's stability theorem[END_REF] and classical dissipation inequalities were proposed (see [START_REF] Vasconcelos | Combination of Lyapunov and Density Functions for Stability of Rotational Motion[END_REF] for an application of such tools to stability analysis of rotational motions). The key idea of the dual approach is to replace Lyapunov functions by suitable density functions and to impose a monotonicity condition on the way these are propagated by the flow. While converse dual Lyapunov results have appeared in the literature short afterwards, [START_REF] Rantzer | An converse theorem for density functions[END_REF], some difficulties in the explicit construction of density functions for systems involving unstable equilibria have also emerged [START_REF] Angeli | Some remarks on density functions for dual Lyapunov methods[END_REF].

More recently, in [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF], the need for conditions involving density functions was removed in the case of systems with exponentially unstable equilibria thanks to a careful application of integral manifold theory. While geometric tools involving manifolds and dimensionality arguments provide a very fine structure to the stability properties, it is also clear that they depart quite fundamentally in spirit from the standard ISS paradigm which is essentially an analytical theory.

In this paper we make the point that the most natural way of relaxing Input-to-State Stability for systems with disconnected invariant sets is in fact to relax the Lyapunov stability requirement Email: D.Angeli@imperial.ac.uk, Dept. of Electrical and Electronic Engineering, Imperial College, London, U.K. and Dip. di Ingegneria dell'Informazione, Università di Firenze, Italy Email: Denis.Efimov@inria.fr, Non-A project @ Inria, Parc Scientifique de la Haute Borne, Villeneuve d'Ascq, France and Dept. of Control Systems and Informatics, University ITMO, Saint Petersburg, Russia [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF] (rather than the global nature of the attractivity property). This, under relatively mild additional assumptions, in order to avoid classical counter-examples of globally attractive systems not admitting smooth Lyapunov functions, [START_REF] Bhatia | Stability Theory of Dynamical Systems[END_REF], will allow the characterization of the ISS property in terms of classical Lyapunov-like inequalities, thus generalizing the standard ISS theory as well as related literature on time-invariant dynamical systems on compact spaces, [START_REF] Nitecki | Filtrations, Decompositions, and Explosions[END_REF].

II. BASIC DEFINITIONS

Let M be an n-dimensional C 2 connected and orientable Riemannian manifold without boundary and D be a closed subset of R m containing the origin. Consider the map:

f (x, d) : M × D → TxM
which we assume to be of class C 1 (TxM denotes as customary the tangent space of M at x). We deal with nonlinear systems of the following form:

ẋ(t) = f (x(t), d(t)) (1)
with state x taking value in M . We denote by X(t, x; d(•)) the uniquely defined solution of (1) at time t fulfilling x(0) = x (under the input d(.) which is a locally essentially bounded and measurable signal). Solutions may fail to be defined for all t ∈ R, however, for the remainder of the Section we assume (without loss of generality) that solutions of the unperturbed system are globally defined backwards and forward in time.

The symbol δ(x1, x2) denotes the Riemannian distance between x1 and x2 in M . We are now equipped to define a notion of convergent dynamics for systems as in [START_REF] Angeli | Some remarks on density functions for dual Lyapunov methods[END_REF]. The unperturbed system is defined by the following set of equations:

ẋ(t) = f (x(t), 0). (2)
We say that S ⊂ M is invariant for the unperturbed system (2) if, for all x ∈ S, X(t, x; 0) ∈ S for all t ∈ R.

A. Decompositions

Let Λ ⊂ M be a compact invariant set for [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF]. In order to highlight the structure of the flow of the unperturbed system it is useful to decompose Λ and explicitly denote existence of solutions travelling between different components of its decomposition, as carried out in the following definitions: However, what qualifies a decomposition for its treatment by means of Lyapunov-like analytical tools is the absence of cycles, as detailed in the definitions that follow. For an open set W ⊂ M , define:

W s = {x0 ∈ M : ∃ t ≥ 0 : X(t, x0; 0) ∈ W }.
Definition 2: If W1, . . . , W k is an open decomposition of Λ, then 1) An r-cycle, for r ≥ 2 is an ordered r-tuple of distinct indices i1, . . . , ir such that

Wi j ∩ W s i j+1 = ∅ (j = 1, . . . , r -1) Wi r ∩ W s i 1 = ∅ 2) A 1-cycle is an index i such that ∃x / ∈ Wi, ∃s, t > 0 : X(-s, x; 0) ∈ Wi and X(t, x; 0) ∈ Wi.
Informally an r-cycle is therefore a collection of r disjoint open sets that are reached from one another in a loop by a suitable concatenation of systems solutions. The above definitions are taken from [START_REF] Nitecki | Filtrations, Decompositions, and Explosions[END_REF] where authors also give the following two definitions and the related proposition. Definition 3: Let Λ ⊂ M be a compact and invariant set for [START_REF] Angeli | An almost global notion of input-to-state stability[END_REF]. A decomposition of Λ is a finite, disjoint family of compact invariant sets Λ1, . . . , Λ k such that

Λ = k i=1 Λi.
For an invariant set Λ, its attracting and repulsing subsets are defined as follows:

A(Λ) = {x ∈ M : |X(t, x, 0)|Λ → 0 as t → +∞}, R(Λ) = {x ∈ M : |X(t, x, 0)|Λ → 0 as t → -∞}.
≺ . . . ≺ Λi r ≺ Λi 1 . 2) A 1-cycle is an index i such that [R(Λi) ∩ A(Λi)] -Λi = ∅. 3) A filtration ordering is a numbering of the Λi so that Λi ≺ Λj ⇒ i ≤ j.
As we can conclude from Definition 4, existence of an r-cycle with r ≥ 2 is equivalent to existence of a heteroclinic cycle for (2) [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF]. And existence of a 1-cycle implies existence of a homoclinic orbit for (2) [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF]. In general, existence of cycles has to be checked on separatrix configurations [START_REF] Markus | Global structure of ordinary differential equations in the plane[END_REF]. Proposition 1: Let W be a compact invariant set containing all α and ω limit sets of (2). Then a necessary and sufficient condition for W to be the maximal invariant set of an open decomposition with no cycles is that W have a decomposition with no cycles.

According to this result, for any compact invariant set W containing all α and ω limit sets, the two notions of decomposition without cycles are equivalent. Notice that in most examples one might be able to choose W = A ∪ R ∪ H, where the set A is composed by locally asymptotically stable invariant sets, the set R contains locally anti-stable invariant sets and H = H + ∩ H -is an hyperbolic invariant set (H + and H -constitute stable and unstable invariant submanifolds for H), some of these sets may be empty. Hyperbolicity is however not a requirement for the subsequent discussion. This makes a large class of systems amenable to be analysed with the techniques described below. Our main assumption on W which will be used throughout is the following: Assumption 1: The compact invariant set W admits a finite decomposition without cycles1 , W = k i=1 Wi for some non-empty disjoint compact sets Wi, which form a filtration ordering of W, as detailed in definitions 3 and 4. An open decomposition of W without cycles will be used in the proof of the main results below, due to Proposition 1 the existence of such an open decomposition follows from Assumption 1. The formulation of Assumption 1 is based on a finite decomposition since verification of this condition is more simple in examples, see Section IV.

B. Robust stability notions for decomposable W

Our object of study is the following robustness notion for the system (1):

Definition 5: The system (1) has the practical asymptotic gain (pAG) property if there exist η ∈ K∞ and q ≥ 0 such that for all x ∈ M and all measurable essentially bounded inputs d(•) solutions are defined for all t ≥ 0 and the following holds:

lim sup t→+∞ |X(t, x; d)|W ≤ η( d ∞) + q. ( 3
)
If q = 0, then we say that the asymptotic gain (AG) property holds. Definition 6: The system (1) has the limit property (LIM) with respect to W if there exists µ ∈ K∞ such that for all x ∈ M and all measurable essentially bounded inputs d(•) solutions are defined for all t ≥ 0 and the following holds:

inf t≥0 |X(t, x; d)|W ≤ µ( d ∞). (4)
Definition 7: We say that the system (1) has the practical global stability (pGS) property with respect to W if there exist β ∈ K∞ and q ≥ 0 such that for all x ∈ M and all measurable essentially bounded inputs d(•) the following holds for all t ≥ 0:

|X(t, x; d)|W ≤ q + β(max{|x|W , d ∞}).
(5)

Note that ( 5) is equivalent to

|X(t, x; d)|W ≤ β(max{|x|W + c, d ∞})
for some β ∈ K∞ and c ≥ 0. We would like to characterize (3) in terms of Lyapunov functions. The following notion is appropriate: [START_REF] Angeli | Some remarks on density functions for dual Lyapunov methods[END_REF] if there exists K∞ functions α1, [α2], α and γ, and q ≥ 0 [and c ≥ 0] such that:

Definition 8: A C 1 function V : M → R is a practical ISS- Lyapunov function for
α1(|x|W ) ≤ V (x) ≤ [α2(|x|W + c)],
and the following dissipation inequality holds for all (x, d) ∈ M ×D:

DV (x)f (x, d) ≤ -α(|x|W ) + γ(|d|) + q. ( 6
)
If ( 6) holds for q = 0, then V is said to be an ISS-Lyapunov function.

If, in addition, the set x∈W i {V (x)} (denoted for short V (Wi)) is a singleton for all i ∈ {1, 2, . . . , k}, then V is said to be an ISS-Lyapunov function constant on invariant sets. Notice that α2 and c are in brackets as their existence follows (without any additional assumptions) by standard continuity arguments.

Under Assumption 1, whenever Wi are recurrent invariant sets of the unperturbed system, any Lyapunov function which is nonincreasing along solutions of (2) also takes constant values on any Wi, 1 ≤ i ≤ k from the decomposition of W.

