SMOOTH OUTPUT-TO-STATE STABILITY FOR MULTISTABLE SYSTEMS ON COMPACT MANIFOLDS

Output-to-State Stability (OSS) is a notion of detectability for nonlinear systems that is formulated in the ISS framework. We generalize the notion of OSS for systems which possess a decomposable invariant set and evolve on compact manifolds. Building upon a recent extension of the ISS theory for this very class of systems [D. Angeli and D. Efimov, IEEE Trans. Autom. Control 60 (2015) 3242–3256.], the paper provides equivalent characterizations of the OSS property in terms of asymptotic estimates of the state trajectories and, in particular, in terms of existence of smooth Lyapunov-like functions. Mathematics Subject Classification. 93B07, 93D05, 93D20, 93D25, 34D23, 34D35, 34D45, 37C70. Received April 9, 2021. Accepted March 25, 2022.

Note that the existence of a filtration ordering modulo-infinity automatically rules out the existence of 1and r-cycles modulo-infinity, namely it rules out the existence of cycles among any of the atoms of W, but it does not rule out the existence of cycles among atoms of Z = Z ∞ ∪ W. While homoclinic and heteroclinic cycles are known to prevent existence of Lyapunov functions which are strictly decreasing along solutions outside the considered invariant set W, we aim here at allowing cycles which involve a specific atom Z ∞ . Their existence is compatible with the dissipation inequality that we will be considering since OSS Lyapunov functions are allowed to increase whenever the output is sufficiently large. Checking existence of the filtration ordering might be difficult, in general, as it entails a global understanding of the layout of solutions of the system "departing from" and "converging to" atoms in W. This is a common difficulty when dealing with Lyapunov -there exist a constant c ≥ 0, and class-K ∞ functions α 1 , α 2 such that: (3.5) -V is differentiable along trajectories of (2.1), namely V (X(t, x)) is differentiable in t ∈ R for all x ∈ M . Furthermore, there exist class-K ∞ functions α, γ y and a constant q 0 ≥ 0 such that d dt V (X(t, x)) ≤ −α(|X(t, x)|W ) + γy(|h(X(t, x))|) + q0 (3.6) for all t ≥ 0 and all x ∈ M .
If (3.6) holds for q 0 = 0, then V is called an OSS-Lyapunov function. If, in addition, the set x∈Wi {V (x)} is a singleton for all i ∈ {1, . . . , K}, then V is called an OSS-Lyapunov function constant on invariant sets. Moreover, if V is smooth in W and dV (x) = 0 for all x ∈ W, then V is called an OSS-Lyapunov function flat on invariant sets.