In addition, if q = 0, then existence of an ISS Lyapunov function as in Definition 8 (viz. strictly decreasing outside W) is possible for a set W admitting a decomposition without cycles only. We are now ready to state our main result: Theorem 1: Consider a nonlinear system as in (1) and let W be a compact invariant set containing all α and ω limit sets of (2) as in Assumption 1. Then the following facts are equivalent:

1) The system enjoys the AG property;

2) The system admits an ISS Lyapunov function;

3) The system admits an ISS Lyapunov function constant on invariant sets; 4) The system admits a practical ISS Lyapunov function; 5) The system enjoys the pAG property; 6) The system enjoys the LIM property and the pGS. The system as in (1) that satisfies these properties will be called ISS with respect to the set W.

Remark 1: The paper [START_REF] Nitecki | Filtrations, Decompositions, and Explosions[END_REF] (Proposition 6) proves 1. ⇒ 2. for the case of systems without disturbances and evolving on compact manifolds. Notice that in this case, extension of the result to systems with inputs is actually trivial and follows by a standard continuity argument:

DV (x)f (x, d) ≤ DV (x)f (x, 0) + |DV (x)[f (x, d) -f (x, 0)]| ≤ -α(|x|W ) + γ(|d|) where γ(r) := max |d|≤r,x∈M |DV (x)[f (x, d) -f (x, 0)]|.
Remark 2: Traditionally ISS is formulated for systems with a single equilibrium. When such systems fulfill the existence of a decomposition without cycles (as required in our set-up) and asymptotic gain, then, they also happen to be stable (in fact globally) and therefore our notions are genuinely equivalent to ISS in the classical set-up.

The equivalences of Theorem 1 appear to be non-trivial new results even in the case of systems evolving in R n . The only straightforward relations given without a proof are: ISS Lyapunov function ⇒ practical ISS Lyapunov function ⇒ pAG.

A diagram for the proof of other implications is given in Fig. 1.

A. Direct Lyapunov argument

To illustrate the consequences of existence of an ISS Lyapunov function we need the following intermediate results.

Claim 1: If the system (1) admits an ISS Lyapunov function, then it admits LIM and pGS properties. Proof. First, the LIM property is satisfied for µ(s) = α -1 • 2γ(s); if this is not the case then for some solution X(t, x; d) and all t 0 we have V (X(t, x; d)) ≤ -α(|X(t, x; d)|W )/2. That ensures the asymptotic convergence of V (X(t, x; d)) to a limit and in turn convergence of α(|X(t, x; d)|W ) to 0, given compactness of sublevel sets of V . The latter implies asymptotic convergence to zero of |X(t, x; d)|W which is a contradiction. This concludes the proof of the LIM property.

As for the practical GS property notice that the Lyapunov dissipation inequality also implies the following stability notion for all x ∈ M : |X(t, x; d)|W ≤ κ(|x|W , t) + σ( d ∞) + q, ∀t ≥ 0, [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF] where κ ∈ KL, σ ∈ K and q > 0. Indeed, α(|x|W /2 + c/2) ≤ α(|x|W ) + α(c) and

DV (x)f (x, d) ≤ -α(|x|W /2 + c/2) + α(c) + γ(|d|) ≤ -α • 1 2 α -1 2 (V (x)) + α(c) + γ(|d|)
that leads to the time-domain estimate [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF] or pGS property.

Claim 2: For the system (1) the following implications hold: AG ⇒ ( LIM and pGS ) ⇒ pAG. Proof. The LIM property follows by its definition and fulfillment of the pGS property is proven in Lemma 1 (see Appendix). Notice that from equation [START_REF] Bhatia | Stability Theory of Dynamical Systems[END_REF] and LIM property we have

lim sup t→+∞ |X(t, x; d)|W ≤ q + σ( d ∞), (8)
with σ(s) = β(max{µ(s), s}).

Claim 3: If the system admits an ISS Lyapunov function constant on invariant sets then it enjoys the AG property. Proof. From claims 1 and 2 the system admits the pAG property [START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF] in this case. This, for any ε > 0, proves AG with respect to input signals d(•) with d ∞ ≥ ε and gain ηε(s) = σ(s)+qs/ε as follows considering the inequality below:

q + σ( d ∞) ≤ q d ∞/ε + σ( d ∞) = ηε( d ∞).
Hence, we only need to show that there exists a sufficiently small ε > 0 such that AG holds for all input signals d with d ∞ ≤ ε. Let us focus, without loss of generality, on inputs with d ∞ ≤ 1. Asymptotically X(t, x; d) enters a compact set, X := {x : |x|W ≤ q + σ(1) + 1}. Let F be:

F := max x∈X ,|d|≤1 |f (x, d)|x < +∞ (9)
where |•|x denotes the norm on TxM which induces the Riemannian metric δ. Notice that F is finite by continuity of f and compactness of X × {d : |d| ≤ 1}. Since X(t, x; d) eventually enters X whenever d ∞ ≤ 1, it holds, |f (X(t, x; d), d)| X(t) ≤ F for all sufficiently large t ≥ 0.

Consider next the minimum distance between the elements of the decomposition:

D := min 1 ≤ i = j ≤ k, xa ∈ Wi, x b ∈ Wj δ(xa, x b ) > 0 (10)
Notice that the minimum exists and it is strictly positive by finiteness of the decomposition and compactness of the Wis. For all ∆ > 0 such that α -1 (2γ(∆)) = µ(∆) ≤ D/4 it holds that the sets Ni(∆) defined below:

Ni(∆) := {x ∈ M : |x|W i ≤ α -1 (2γ(∆))}
are disjoint and at least at distance D/2 from each other. By LIM property for d ∞ ≤ ∆ < 1, for all x ∈ M there is a time instant t ≥ 0 such that x(t ) ∈ Ni(∆) for some 1 ≤ i ≤ k. Next, for all sufficiently large times, solutions take at least D/2F in order to travel between two of the Ni(∆) sets. Notice moreover that for all x ∈ X \ k i=1 Ni(∆) and all d with |d| ≤ ∆ ≤ 1 it holds:

DV (x)f (x, d) ≤ -α(|x|W ) + γ(|d|) ≤ -γ(∆),
where the last inequality follows considering that for all x ∈ M there exists ix so that |x|W = |x|W ix . Hence the Lyapunov function V (x) along any solution that travels between two distinct sets Ni(∆) and Nj(∆) decreases at least by Dγ(∆)/2F between the time that it last leaves Ni(∆) and the one that it first enters Nj(∆).

Consider next the function G : [0, +∞) → [0, +∞) defined below:

G(r) := max i∈1,...,k max x∈N i (r) V (x) -min x∈N i (r) V (x) + r. (11)
Notice that G is continuous and increasing, moreover G(0) = 0 as V (Wi) is a singleton for all i ∈ {1, 2, . . . , k}. This function is a bound from above to the difference in values of V between points within the same neighborhood of radius r of the Wis. For any ∆ > 0 one may pick

∆1 ≤ γ -1 α 2 α -1 (2γ(∆)) 2 ≤ ∆
positive and sufficiently small so as to fulfill G(∆1) ≤ Dγ(∆)/4F for instance by letting:

∆ 1 := min G -1 Dγ(∆) 4F , γ -1 α 2 α -1 (2γ(∆)) 2 := γ(∆).
The rationale for this peculiar expression will be clearer after the following arguments are developed. We claim that with such a choice, all solutions corresponding to input signals with d ∞ ≤ ∆1 cannot visit twice the same neighborhood Ni(∆1) if in between they have visited another set Nj(∆1). The proof of this fact is sketched below.

Notice that for inputs with infinity norm less than ∆1, V (x) decreases outside i Ni(∆1), moreover as Ni(∆1) ⊂ Ni(∆) for all is, the Lyapunov function decreases at least by Dγ(∆)/2F when traveling between two distinct sets Ni(∆1) and Nj(∆1). On the other hand, while inside such sets, the Lyapunov function can at most grow by G(∆1) ≤ Dγ(∆)/4F . Overall, if a solution could visit in a cycle a number of distinct sets Ni 1 (∆1), Ni 2 (∆1), . . . , Ni 1 (∆1) this would lead to a net decrease of V which is larger than the maximum gap G(∆1) allowed between values of V (x) within the set Ni 1 (∆1). This is a contradiction and therefore we may conclude that all solutions corresponding to input signals d with d ∞ ≤ ∆1 eventually keep visiting a single set Ni(∆1). Notice that:

∆ 1 ≤ γ -1 α 2 α -1 (2γ(∆)) 2 ⇒ α -1 (2γ(∆)) 2 ≥ α -1 (2γ(∆ 1 )).
Consider next the function ρ(r) := α -1 (2γ(r))/2. Our choice of ∆1 implies that:

ρ(∆) = α -1 (2γ(∆)) - α -1 (2γ(∆)) 2 ≤ α -1 (2γ(∆)) -α -1 (2γ(∆ 1 )),
so that ρ(∆)/F is a lower bound to the travel time between any two points at distance α -1 (2γ(∆1)) and α -1 (2γ(∆)) respectively from any of the Wis. Pick next 0 < ∆2 ≤ ∆1 so as to fulfill:

G(∆2) γ(∆1) ≤ ρ(∆)/2F. (12)
The left-hand side is an upper bound to the time that solutions corresponding to inputs of amplitude less than ∆2 can consecutively spend outside the set Ni(∆1). In particular as this upper-bound is smaller than the minimum time to reach the boundary of Ni(∆) (from the boundary of Ni(∆1) ), then any such solution (leaving a set Ni(∆2)) will never leave the set Ni(∆). This fact allows us to establish existence of a suitable asymptotic gain function for all inputs of sufficiently small amplitude. In fact we may notice that ∆ = γ-1 (∆1) and G(∆2) ≤ ρ(γ -1 (∆1))γ(∆1)/2F . Thus for all sufficiently small ∆1 we have:

∆2 ≤ G -1 (ρ(γ -1 (∆1))γ(∆1)/2F ) := γ(∆1).
A suitable asymptotic gain function for d ∞ ≤ ∆2 is therefore given by:

η( d ∞) = α -1 (2γ(γ -1 (γ -1 ( d ∞)))).
Take η(s) = max{η(s), η ∆ 2 (s)}, this concludes the proof of Claim 3 and of the implication 3. ⇒ 1. of Theorem 1.

Remark 3: As it has been shown, existence of an ISS Lyapunov function for the system (1) implies the AG property (i.e. the inequality (3) is true for q = 0) and estimate [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF], which is satisfied for all t ≥ 0. One might wonder if it is possible to combine both results, in order to obtain another estimate

|X(t, x; d)|W ≤ κ (|x|W + q , t) + σ ( d ∞), ∀t ≥ 0,
where κ ∈ KL, σ ∈ K and q > 0, which would mimic the conventional ISS theory. However, it is possible to show that in the general framework considered, even for q > 0, existence of an estimate like this is not possible.

B. Converse Lyapunov theorem for (2)

The objective of this subsection is to present an auxiliary useful result on existence of Lyapunov functions for the system (2), and in particular to provide an extension to non-compact manifolds of the following result from [START_REF] Nitecki | Filtrations, Decompositions, and Explosions[END_REF]:

Proposition 2: Let Assumption 1 be satisfied for a set W and suppose for all

x0 ∈ M lim t→+∞ |X(t, x0, 0)|W = 0.
Then, there exists a smooth Lyapunov function L : M → R+ such that:

• DL(x) = 0 for all x ∈ W, L(xi) = L(x i ) = L(xj) for any xi, x i ∈ Wi, xj ∈ Wj and all 1 ≤ i = j ≤ k; • υ(|x|W ) ≤ L(x), DL(x)f (x, 0) ≤ -(|x|W ) for all x ∈ M
and some υ ∈ K and a positive definite : R+ → R+.

If the set W contains only purely attracting or repelling subsets Wi, then this proposition has been proven in [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF]. Below a general case is studied adapting the arguments proposed in [START_REF] Nitecki | Filtrations, Decompositions, and Explosions[END_REF].

C. Converse Lyapunov arguments for (1)

The remaining part of the Section will be devoted to the main steps in the proof of the implication 1. ⇒ 3. of our Theorem 1. This will be carried out in several steps:

• First of all we remark that existence of an ISS Lyapunov function as in Definition 8 is equivalent to the following type of dissipation inequality:

|x|W ≥ χ(|d|) ⇒ DV (x)f (x, d) ≤ -ρ(|x|W )
with χ of class K∞ and ρ positive definite (or equivalently K∞). This follows by a standard continuity argument and it is shown for instance in [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] for the case of systems defined in Euclidean space and W being a single equilibrium. The same proof applies here, as the K∞ upper-bound of V (x) in terms of |x| is never needed in the proof. • From Lemma 2 (see appendices) it follows that there exists β of class K∞ such that differential inclusion

ż ∈ v∈D:|v|≤β -1 (|z| W /2) f (z, v). ( 13
)
fulfills a global attractivity property:

lim sup t→+∞ |Z(t, z0)|W = 0.
Notice that, due to the lack of stability and of KL bounds we cannot assume β to be directly related to the asymptotic gain η, see Lemmas 1 and 2. In other words the gain margin may be much slimmer than the asymptotic gain (in fact transient overshoots could be much larger than the asymptotic gain). • Consider a monotonically increasing sequence of compact subsets M1 ⊂ M2 ⊂ . . . ⊂ Mn . . . ⊂ M with the property that:

M = +∞ n=1 int(Mn),
where int(S) denotes the interior of the set S.

We denote by Z(t, S) the attainable set of (13) at time t from initial conditions in S. Consider next:

D = +∞ n=1 t≥0 Z(t, Mn) (14)
• The following properties of D should be crucial (see Lemma 3 for a proof):

1) D is positively invariant, 2) D is compact,
3) D is globally asymptotically stable for (13), 4) W ⊂ D 5) moreover due to compactness of D the following is also true: |x|D ≤ |x|W ≤ |x|D + c for some non-negative real c.

• By a standard converse Lyapunov argument (see for instance [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF] where the Euclidean case is treated) we show existence of a smooth W : M → R such that for all |v| ≤ 1:

DW (x)f (x, β -1 (|x|W /2)v) ≤ -α(|x|D)
for some α ∈ K∞ and α1, α2 ∈ K∞ which bound from above and below W as follows:

α1(|x|D) ≤ W (x) ≤ α2(|x|D).
• From Proposition 2 under the stated assumptions there exists a Lyapunov function U (x) (constant on invariant sets Wi) for the unperturbed system (2) fulfilling:

DU (x)f (x, 0) ≤ -(|x|W ). (15)
and υ (|x|W ) ≤ U (x) with υ ∈ K and a positive definite : R+ → R+. • By continuity of the function DU (x)f (x, d) and the equation [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] we see that the following holds for (1):

DU (x)f (x, d) ≤ -(|x|W ) + γ1(1 + |x|W )γ2(|d|).
for suitable γ1, γ2 of class K∞, for all x ∈ M and all d ∈ D. • As U (x) is a semi-proper function (it may be bounded), we need to add U and W in order to obtain a function with a class K∞ lower-bound with respect to |x|W . Define Ũ (x) = U (x) + W (x), then υ(|x|W ) ≤ Ũ (x), where

υ(s) = υ (s) if s ≤ c υ (c) + α1(s -c) if s > c
is a function from class K∞ (since α1 is from this class), and

D Ũ (x)f (x, 0) ≤ -α(|x|W ), (16)
where

α(s) = s 1+s inf s≤r≤c (r) if s ≤ c c 1+c (c) + α(s -c) if s > c
is also a function from class K∞ (since α ∈ K∞). Moreover Ũ is constant on invariant sets Wi as U is such and W is identically 0 on D ⊃ W.

• Next we want to add Ũ and W (suitably rescaled) in order to get a Lyapunov function as desired. To this end consider any Lipschitz continuous K∞ function δ fulfilling:

δ(r) ≤ γ-1 2 (α(r -c)) if r ≥ 2c γ-1 2 α(r) 2γ 1 (1+2c) if r ≤ 2c .
Rescaling W as

W (x) = W (x) 0 q(r)dr,
where the function q ∈ K will be defined later, yields:

D W (x)f (x, β -1 (|x|W /2)v) ≤ -q(W (x))α(|x|D)
≤ -q(α1(|x|D))α(|x|D).

• Define V (x) = W (x) + Ũ (x). Clearly V (x) is lower-bounded by a K∞ function of |x|W . Moreover, it fulfills for all |v| ≤ 1:

DV (x)f (x, min{β -1 (|x|W /2), δ(|x|W )}v) ≤ -α(|x|W )/2,
which follows considering separately two cases: 1) Case |x|W ≤ 2c:

DV (x)f (x, min{β -1 (|x|W /2), δ(|x|W )}v) ≤ D Ũ (x)f (x, min{β -1 (|x|W /2), δ(|x|W )}v) ≤ -α(|x|W ) + γ1(1 + 2c)γ2(δ(|x|W )) ≤ -α(|x|W ) + α(|x|W )/2 = -α(|x|W )/2.
2) Case |x|W ≥ 2c:

D W (x)f (x, min{β -1 (|x|W /2), δ(|x|W )}v) ≤ -q(α1(|x|D))α(|x|D) ≤ -q(α1(|x|W -c))α(|x|W -c).
Moreover:

D Ũ (x)f (x, min{β -1 (|x|W /2), δ(|x|W )}v) ≤ -α(|x|W ) + γ1(1 + |x|W )γ2(δ(|x|W )) ≤ -α(|x|W ) + γ1(1 + |x|W )α(|x|W -c).
Hence, it is enough to take:

q(α1(r -c)) = γ1(1 + r)
for all r ≥ 2c, in order to get:

DV (x)f (x, min{β -1 (|x|W /2), δ(|x|W )}v) ≤ -α(|x|W )
for all |v| ≤ 1 and |x|W ≥ 2c.

• The obtained inequality for V implies that for all x ∈ M and

d ∈ R m |x|W ≥ χ(|d|) ⇒ DV (x)f (x, d) ≤ -α(|x|W )/2 for χ -1 (s) = min{β -1 (s/2), δ(s)}, therefore V is an ISS Lyapunov function constant on invariant sets.
In order to complete the proof of Theorem 1 we just need to show that practical AG implies AG. This is discussed below.

Claim 4: If the system (1) enjoys the pAG property, then it also fulfills the AG property. Proof. Let us consider input signals with infinity norm less or equal to 1. From the definition of pAG property, there exists η ∈ K∞ and q ≥ 0 such that the set Ω = {x ∈ M : |x|W ≤ η(1) + q + 1} traps in finite time all solutions of (1). Moreover, in the proof above it has been established that for the set W satisfying restrictions of Assumption 1 there is a smooth Lyapunov function Ũ : M → R n (constant on invariant sets) such that ( 16) is true for the unperturbed system (2) with υ, α ∈ K∞. Thus by standard continuity arguments for the system (1) the estimate

D Ũ (x)f (x, d) ≤ -α(|x|W ) + γ1(1 + |x|)γ2(|d|)
is satisfied for some γ1, γ2 ∈ K∞. Since |x| ≤ |x|W + c for some c ≥ 0, then for all x ∈ Ω and all d ∈ D with |d| ≤ 1, the following inequality is valid:

D Ũ (x)f (x, d) ≤ -α(|x|W ) + γ1(η(1) + q + c + 2)γ2(|d|). (17)
Thus, in this case, Ũ is a kind of local ISS Lyapunov function for [START_REF] Angeli | Some remarks on density functions for dual Lyapunov methods[END_REF]. Hence given any initial condition x0 ∈ M and any input d with d ∞ ≤ 1, equation ( 17) eventually holds along the solution and an asymptotic gain estimate η1 follows along the steps of the proof of Claim 3. For inputs of larger infinity norm one may use the upperbound provided by the pAG property. Overall a single K∞ asymptotic gain exists, for instance max{η1(r), η(r) + qr}, just by patching the two cases.