Results on sufficiency part
In this Section, we provide a list of Lemmas which show how the existence of an OSS-Lyapunov function characterizes the asymptotic behavior of the state of system (2.1) with respect to its output signal y and the compact invariant set W. Proof. We are first going to prove the O-LIM property. Consider the case x ∈ M such that y = 0. Inequality (3.6) then reads asV ≤ −α(|X(t, x)| W ) which, together with boundedness from below (3.5) and continuity of V along trajectories, satisfies the hyphothesis of LaSalle's invariance principle and thus implies lim t→+∞ |X(t, x)| W = 0. It then follows that inf t≥0 |X(t, x)| W = 0, thus proving the O-LIM property.
The following lemma is obtained by adapting the arguments of Lemma 3.4 in [3]. Proof. Let µ y , β 0 , β y be as in (3.3) and (3.4). For any x ∈ M and any ε > 0, by the O-LIM property, there is some T ≥ 0 such that |X(T, x)| W ≤ µ y ( y ) + ε .
As ε > 0 is arbitrary, we see that: lim sup t→+∞ |X(t, x)| W ≤ η y ( y ) + q , with η y (s) := sup {β 0 (µ y (s)), β y (s)}. η y ( y ) + q ≤ η y ( y ) + q y ε ≤η y ( y ), withη y, (s) := η y (s) + qs/ε. Therefore, it is enough to show that there exists a sufficiently smallε > 0 such that O-AG holds for all trajectories such that y <ε. Without loss of generality, assumeε = 1 at first. The latter assumption and the O-pAG property imply that X(t, x) enters a compact set X := {x : |x| W ≤ q + η y (1) + 1, |h(x)| ≤ 1} in finite time. Let F be: F := max x∈X |f (x)| g < +∞. (4.2) Observe that F is finite by continuity of f and compactness of X . Since X(t, x) enters X whenever y ≤ 1, |f (X(t, x))| g ≤ F for all sufficiently large t ≥ 0. Consider next the minimum distance between the elements of the decomposition: The minimum exists and it is striclty positive by finiteness of the decomposition and compactness of the W i s. Pick a 0 < ∆ < 1 such that α −1 (2γ y (∆)) ≤D/4. Define the sets N i (∆) as: By the O-LIM property, it holds for all x ∈ M such that y ≤ ∆ that there exists a t ≥ 0 such that: for some 1 ≤ i ≤ K. By proceeding along the lines of the proof of Claim 4 in [2], it is possible to find an asymptotic gain function η(·) and a thresholdε = ∆ 2 such that O-AG holds with gain η y (s) := max η y,ε (s), η(s) . Proof. Consider the class-K ∞ function η y as in Definition 3.2. Pick any x ∈ M . If y = sup t≥0 |h(X(t, x))| = +∞, then O-AG trivially follows. If y < +∞, then, by definition of lim sup, it follows that trajectory X(t, x) enters the set Ω := {x ∈ M : |x| W ≤ η y ( y ) + q + 1} in finite time 0 ≤ T < +∞. By compactness of W, solutions are bounded for all t ≥ T . Then, Markus's result in [9] yields lim sup t→+∞ |X(t, x)| W = 0 which trivially implies the O-AG property.
Lemma 4.5. If system (2.1) admits a practical OSS-Lyapunov function, then it satisfies the O-pAG property.

Preliminary results
Throughout this Section, we discuss the steps of the proof leading to existence of OSS-Lyapunov functions as a consequence of item (1) of Theorem 3.9. Hence, we make the following Assumption 5.1. System (2.1) satisfies the O-AG property with respect to W for some class-K ∞ function η y .
We recall the following preliminary result whose proof is given in Appendix A.
Lemma 5.2. The O-AG implies the GATTMO property with the class-K ∞ function ρ being smooth on (0, +∞) and fulfilling: Select a class-K ∞ function ρ according to Lemma 5.2. Then, consider the following sets: The set E 0 will be alternatively referred to as the output set and represents the region of the state space where the distance from W dominates the norm of the output.