IV. EXAMPLES AND COUNTEREXAMPLES

The main results of this note largely improve the range of systems to which Input-to-State stability techniques can be applied. We illustrate this point through simple and effective examples that show the power of this extended framework.

A. Multistable systems

Multistable systems cannot be treated by standard ISS theory as this only applies to globally asymptotically stable attractors. The presence of multiple stable equilibria, in fact, typically entails also existence of unstable equilibria or other attractors which cannot be accomodated by the standard theory. We illustrate this through a simple scalar example, although similar considerations apply to much more general systems:

ẋ = -x 3 + x + d. (18)
The state manifold M is in this case the Euclidean line R. The unperturbed system has 3 equilibria in -1, 0 and 1 respectively. The equilibria in 1 and -1 are locally asymptotically stable and almost-globally attractive. A standard ISS argument would necessarily involve the interval I = [-1, 1] which is the only compact globally asymptotically stable set. In particular, a Lyapunov function such as |x| 2 I could be used to prove ISS with respect to the set I. This however provides a very rough estimate of where solutions belong asymptotically, especially for zero or small disturbance amplitudes. In higher dimensional examples this issue becomes even more critical as explicit knowledge of the smallest globally asymptotically stable attractor is normally not possible. On the contrary, Theorem 1 allows to analyze system's robustness by letting W = {-1, 0, 1}. Assumption 1 is trivially satisfied for this set. A suitable candidate ISS-Lyapunov function is V (x) = (x -1) 2 (x + 1) 2 . Notice that |x|W = min{|x -1|, |x|, |x + 1|}, and therefore: |x| 4 W ≤ V (x). Moreover, taking derivatives along solutions of (18), we have:

∂V ∂x (x)[-x 3 + x + d] = -2(x -1) 2 (x + 1) 2 x 2 + 4dx(x -1)(x + 1) ≤ -(x -1) 2 (x + 1) 2 x 2 + 4d 2 ≤ -|x| 6 W + 4d 2 .
This proves that V is an ISS-Lyapunov function and by Theorem 1 the system enjoys the AG property. System ( 18) is also amenable to be

( ) 5sin(15t) d t = ( ) 35sin(15t) d t = 0 2 4 6 8 t 2 - 1 - 0 1 2 x 0 2 4 6 8 t 4 - 2 - 0 2 4
x Fig. 2. The results of simulation for [START_REF] Wilson | Smoothing derivatives of functions and applications[END_REF] analyzed along the lines of [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF], in particular, due to the exponential instability of the equilibrium in 0, it is possible to conclude that almost all solutions converge to a neighborhood of {-1, 1} of size "proportional" to the infinity norm of d.

Next we consider a slight modification of the system (18):

ẋ = -x 5 + x 3 + xd. ( 19
)
The unperturbed system still exhibits 3 equilibria, in -1, 0, 1, two of which (those in -1 and 1 ) are asymptotically stable. The equilibrium in 0 is antistable, but non-hyperbolic. This makes the almost global stability theory of [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF] unapplicable. In fact a small disturbance is capable of locally stabilizing the 0 equilibrium and creating a basin of attraction of non-zero measure around it. Despite this, let W = {-1, 0, 1} and define V (x) = (x -1) 2 (x + 1) 2 . One may verify that V still serves as an ISS-Lyapunov function for equation [START_REF] Wilson | Smoothing derivatives of functions and applications[END_REF].

∂V ∂x (x)[-x 5 +x 3 +xd] = -4x 4 (x-1) 2 (x+1) 2 +4x 2 (x-1)(x+1)d ≤ -2x 4 (x -1) 2 (x + 1) 2 + 2d 2 ≤ -2|x| 8 W + 2d 2 .
Finally, we would like to show by simulation that even a "large" perturbation d may stabilize the unstable equilibrium at the origin. The results of simulation of [START_REF] Wilson | Smoothing derivatives of functions and applications[END_REF] for different initial conditions and d(t) = 5 sin(15t) or d(t) = 35 sin(15t) are shown in Fig. 2. As we can see, for a "small" perturbation d(t) = 5 sin(15t) almost all trajectories converge to neighborhoods of equilibriums -1 and 1 (size of the neighborhoods is proportional the amplitude of d), but a "big" perturbation d(t) = 35 sin(15t) makes the origin attractive, which is not an intuitively awaited behavior.

B. A planar example: pendulum with friction

Consider the following set of differential equations, describing the motion of a forced pendulum with friction which was also used as a motivating example in [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF]:

θ = ω ω = -a sin(θ) -bω + d. ( 20
)
We regard them as a system with state x = [θ, ω] taking values on the cylinder M := S × R affected by some exogenous disturbance d(t), whereas a, b are constant positive parameters. Overall the unperturbed system admits two equilibria [0, 0] and [π, 0] the latter being a saddle-point. It is shown in [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF] that this system is almost-globally Input-to-State Stable with respect to the downwards pendulum equilibrium x = [0, 0] . The same Lyapunov functions used in [START_REF] Angeli | Stability robustness in the presence of exponentially unstable isolated equilibria[END_REF] can be used in order to prove Input-to-State stability with respect to the set W = {[0, 0], [π, 0]}. Consider the mechanical energy of the pendulum, that is V (x) = ω 2 /2 -a cos(θ) + a.

For the case d = 0 the dissipation equality V = -bω 2 holds, which together with the fact that there is no trajectory on the line ω = 0 connecting the equilibriums imply Assumption 1. Notice that |x|W = ω 2 + min{|θ|, |θ -π|} 2 . Therefore ε|x| 2 W ≤ V (x) for some sufficiently small ε > 0. Moreover, taking derivatives of V along solutions of (20) yields:

V (x) = -bω 2 + ωd ≤ - b 2 ω 2 + 1 2b d 2 = - b 2 V (x) - ab 2 cos(θ) + 1 2b d 2 ≤ - b 2 V (x) + c + 1 2b d 2
with constant c := ab/2. Therefore system (20) admits a practical ISS Lyapunov function. By Theorem 1 system (20) is Input-to-State Stable with respect to W.

C. Non decomposable invariant set

Consider the system:

θ = 1 -cos(θ) + d ż = -z + d (21)
with state x = [θ, z] taking values in the cylinder S×R. In the absence of disturbances all solutions converge to the unique equilibrium x = [0, 0] (up to multiples of 2π in the first coordinate) as it is easy to check by considering separately the θ and z equation. The equilibrium [0, 0] is not asymptotically stable, however, and in fact, the singleton {[0, 0]} does not admit a decomposition without cycles. This means that in order to apply our main result we need to enlarge the set W. We can in fact define W = S × {0}. This is an invariant and asymptotically stable set for the unperturbed system. Moreover, letting V (x) = z 2 , yields along solutions:

V (x) = 2z(-z + d) ≤ -z 2 + d 2 = -|x| 2 W + d 2
where the last equality follows since |x|W = |z|. Moreover, |x| 2 W ≤ V (x); therefore, the system admits an ISS Lyapunov function and thanks to Theorem 1 is Input-to-State Stable with respect to the set W. One may wonder whether tighter characterizations of the robustness properties of system (21) could be possible, for instance if ISS with respect to the set {0} be fulfilled regardless of it exhibiting homoclinic cycles. To show that this is not possible take any positive and vanishing disturbance which is not in L1, such as d(t) = 1/(1 + t), and consider the solution [θ(t), z(t)] := X(t, [0, 0]; d). If ISS would be true, solutions should converge to 0 by the converginginput converging-state property. While it is easy to see that z(t) → 0 as t → +∞ it is also clear that

+∞ 0 | θ(t)|dt = +∞ 0 1 -cos(θ(t)) + d(t) dt ≥ +∞ 0 d(t)dt = +∞,
thus proving that solutions never stop describing full rotations around the circle. Hence, the choice of W = S × {0} is in fact the tightest possible.

D. Van der Pol oscillator

Consider the following set of equations, describing the state evolution of a system evolving in M = R 2 :

ẋ1 = x2 ẋ2 = -x1 -ϕ(x2) + d (22)
where ϕ : R → R satisfies ϕ(0) = 0, ϕ (0) < 0 and ϕ(y) → +∞ as y → +∞, ϕ(y) → -∞ as y → -∞.

This set of equations encompasses, for a peculiar choice of ϕ, the so called Van der Pol oscillator. It was shown in [START_REF] Arcak | Input-to-State Stability for a class of Lurie systems[END_REF] that a practical asymptotic gain property holds for (22) (as well as for a broader class of systems). For ϕ(x2) = x2 -

x 3 2
3 a Lyapunov function to establish the practical asymptotic gain property with respect to the origin is given in [START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF]. Moreover, it is well known that Van der Pol oscillator in the absence of external forcing admits 2 invariant sets, namely the origin and the limit cycle L. Hence, we may define W = {0}∪L. As any solution converges to L, except for the one initiated at 0, which is antistable, we can conclude that W admits a decomposition without cycles. As a consequence we can claim existence of an ISS-Lyapunov function and of a class K∞ function γ such that:

lim sup t→+∞ |X(t, ξ; d)|W ≤ γ( d ∞).

E. FitzHugh-Nagumo model

This model is a two-dimensional simplification of the Hodgkin-Huxley model of spike generation:

ẋ1 = x1 -x 3 1 -x2 + d ẋ2 = τ -1 (x1 -a -bx2) (23)
where x1 ∈ R is the membrane potential, x2 ∈ R is a recovery variable, and d is the magnitude of stimulus current. The model ( 22) is a particular case of (23) for a = b = 0. It is well known fact that for any constant d this model has an equilibrium x d and almost globally attracting set Γ containing oscillating trajectories (depending on values of parameters). Thus Assumption 1 is satisfied for W = {x d }∪Γ, but as for L in (22), it is hard to find an analytical expression for characterization of Γ as a function of x. Therefore, similarly to (22) we can establish the practical asymptotic gain property with respect to the origin using the function

V (x) = 0.5(x 2 1 +τ x 2 2 ), V ≤ -0.5(x 2 1 +bx 2 2 )+1+0.5a 2 /b+0.5d 2 ,
then the system (23) admits the practical asymptotic gain property with respect to W since |x|W ≤ |x|. Next, according to result of Theorem 1 the FitzHugh-Nagumo model is ISS with respect to W considering stimulus current as input.

A similar consideration can be repeated for the Hindmarsh-Rose model of neuronal activity.

F. A model with a continuum of equilibria

Consider the system defined on the cylinder M = R × S by the following set of differential equations

ż = -z + 2z cos 2 (θ) + d θ = z 2 cos(θ). ( 24
)
with state x = [z, θ] expressed in coordinates, with the usual convention that points x whose second coordinate differs by a multiple of 2π are identified. It is easy to prove that the set W = {0} × S is the equilibrium set for the unperturbed system. This is an invariant connected set that does not admit finer decompositions. Take the Lyapunov function candidate V given below:

V (z, θ) = z 2 + 4 -4 sin(θ).
Clearly, V (x, θ) ≥ z 2 = |x| 2 W . Moreover, differentiating V along solutions of (24) yields:

V (z, θ) = 2z(-z + 2z cos 2 (θ) + d) -4z 2 cos(θ) 2 = -2z 2 + 2zd ≤ -z 2 + d 2 = -|x| 2 W + |d| 2 .
Hence V is an ISS Lyapunov function and the system is ISS with respect to W. Notice that V (W) = [0, 8] and is therefore not a singleton. Finding a Lyapunov function constant on invariant sets appears to be more challenging. In particular, V (x) = z 2 is not appropriate as its derivative is not negative definite already for the unperturbed dynamics.

V. CONCLUSION

The paper proposes definitions and characterizations of input-tostate stability for systems with multiple attractors, in the cases when the invariant sets of the system are not connected by homoclinic and heteroclinic trajectories or, alternatively, these are included in the invariant set themselves. The invariant sets under consideration may contain disjointed subsets, some of which may be unstable in the Lyapunov sense. It is shown that under such assumptions the practical stability notions are equivalent to "conventional" ones. Necessary and sufficient characterizations of input-to-state stability in terms of Lyapunov function existence are given. Applicability of the proposed framework is demonstrated on several examples of popular systems.

APPENDIX

A. Some technical Lemmas

Lemma 1: Consider a nonlinear system as in [START_REF] Angeli | Some remarks on density functions for dual Lyapunov methods[END_REF], and assume that it enjoys the AG property (3). Then, there exists β of class K∞ and q ≥ 0 such that for all x ∈ M and all measurable inputs d(•) the pGS property holds. Proof. Define, for all r ≥ 0 the following open set:

Ω(r) := {x ∈ M : |x|W < η(r) + 1} ( 25
)
where η denotes the asymptotic gain. For all solutions X(•, x; d) we define the hitting time as follows: (notice that the corollary is stated for systems evolving in Euclidean space, but the same proof applies to systems on manifolds). Define next the reachable set:

τ x,d (r) = inf{t ≥ 0 : X(t, x; d) ∈ Ω(r)}. ( 26
R T (C, r) := {z ∈ M : ∃ t ∈ [0, T ], ∃ x ∈ C, ∃ d(•) : d ∞ ≤ r : z = X(t, x; d)}.
Clearly, if C ⊃ Ω(r) we have: Notice that by construction Cr ⊃ Ω(r). Let, for any r ≥ 0, the function Γ be defined as follows:

R(C) := T ≥0 R T (C, r) = R T C,r (C, r). (27)
Γ(r) := sup z∈Cr ,d(•): d ∞≤r, t≥0 |X(t, z; d)|W (28)
Hence, by boundedness of R T Cr ,r (Cr, r), Γ is a well defined nondecreasing function. Moreover, for any x ∈ M and any bounded input d we may let r = max{|x|W , d ∞} and by equation ( 28) we see that:

|X(t, x; d)|W ≤ Γ(max{|x|W , d ∞}) ∀ t ≥ 0.
Hence, for some K∞ function β and q = Γ(0):

|X(t, x; d)|W ≤ q + β(max{|x|W , d ∞}) ∀ t ≥ 0.
Lemma 2: Consider a system as in [START_REF] Angeli | Some remarks on density functions for dual Lyapunov methods[END_REF] and fulfilling all assumptions of Lemma 1. Then, provided β(•) ≥ η(•) (which can be assumed without loss of generality) the differential inclusion:

ż ∈ v∈D:|v|≤β -1 (|z| W /2) f (z, v). ( 29
)
has uniformly bounded solutions. Moreover, all solutions converge asymptotically to W. Proof. Consider z ∈ M arbitrary and let Z(t, z) denote any solution of (29) (maximally defined over some open interval I including 0). By selection, there exists µ, measurable and µ ≤ 1 such that, Z(t, z) is a solution of:

ż = f (z, β -1 (|z|W /2)µ).
By Lemma 1, Z fulfills for all t ≥ 0 in I:

|Z(•, z)|W [0,t] ≤ q + β(max{|z|W , β -1 ( |Z(•, z)|W [0,t] /2)}).
Then we have for all t ∈ I:

|Z(t, z)|W ≤ 2q + 2β(|z|W ). (30)
Hence, solutions of (29) are uniformly bounded and defined for all t ≥ 0. It follows from the asymptotic gain property that: Applying this inequality to the solution Z previously defined we have:

lim sup t→+∞ |Z(t, z)|W ≤ η lim sup t→+∞ β -1 (|Z(t, z)|W /2) ≤ lim sup t→+∞ |Z(t, z)|W /2.
Hence, |Z(t, z)|W → 0 as t → +∞. Lemma 3: The set D defined in ( 14) is bounded, closed, forward invariant and globally asymptotically stable. Moreover it contains W. Proof. Consider the differential inclusion (29) and let R T (S) denote its reachable set up to time T for initial conditions in S, and R(S) := T ≥0 R T (S); by Lemma 2 each solution converges to W and therefore enters in finite time the set Z := {z ∈ M : |z|W ≤ 1} as well as a suitable compact subset of int(Z).

There exists index n such that Z ⊂ Mn. Indeed, {int(Mn)} +∞

n=1 is an open cover of M , and therefore, due to paracompactness of M , it admits a locally finite refinement, i.e. any point x in M has a neighboorhood Ux that intersects only finitely many sets in the refinement {V k } +∞ k=1 . Denote by n k the integer (as a function of k) such that V k ⊂ int(Mn k ). Any compact subset K of M is covered by the {Ux : x ∈ K}, and in particular (by compactness) by a finite number of them:

K ⊂ Ux 1 ∪ Ux 2 ∪ . . . ∪ Ux N .
Each one of the Ux i s is in its turn contained in a finite number of sets from the refinement {V k } +∞ k=1 so that, overall, every compact set K is covered by a finite number of Vns. Let n be the maximum of the indexes ks involved in such a cover. Then

K ⊂ n k=1 V k ⊂ n k=1 int(Mn k ) = M N , where N = max k∈{1...n} n k
and the last equality follows by monotonicity of the Mn sequence. This shows that every compact set is contained in some Mn. Without loss of generality assume M1 ⊃ Z. By virtue of Corollary III.3 in [START_REF] Sontag | New characterizations of input-to-state stability[END_REF] for each n ∈ N there exists Tn < +∞, such that for all z ∈ Mn and any solution Z(•, z) of (29) there exists some time t Z(•),n ≤ Tn, such that Z(t Z(•),n , z) ∈ Z. Hence, for all t ≥ Tn the following holds:

Z(t, Mn) ⊂ T ≥0 R T (Z) = R T 1 (Z)
where the last equality follows since M1 ⊃ Z. Notice that R T 1 (Z) is a compact set, by forward completeness of (29) and compactness of Z. This entails that R(Z) is also compact.

Hence, t≥0 Z(t, Mn) ⊂ R T 1 (Z) and

D = +∞ n=1 t≥0 Z(t, Mn) ⊂ +∞ n=1 R T 1 (Z) = R T 1 (Z), thus showing boundedness of D.
By boundedness of D, there exists n ∈ N such that for all integers n ≥ n it holds Mn ⊃ D and Mn ⊃ W. Moreover, assuming without loss of generality Lipschitzness of the differential inclusion (29) (this can be done as any K∞ function admits a Lipschitz and K∞ lower bound), it holds Z(t, W) = W, for all t ≥ 0. Hence:

D = n≥n t≥0 Z(t, Mn) ⊃ n≥n t≥0 Z(t, W) = W. (31)
Notice that, for all n large enough, (say larger than n) it holds:

Mn ⊃ R(Z).
Moreover, we see that:

t≥0 Z(t, R(Z)) ⊂ t≥0 Z(t, Mn) ⊂ t≥Tn Z(t, Mn) ⊂ t≥0 Z(t, R(Z)).
where the last inclusion holds since any solution originating in Mn ends up in R(Z) at time Tn (having touched Z in the meanwhile). Hence, the following holds for all n large enough:

t≥0 Z(t, R(Z)) = t≥0 Z(t, Mn).
We exploit this fact by remarking that:

D = n≥n t≥0 Z(t, Mn) = n≥n t≥0 Z(t, R(Z)) = t≥0 Z(t, R(Z)).
Notice the following monotonicity property, Z(t, R(Z)) ⊂ R(Z) (for t ≥ 0), as any solution initialted in the reachable set R(Z) is after t units of time still inside R(Z). More generally:

t1 ≥ t2 ≥ 0 ⇒ Z(t1, R(Z)) ⊂ Z(t2, R(Z))
as it follows considering that:

Z(t1, R(Z)) = Z(t2, Z(t1 -t2, R(Z))) ⊂ Z(t2, R(Z)).
This can be used in order to see that for all τ ≥ 0:

Z(τ, D) = Z(τ, t≥0 Z(t, R(Z))) = t≥τ Z(t, R(Z)) = t≥0 Z(t, R(Z)) = D.
Finally, D is closed as it can be written as an intersection of closed sets, (viz. Z(t, R(Z)) for t ≥ 0, where each of the set is closed by forward completeness of the flow and compactness of R(Z). In order to show Global Asymptotic Stability it is enough to remark that this is equivalent to uniform Attraction (as in [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF]). Notice also that the latter follows Corollary 3.3 in [START_REF] Sontag | New characterizations of input-to-state stability[END_REF].