Sectors
The following Lemma highlights a crucial property of solutions of systems evolving on a compact manifold. Lemma 5.3. M is the disjoint union of the attracting (or repulsing) subsets of the decomposition of Z, namely: Proof. We recall here the proof of Lemma 2.2 in [13]. We prove the first equality of (5.2) and the second one will follow by considering the backward flow. Since the vector field f is locally Lipschitz continuous and due to the Z i s being compact and disjoint, we can select open neighborhoods U i of Z i such that for all t, s ∈ [−1, 1] and all i 1 , i 2 ∈ {1, . . . , K + 1} such that i = j. By virtue of Assumption 2.2, for any U i for all t > T , with T > 0 large enough. In particular, due to (5.4), the solution X(t, x) is eventually contained in one particular U i . Choosing the U i s arbitrarily close to the Z i , we see that x ∈ A(Z i ).
In the remaining of this Section, let Assumptions 2.2, 2.6, and 5.1 hold true.
Under aforementioned assumptions, the behavior of solutions with respect to the output set is characterized as follows. Define for all x ∈ E 0 the first hitting times λ(x) and λ (x) as: Proof. Assume by contradiction that there exists x ∈ Z ∞ ∩ E 0 . Assumption 2.2 guarantees that {x} is an invariant set, therefore X(t, x) ∈ E 0 for all t ≥ 0. By virtue of the GATTMO property, the omega limit set of x satisfies ω(x) ⊆ W. However, lim t→+∞ |X(t, x)| W = 0 contradicts invariance of {x} ⊆ Z ∞ .
The next result is a direct consequence of Lemma 5.6.
Consider the following sets: with 1 ≤ i ≤ K. We furthermore define: In particular, we will focus our attention on the attractivity property modulo-output of the sets: Proof. Backward (respectively forward) invariance ofĀ i (respectivelyB i ) trivially follows by backward (respectively forward) invariance of L −∞,0 and R(W j ) (respectively L 0,+∞ and A(W j )). We prove in Section E that A i ,B i are closed, thus compact by compactness of the manifold M . Complementarity of B i , C i and of A i , D i in M follows by Lemma 5.3. Disjointness is proved by the existence of a filtration ordering modulo-infinity in Assumption 2.6.
The attractivity property is proved as follows. By definition of C i , for any x ∈ C i it holds lim t→+∞ |X(t, x)| j≤i Wj = 0 or lim t→+∞ |X(t, x)| Z ∞ = 0. In the first case, since W ∩ E 0 and j≤i W j ⊆ A i and all W i s are invariant, we have that j≤i W j ⊆Ā i , which implies |X(t, x)| j≤i Wj ≥ |X(t, x)|Ā i , and thus lim t→+∞ |X(t, x)|Ā i = 0. In the second case, Corollary 5.7 proves that λ(x) < +∞.
The repulsion property follows along the same lines of the proof of the attractivity property.
The definitions of setsĀ i ,B i provide an appropriate partition of the output set E 0 , in the following sense.
Lemma 5.9. Let Assumptions 2.2, 5.1, and 2.6 hold true. For any x / ∈ W ∪ Z ∞ , there exists an index i(x) ∈ {0, 1, . . . , K} such that x / ∈Ā i(x) ∪B i(x) . In particular: . . , K}, then the existence of a filtration ordering modulo-infinity implies that j > h, and Lemma 5.3 thus implies x / ∈ A h ; (ii) the statement easily follows from Lemma 5.3; Proof. SinceĀ i andB i are closed and disjoint, it is possible to select a closed set A i ⊂ M \B i and a closed set By the definition ofĀ i and due to inclusionB i ⊇ j>i W j , it holds that A i ∩B i = ∅, and thus A i ∩ W j = ∅, for all j > i. Similarly, it follows that B i ∩ W j = ∅, for all j ≤ i.

OSS-Lyapunov functions in sectors
In this Section, we show that sectorsĀ i ,B i as defined in 5.2 satisfy a stability property which is highly reminiscent of the so-called global asymptotic stability modulo-output (GASMO), as defined in [6] and reported in the following.
Definition 5.11 (GASMO). Let Q, S be two subsets of M such that Q is closed and Q ∈ int S. The set Q is said to satisfy the global asymptotic stability modulo-output property (GASMO) on S if there exists a class-KL function β such that |X(t, x)| Q ≤ β(|x| Q , t). for all x ∈ S and all t ∈ [0, λ(x)).
GASMO has been shown to be equivalent to OSS in the classical framework [6] and instrumental in the construction of a smooth OSS-Lyapunov function. Similarly, the following Lemmas 5.12 and 5.13 prove that sectorsĀ i ,B i satisfy a GASMO property with respect to two measures, which will be instrumental in the construction of a smooth OSS-Lyapunov function on A i , B i respectively.
Let Assumptions 2.2, 2.6, 3.1, and 5.1 hold true. For all i ∈ {0, 1, . . . , K}, select closed sets A i and B i according to Lemma 5.10. Then, for any i ∈ {0, 1, . . . , K}, the following five Lemmas hold true.  We bring to the attention of the reader that the type of stability property being satisfied in A i and B i respectively allows the construction of classical OSS-Lyapunov functions in A i and B i which follows along the lines of [16]. Let E 3 := {x ∈ M : |x| W ≥ 3ρ (|h(x)|)}.
Lemma 5.14. (Existence of an OSS-Lyapunov function in A i ) There exists a smooth function V 1 : A i → R ≥0 and class-K ∞ functions α 1 , α 2 , β 1 such that the following two properties hold: In particular, dV 1 (x) = 0 whenever x ∈Ā i .
Proof. See Appendix D.
Lemma 5.15. (Existence of an OSS-Lyapunov function in B i ) There exists a smooth function V 2 : B i → R ≥0 and of class-K ∞ functions α 3 , α 4 , β 2 such that the following two properties hold: In particular, dV 2 (x) = 0 whenever x ∈B i .
Proof. The proof is symmetrical with respect to the proof of Lemma (5.14) and thus follows along the lines of Appendix D.
Lemma 5.16. There exists a smooth function L i : M → R ≥0 which satisfies the following two properties: 1. there exists a class-K function α i 1 such that: In particular, L i (x) = 0 whenever x ∈Ā i and there exists a constant b i > 0 such that class-K ∞ function β i such that: Proof. See Appendix F.