B. Converse Lyapunov theorems for dichotomy systems on manifolds

In the proof of Proposition 2 (see Appendix C) we will need the following auxiliary lemmas dealing with two forward invariant sets Ai, Bi ⊂ M (the case of a compact set Ai has been treated in [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF]). Define ρ i max = infx∈A i ,y∈B i δ(x, y). As in Proposition 2, it is assumed that Assumption 1 is satisfied, then the set Ai ∪ Bi admits a finite decomposition without cycles.

Lemma 4: Let Ai, Bi be forward invariant sets, which are asymptotically stable for forward and backward flows of (2) in M \Bi and M \ Ai respectively. Then for any 0 < ρ < ρ i max there exists a locally Lipschitz continuous function V : M \Dρ → R+, Dρ = {x ∈ M : |x|B i < ρ} (continuous on the set Ai) such that α1(|x|A i )

V (x) for all x ∈ M \Dρ for α1 ∈ K∞, and

DV (x)f (x, 0) -β(V (x), |x|), β ∈ KL for a.e. x ∈ M \Dρ. Proof. For any x0 ∈ M \Dρ, define v(x0) = sup t 0 |X(t, x0, 0)|A i , by construction |x0|A i v(x0
) and v(x) = 0 for x ∈ Ai due to forward invariance of Ai. From attractivity of Ai and continuity of X(t, •, 0), for any x0 ∈ M \Dρ there exists Tx

0 ∈ R+ such that v(x0) = sup 0 t Tx 0 |X(t, x0, 0)|A i . To analyze continuity of the function v, consider |v(x 1 ) -v(x 2 )| = | sup t 0 |X(t, x 1 , 0)| A i -sup t 0 |X(t, x 2 , 0)| A i | = | sup 0 t Tx 1 |X(t, x 1 , 0)| A i - sup 0 t Tx 2 |X(t, x 2 , 0)| A i | sup 0 t T ||X(t, x 1 , 0)| A i -|X(t, x 2 , 0)| A i |,
where T = max{Tx 1 , Tx 2 } and x1, x2 ∈ M \ Dρ. Due to Lipschitz continuity of the system (2), for any compact set of initial conditions E ⊂ M \ Dρ and any time 0

T < +∞, there exist K ∈ R+, L ∈ R+ such that δ(X(t, x1, 0), X(t, x2, 0)) K δ(x1, x2) and ||X(t, x1, 0)|A i -|X(t, x2, 0)|A i | L δ(x1, x2),
for all 0 t T and any x1, x2 ∈ E. For all 0 < ρ < ρ i max and for any compact E ⊂ M \ Dρ there exists Tρ = sup x 0 ∈E Tx 0 with the property Tρ < +∞ (due to local repelling property of the set Bi, for any 0 < ρ < ρ i max there exists 0 < ρ ρ such that trajectories initiated into the set E never reach the set D ρ ). Keeping this in mind we see that

|v(x1) -v(x2)| sup 0 t Tρ ||X(t, x1, 0)|A i -|X(t, x2, 0)|A i | L δ(x1, x2)
for all x1, x2 ∈ E, and the function v is locally Lipschitz continuous on the set E for any fixed 0 < ρ < ρ i max as claimed. Moreover, the function v is not increasing on any trajectory of the system ( 2

V (x0) = sup t 0 {v(X(t, x0, 0))k(t)},
where k : R+ → R+ is a continuously differentiable function satisfying 0 < κ1 k(t) κ2 < +∞ and k(t) κ3(t) > 0 for all t 0, where κ3 is a monotonically decreasing function. An example of such a function is

k(t) = (κ1 + κ2t)(1 + t) -1 , k(t) = (κ2 -κ1)(1 + t) -2 .
The function V has a lower bound κ1|x0|W V (x0) and V (x) = 0 for all x ∈ Ai. Again, for any x0 ∈ M \ Dρ there exists Tx 0 ∈ R+ such that V (x0) = sup 0 t Tx 0 {v(X(t, x0, 0))k(t)}. This claim follows from the non-strict decreasing of the function v(X(t, x0, 0)) to zero with t → +∞. Next, for all x1, x2 ∈ M \ Dρ

|V (x 1 ) -V (x 2 )| = | sup t 0 {v(X(t, x 1 , 0))k(t)} -sup t 0 {v(X(t, x 2 , 0))k(t)}| = | sup 0 t T {v(X(t, x 1 , 0))k(t)} - sup 0 t T {v(X(t, x 2 , 0))k(t)}| sup 0 t T |k(t)[v(X(t, x 1 , 0)) -v(X(t, x 2 , 0))]| κ 2 sup 0 t T |v(X(t, x 1 , 0)) -v(X(t, x 2 , 0))|,
where T = max{Tx 1 , Tx 2 }. For all 0 < ρ < ρ i max and for any compact E ⊂ M \ Dρ there exists Tρ = sup x 0 ∈E Tx 0 such that Tρ < +∞ and

|V (x1) -V (x2)| κ2 sup 0 t Tρ |v(X(t, x1, 0)) -v(X(t, x2, 0))| κ2L δ(X(t, x1, 0), X(t, x2, 0)) κ2LK δ(x1, x2)
for all x1, x2 ∈ E. The function V is locally Lipschitz continuous on the set M \Dρ for any 0 < ρ ρ i max and strictly decreasing for any x0 ∈ M \{Ai ∪ Dρ}:

V (X(t, x0, 0)) = sup τ 0 {v(X[τ, X(t, x0, 0), 0])k(τ )} = sup τ t {v(X[τ, x0, 0])k(τ -t)} < sup τ 0 {v(X[τ, x0, 0])k(τ )} = V (x0),
In addition, V equals zero on the set Ai. Denote by for all r ∈ R+ and some ϕ0 ∈ R+, ϕ1 ∈ K. By definition

L f (x 0 ,0) V (x0) := lim sup h→0 h -1 [V (X(h, x0, 0)) -V (x0)], then L f (x 0 ,0) V (
V (X(h, x 0 , 0)) = sup t 0 {v(X[t, X(h, x 0 , 0), 0])k(t)} = sup h t T |x 0 | {v(X[t, x 0 , 0])k(t -h)} = sup h t T |x 0 | {v(X[t, x 0 , 0])k(t)k(t) -1 k(t -h)} ≤ sup h t T |x 0 | {v(X[t, x 0 , 0])k(t)} sup h t T |x 0 | {k(t) -1 k(t -h)} V (x 0 ) sup h t T |x 0 | {k(t) -1 k(t -h)}
for a.e. x0 ∈ M \{Ai ∪ Dρ}. Further

lim h→0 h -1 [V (X(h, x 0 , 0)) -V (x 0 )] = lim h→0 h -1 [V (x 0 ) sup h t T |x 0 | {k(t) -1 k(t -h)} -V (x 0 )] = V (x 0 ) lim h→0 h -1 [ sup h t T |x 0 | {k(t) -1 k(t -h)} -1] = V (x 0 ) lim h→0 h -1 sup h t T |x 0 | k(t) -1 {k(t -h) -k(t)} V (x 0 ) sup 0 t T |x 0 | k(t) -1 lim h→0 h -1 {k(t -h) -k(t)} = V (x 0 ) sup t 0 {-k(t) -1 k(t)} -κ -1 2 κ 3 (T |x 0 | )V (x 0 ).
Due to properties of the function κ3 (it is a strictly decreasing function from a constant (κ2 -κ1) to zero), the inequality

L f (x,0) V (x) -κ -1 2 κ3(ϕ0 + ϕ1[|x|])V (x)
has been approved for a.e. x ∈ M \{Ai ∪ Dρ}. This inequality is additionally valid on the set Ai, then we obtain

L f (x,0) V (x) -β(V (x), |x|)
for all x ∈ M \Dρ for a function β ∈ KL.

Lemma 5: Let Ai, Bi be forward invariant sets, which are asymptotically stable for forward and backward flows of (2) in M \ Bi and M \ Ai respectively. Then for any 0 < ρ < ρ i max there exists a locally Lipschitz continuous function V : Dρ → R+, Dρ = {x ∈ M : |x|B i < ρ} (continuous on the set Bi) such that α1(|x|B i ) V (x) α2(|x|B i ) < 1, L f (x,0) V (x) β(V (x), |x|), for α1, α2 ∈ K, β ∈ KL and a.e. x ∈ Dρ. Proof. For any 0 < ρ < ρ i max and all x0 ∈ Dρ\Bi there exists T ρ

x 0 ∈ R+ such that X(t, x0, 0) / ∈ Dρ for all t T ρ x 0 (the set Ai is asymptotically stable). Then there exists 0 ≤ Tx 0 ≤ T ρ x 0 such that inf t≥0 |X(t, x0, 0

)|B i = inf 0 t Tx 0 |X(t, x0, 0)|B i , define v(x0) = inf 0 t Tx 0 |X(t, x0, 0)|B i , by construction v(x0) |x0|B i < ρ and v(x) = 0 iff x ∈ Bi To analyze continuity of the function v, consider |v(x 1 ) -v(x 2 )| = | inf 0 t Tx 1 |X(t, x 1 , 0)| B i -inf 0 t Tx 2 |X(t, x 2 , 0)| B i | sup 0 t T ||X(t, x 1 , 0)| B i -|X(t, x 2 , 0)| B i |,
where T = max{Tx 1 , Tx 2 }. Due to Lipschitz continuity of solutions of the system (2), for any compact set of initial conditions E ⊂ M and time 0 T < +∞, there exist K ∈ R+, L ∈ R+ such that δ(X(t, x1, 0), X(t, x2, 0)) K δ(x1, x2),

||X(t, x1, 0)|B i -|X(t, x2, 0)|B i | L δ(x1, x2),
for all 0 t T and any x1, x2 ∈ E. For any compact set E ⊂ M define Tρ = sup x 0 ∈E∩Dρ Tx 0 , then

|v(x 1 )-v(x 2 )| sup 0 t Tρ ||X(t, x 1 , 0)| B i -|X(t, x 2 , 0)| B i | L δ(x 1 , x 2 )
for all x1, x2 ∈ E∩Dρ, and the function v is locally Lipschitz continuous on the set Dρ\Bi and continuous on Dρ. The function v is not decreasing on any trajectory of the system (2), indeed for any

x0 ∈ Dρ: v(X(t, x0, 0)) = inf 0 τ T X(t,x 0 ,0) |X(τ, X(t, x0, 0), 0)|B i = inf t τ Tx 0 |X(τ, x0, 0)|B i inf 0 τ Tx 0 |X(τ, x0, 0)|B i = v(x0). Therefore, δ (|x0|B i ) v(x0) for δ (s) = s(1 + s) -1 inf |x| B i =s v(x), δ ∈ K and all x0 ∈ Dρ.
Now, define a new function for all x0 ∈ Dρ:

V (x0) = inf 0 t Tx 0 {v(X(t, x0, 0))k(t)},
where k : R+ → R+ is a continuously differentiable function with properties 0 < κ1 k(t) κ2 < +∞ and k(t) -κ3(t) < 0 for all t 0, where κ3 is a monotonically decreasing function. An example of such a function is

k(t) = κ1 + (κ2 -κ1)e -t , k(t) = (κ1 -κ2)e -t .
The function V has bounds κ1δ (|x0|B i ) V (x0) κ2|x0|B i and V (x) = 0 for all x ∈ Bi. Next, for all x1, x2 ∈ Dρ,

|V (x 1 ) -V (x 2 )| = | inf 0 t Tx 1 {v(X(t, x 1 , 0))k(t)} - inf 0 t Tx 2 {v(X(t, x 2 , 0))k(t)}| = | inf 0 t T {v(X(t, x 1 , 0))k(t)} - inf 0 t T {v(X(t, x 2 , 0))k(t)}| sup 0 t T |k(t)[v(X(t, x 1 , 0)) -v(X(t, x 2 , 0))]| κ 2 sup 0 t T |v(X(t, x 1 , 0)) -v(X(t, x 2 , 0))|,
where T = max{Tx 1 , Tx 2 }. For any compact set E ⊂ M define Tρ = sup x 0 ∈E∩Dρ Tx 0 as before and

|V (x1) -V (x2)| κ2 sup 0 t Tρ |v(X(t, x1, 0)) -v(X(t, x2, 0))| κ2L δ(X(t, x1, 0), X(t, x2, 0)) κ2LK δ(x1, x2)
for all x1, x2 ∈ E∩Dρ. Then the function V is locally Lipschitz continuous on the set Dρ\Bi and continuous on Dρ. This function is strictly increasing for any x0 ∈ Dρ\Bi:

V (X(t, x 0 , 0)) = inf 0 τ T X(t,x 0 ,0) {v(X[τ, X(t, x 0 , 0), 0])k(τ )} inf t τ Tx 0 {v(X[τ, x 0 , 0])k(τ -t)} > inf 0 τ Tx 0 {v(X[τ, x 0 , 0])k(τ )} = V (x 0 ), V (
t) equals zero on any trajectories into the set Bi, then for all r ∈ R+ and some ϕ0 ∈ R+, ϕ1 ∈ K. By definition

L f (x 0 ,0) V (x0) = lim h→0 h -1 [V (X(h, x0, 0)) -V (
V (X(h, x 0 , 0)) = inf 0 t T X(h,x 0 ,0) {v(X[t, X(h, x 0 , 0), 0])k(t)} = inf h t T |x 0 | {v(X[t, x 0 , 0])k(t -h)} = inf h t T |x 0 | {v(X[t, x 0 , 0])k(t)k(t) -1 k(t -h)} inf h t T |x 0 | {v(X[t, x 0 , 0])k(t)} inf h t T |x 0 | {k(t) -1 k(t -h)} V (x 0 ) inf h t T |x 0 | {k(t) -1 k(t -h)}.
Finally,

lim h→0 h -1 [V (X(h, x 0 , 0)) -V (x 0 )] lim h→0 h -1 [V (x 0 ) inf h t T |x 0 | {k(t) -1 k(t -h)} -V (x 0 )] = V (x 0 ) lim h→0 h -1 [ inf h t T |x 0 | {k(t) -1 k(t -h)} -1] = V (x 0 ) lim h→0 h -1 inf h t T |x 0 | k(t) -1 {k(t -h) -k(t)} V (x 0 ) inf 0 t T |x 0 | k(t) -1 lim h→0 h -1 {k(t -h) -k(t)} = V (x 0 ) inf 0 t T |x 0 | {-k(t) -1 k(t)} κ -1 2 κ 3 (T |x 0 | )V (x 0 ).
for a.e. x ∈ Dρ (the inequality is additionally valid on the set Bi). Since κ3 is a monotonically decreasing function, the following inequality has been substantiated:

L f (x,0) V (x) κ -1 2 κ3(ϕ0 + ϕ1(|x|))V (x) ≥ β(V (x), |x|)
for a.e. x ∈ Dρ for some β ∈ KL. Remark 4: Note that if the set Ai (or Bi) is compact, then there exists constant mA i > 0 (mB i > 0) such that for any x ∈ M it holds |x| ≤ |x|A i + mA i (|x| ≤ |x|B i + mB i ) and therefore

DV (x)f (x, 0) -β(V (x), |x|) ≤ -β(V (x), |x|A i + mA i ) (DV (x)f (x, 0) β(V (x), |x|) ≥ β(V (x), |x|B i + mB i )) .
Therefore, we can introduce a function Ṽ (x) = V (x) 0 σ(r)dr for a suitably defined σ ∈ K∞ such that

α1(|x|A i ) Ṽ (x), D Ṽ (x)f (x, 0) -β( Ṽ (x)) α1(|x|B i ) Ṽ (x) α2(|x|B i ), D Ṽ (x)f (x, 0) β( Ṽ (x))
for a.e. x ∈ M \Dρ (x ∈ Dρ) and some α1, α2, β ∈ K∞, see also Lemma 3 in [START_REF] Efimov | Global Lyapunov analysis of multi-stable nonlinear systems[END_REF].

C. Proof of Proposition 2

In the following assume, without loss of generality, that the unperturbed system ẋ = f (x, 0) is backwards and forward complete, viz. its solutions are globally defined in R. If this is not the case just replace it by the system

ẋ = f (x, 0) = f (x, 0)/(1 + |f (x, 0)|x).
The dissipation inequality proved for f will a fortiori hold for f . Consider the following sets for any

1 ≤ i ≤ k: Ai = j≤i R(Wj), Bi = i<l A(W l ).
We have the following properties for these sets: A Since the sets Wi, 1 ≤ i ≤ k create a filtration ordering of W, then the set Ai represents the initial conditions attracted in the backward time by Wi and all its sinks Wj (Wj ≺ Wi for j < i).

The set Ai is a union of "repulsion" sets R(Wj) of Wj with j ≤ i; if Wi is a purely attracting set, that is always the case for i = 1 under conditions of the proposition, then Ai = Wi. The set Bi corresponds to the initial conditions attracted by Wi+1 and all its sources W l (Wi+1 ≺ W l for i+1 < l) in the forward time. B The sets Ai and Bi are forward invariant for (2) and Ai ⊂ Ci, where Ci = j≤i A(Wj) is a forward invariant set. Indeed, if a set Wi is purely attracting then R(Wi) = Wi ⊂ A(Wi) ⊂ Ci. If a set Wi is not purely attracting, then by the definition of filtration ordering (see Definition 4) all its sinks Wj have j < i, therefore R(Wi) ⊂ j<i A(Wj) ⊂ Ci. C Take an x ∈ clos[R(Wi)], then X(t, x, 0) has its α-limit set inside some Wj with j ≤ i by the filtration ordering definition, therefore clos[R(Wi)] ⊂ Ai for any i ≥ 1. Now let us apply these arguments to the reverse flow X(-t, x, 0). We see that: j≤i A(Wj) \ Wj and it does not contain limit invariant solutions of the system (2), therefore all trajectories initiated at Ci \ Ai go to Ai, which contains all Wj for j ≤ i. Thus for any ε > 0 and all x0 ∈ Ci there is a 0 ≤ Tx 0 ,ε < +∞ such that |X(t, x0, 0)|A i ≤ ε for all t ≥ Tx 0 ,ε (note that X(t, x0, 0) ∈ Ci for all t ≥ 0 since Ci is invariant). E The set Ai is locally Lagrange stable, i.e. for any δ > 0 there is an > 0 such that |X(t, x0, 0)|A i ≤ for all t ≥ 0 and for all x0 ∈ Ci with |x0|A i ≤ δ. Indeed, take a set Dρ = {x ∈ Ci : |x|A i ≤ ρ} for some ρ > 0 and consider ν(ρ) = sup x 0 ∈Dρ sup t≥0 |X(t, x0, 0)|A i . Remark that ν(ρ) is a monotone function. Assume that ν(ρ) < +∞ for any such ρ > 0, then the set Ai is stable with δ = ρ and = ν(ρ). On the contrary, assume that there is a ρ > 0 such that ν(ρ ) = +∞, then it means that there exists a sequence of points x q ∈ D ρ , q ∈ N+ such that sup q≥0 sup t≥0 |X(t, x q , 0)|A i = +∞. Take an ε < ρ , then by attractiveness of Ai there are 0 ≤ T x q ,ε < +∞ such that |X(t, x q , 0)|A i ≤ ε for all t ≥ T x q ,ε . Define T = sup q≥0 T x q ,ε , let T < +∞, then the trajectories X(t, x q , 0) leave to infinity and approach a vicinity of the set Ai in a finite time T , that is a contradiction due to the system continuity and its forward completeness. Finally, assume that T = +∞, then it means that there is a point x∞ ∈ Ci such that x q → x∞ when q → +∞ such that T x∞,ε/2 = +∞, which contradicts the set Ai attractiveness in Ci. Therefore, ν(ρ) < +∞ for any such ρ > 0 and the set Ai is locally Lagrange stable in Ci. F The set Ai is locally Lyapunov stable, i.e. for any > 0 there is a δ > 0 such that |X(t, x0, 0)|A i ≤ for all t ≥ 0 and for all x0 ∈ Ci with |x0|A i ≤ δ. Indeed, by the definition above, ν(ρ1) ≤ ν(ρ2) for ρ1 ≤ ρ2. In addition limρ→0 ν(ρ) = 0, assume it is not, then it means that there exists an υ > 0 such that for any ι > 0 there is x0 ∈ Ci with |x0|A i ≤ ι such that |X(t , x0, 0)|A i ≥ υ for some t ≥ 0. Since ι > 0 can be chosen arbitrary, it implies that there exist some trajectories, which exit from Ai into Ci and return back to Ai (the set Ai is attractive in Ci). Therefore, the set Ai has 1-cycle, that is a contradiction and limρ→0 ν(ρ) = 0. Thus there is a function ν ∈ K∞ such that ν(ρ) ≤ ν(ρ), then the set Ai is locally Lyapunov stable in Ci: for any > 0 if x0 ∈ D ν-1 ( ) then |X(t, x0, 0)|A i ≤ . Since Ai is also attractive by the consideration above, the set Ai is asymptotically stable with the domain of attraction Ci [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF]. G Note that M = k i=1 A(Wi) and Ci = M \ Bi, therefore the set Ai is uniformly asymptotically stable with the domain of attraction M \Bi. Applying the above arguments for the flow of (2) in the backward time (i.e. X(-t, •, 0)) we can prove that Bi is asymptotically stable for the backward flow with the domain of attraction M \ Ai. Now we need to recall Lemma 4 and Lemma 5, applying these lemmas to the sets Ai and Bi, and using the smoothing arguments from [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF] (Theorem B1) or [START_REF] Wilson | Smoothing derivatives of functions and applications[END_REF] for 0 < ρ1 < ρ2 < ρ i max we may obtain two smooth functions V1 : M \Bρ 1 → R+, V2 : Bρ 2 → R+ such that:

clos[ j>i A(Wj)] = j>i clos[A(Wj)] ⊆ j>i l≥j A(W l ) = j>i A(Wj).
• α1(|x|A i ) V1(x) for all x ∈ M \Bρ 1 for α1 ∈ K∞; • α2(|x|B i ) V2(x) α3(|x|B i ) < 1 for all x ∈ Bρ 2 for α2, α3 ∈ K∞; • DV1(x)f (x, 0) -β1(V1(x), |x|), β1 ∈ KL for all |x|B i ρ1;

• DV2(x)f (x, 0) β2(V2(x), |x|), β2 ∈ KL for all |x|B i ≤ ρ2. Note that DV1(x)f (x, 0) -β1(α1(|x|A i ), |x|), D[1 -V2(x)]f (x, 0) -β2(α2(|x|B i ), |x|).

Next, it is necessary to unite the functions V1 and V2 using the covering property of these functions into the set Υ = {x ∈ M : ρ1 < |x|B i < ρ2}. The obstacle there is that Bi can be non-compact in a general case, and the function V1 may take unbounded values on Υ (recall that V1(x) ≤ α4(|x| + m) for some m > 0). To avoid this issue let us introduce a semi-proper function V3(x) = θ(V1(x)), where θ(s) = s 0 (1 + χ(r)) -1 dr is a bounded function for a suitable selected χ ∈ K∞, then for all |x|B i ρ1 α5(|x|A i ) V3(x), DV3(x)f (x, 0) -β3(|x|A i , |x|) for α5(s) = θ • α1(s) ∈ K and β3(s, r) = β1(α1(s), r)/(1 + χ(α4(r + m))) ∈ KL. There exist v1, v2 ∈ R+(under an appropriate rescaling of V2) such that Υ = {x ∈ M : v1 < V2(x) < v2 < 0.5} ⊂ Υ, define v3 = sup x∈ΥV3(x) (by construction v3 < +∞) and take a smooth function φ : R+ → R+ such that φ(s) = 0 for s ≤ v1, φ(s) = 1 for s ≥ v2 and dφ(s)/ds > 0 for s ∈ (v1, v2) then Li(x) = v4φ(V2(x))V3(x) + (1 -φ(V2(x)))(1 -V2(x)) with v4 = (0.5 -v2)v -1 3 > 0 is a smooth Lyapunov function for the system. Indeed, Li(x) = v4V3(x) for V2(x) ≥ v2 and Li(x) = 1 -V2(x) for V2(x) ≤ v1, while on the set Υ we have and all terms are negative in the last expression since v4V3(x) -1 + V2(x) ≤ v4v3 -1 + v2 = -0.5. Thus we have proven the following auxiliary result. Lemma 6: Let Ai, Bi be forward invariant sets asymptotically stable for forward and backward flows of (2) in M \ Bi and M \ Ai respectively. Then there exists a smooth function Li : M → R+ such that

  For a set S ⊂ M define | • |S as |x|S = inf a∈S δ(x, a). For a point x0 ∈ M selected as "the origin" on M , denote |x| = |x| {x 0 } . For a measurable function d : R+ → R m define its infinity norm d ∞ = ess sup t≥0 |d(t)|.

Definition 1 :

 1 An open decomposition for Λ is a finite, disjoint family of open sets, W1, W2, . . . , W k ⊂ M such that Λ ⊂ k i=1 Wi. An open decomposition can be associated with any invariant set Λ. The roughest decomposition is M itself, while the finest is made up by open neighborhoods of each connected component of Λ.

  Define a relation on W ⊂ M and D ⊂ M by W ≺ D if A(W) ∩ R(D) = ∅ (this relation implies that there is a solution connecting set D with set W). Definition 4: Let Λ1, . . . , Λ k be a decomposition of Λ, then 1) An r-cycle (r ≥ 2) is an ordered r-tuple of distinct indices i1, . . . , ir such that Λi 1

Fig. 1 .

 1 Fig. 1. The road-map of the main result proof

)

  Notice that, for all d with infinity norm less or equal to r, τ x,d (r) < +∞ by virtue of the asymptotic gain property. By virtue of Corollary III.3 in [16], given any compact set C ⊂ M of initial conditions, TC,r := sup x∈C, d(•): d ∞ ≤r τ x,d (r) < +∞.

By virtue of Proposition 5 .

 5 1 in [10], R T C,r (C, r) is bounded.Define, the set Cr as follows, Cr = {z ∈ M : |z|W ≤ max{r, η(r) + 1}} .

  , x; d)|W ≤ η lim sup t→+∞ |d(t)| .

  ), indeed for any x0 ∈ M \ Dρ: v(X(t, x0, 0)) = sup τ 0 |X(τ, X(t, x0, 0), 0)|A i = sup τ t |X(τ, x0, 0)|A i sup τ 0 |X(τ, x0, 0)|A i = v(x0). Now, define a new function for all x0 ∈ M \ Dρ:

  x0) < 0 for a.e. x0 ∈ M \{Ai ∪ Dρ}. Define for some r ∈ R+ the set Gr = {x ∈ M : |x| ≤ r} \ Dρ and the time Tr = sup x 0 ∈Gr Tx 0 . The time Tr is well defined and finite since the set Gr is compact, in addition Tr ≤ ϕ0 + ϕ1(r)

  x0)] > 0 for a.e. x0 ∈ Dρ\Bi. Define for some r ∈ R+ the set Gr = {x ∈ M : |x| ≤ r} ∩ Dρ and the time Tr = sup x 0 ∈Gr Tx 0 . The time Tr is well defined and finite since the set Gr is compact, in addition Tr ≤ ϕ0 + ϕ1(r)

  Then the set j>i A(Wj) is closed and the set Ci = M \ j>i A(Wj) is open. D The set Ai is attractive for all initial conditions in an open and forward invariant set Ci. Indeed, the set Ci \ Ai ⊂

  x)DV 2 (x)f (x, 0) + v 4 φ(V 2 (x))DV 3 (x)f (x, 0) V 3 (x) -1 + V 2 (x)] DV 2 (x)f (x, 0) +v 4 φ(V 2 (x))DV 3 (x)f (x, 0) -(1 -φ(V 2 (x)))DV 2 (x)f (x, 0)

This rules out cycles of any length.

Acknowledgements The authors are grateful to Prof. Laurent Praly for enlightening discussions during the preparation of the note and for pointing out the reference [12]. This work was supported in part by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).

• α (|x|A i ) ≤ Li(x) for all x ∈ M and some α ∈ K;

, DLi(x)f (x, 0) = 0 for all x ∈ Gi = Ai ∪ Bi;

• DLi(x)f (x, 0) ≤ -β (|x|G i , |x|) for all x ∈ M and some β ∈ KL. Finally to prove Proposition 2 select L(x) = Σ k-1 i=1 Li(x), which fulfils all our requirements. Indeed, k i=1 Ai ⊂ W and W = k i=1 Ai ∪ k i=1 Bi, then there are functions υ ∈ K and ∈ KL such that

for all x ∈ M , while the sets Wi contain critical points of L and belong to different constant levels of L. Since the set W is compact there is y ∈ R+ such that |x| ≤ |x|W + y, then DL(x)f (x, 0) ≤ -(|x|W ) for all x ∈ M and some positive definite function .