An OSS-Lyapunov function wrt W on E 3
Lemma 5.16 has proved for any i ∈ {0, 1, . . . , K} the necessity of the existence of functions L i whenever Assumptions 2.2 and 2.6, and 5.1 hold for system (2.1). In this Subsection, we are going to aggregate the contributions of the L i s in order to construct an OSS-Lyapunov function with respect to the set W which decreases along the trajectories evolving on E 3 ∪ W.
In particular, we will be focusing on the cases i = 0 and i = K. Indeed, we have: Lemma 5.14 proves the existence of a smooth Lyapunov function V K : M → R ≥0 and of class- for all x ∈ M and for all x ∈ E 3 ∪Ā K . In particular, dV K (x) = 0 whenever x ∈Ā K . Lemma 5.15 proves the existence of a smooth Lyapunov function V 0 : M → R ≥0 and of class-K ∞ functions α 0 1 , α 0 2 ,β 0 such that for all x ∈ M and for all x ∈ E 3 ∪B 0 . In particular, dV 0 (x) = 0 whenever x ∈B 0 . Consider now the case i = 1. By virtue of Lemma 5.16, we have that L 1 (x) ≥ α 1 1 (|x|Ā 1 ). SinceĀ 1 is compact and due to W 1 ⊆Ā 1 , it also holds that |x|Ā 1 ≤ |x| W1 ≤ |x|Ā 1 +ν for some constantν ≥ 0, and thus α 1 1 (0.5 |x| W1 ) ≤ α 1 1 (|x|Ā 1 ) + ν with ν := α 1 1 (ν). Let L : M → R ≥0 be the Lyapunov function defined as: It is then immediate to obtain the following lower bound: Note that α 1 1 (·) is a class-K function while α K 1 is a class-K ∞ . Furthermore, by compactness of W andĀ K , it holds that |x| W ≤ |x|Ā K + ν K for some ν K ≥ 0. Therefore, we can select: It follows that α 1 is a class-K ∞ function which satisfies: By Lemmas 5.16, 5.14, and 5.15, it holds that dV 0 (x) = 0 for any x ∈B 0 = G 0 , dV K (x) = 0 for any x ∈ A K = G K , and dK i (x) = 0 for any i ∈ {1, . . . , K − 1} and any x ∈ G i . Furthermore, for any i, j ∈ {0, 1, . . . , K} with j = 0, it holds W j ∈ G i . For these reasons, we can conclude that dL(x) = 0 whenever x ∈ W, namely L is flat on W.
By virtue of property 1. in Lemma 5.16 and inequality (5.8), the L i s and V K vanish on theĀ i s andĀ K respectively, and, moreover, the L i s take constant values on the B i s. It then follows that the L i s and V K take constant values on W j , and we thus conclude that L is constant on any W j .
Moreover, the definition of G i :=Ā i ∪ B i implies that G K =Ā K and, in particular, that G i ⊇ W for any i ∈ {1, . . . , K}. Then, by making use of property 2) in Lemma 5.16 and dissipation (5.15), we conclude that: (5.20) Lemma 5.9 has proved that, if x / ∈ W, there exists some i ∈ {0, 1, . . . , K} such that |x| Gi > 0. Viceversa, for any i ∈ {0, 1, . . . , K}, inclusion W ⊆ G i implies |x| Gi ≤ |x| W , which in turn implies that x / ∈ W whenever x / ∈ G i . Therefore, there exist a continuous positive definite function such that for all x ∈ M , where last inequality embeds the property |x| ≤ |x| W + ν W with ν W ≥ 0. In order to obtain a class-K ∞ dissipation function for the time derivative of L, consider the property |x| G K ≤ |x| W + ν K with ν K ≥ 0 and define the following K ∞ function: Then, it holds that: (5.21)

Extending the dissipation to the rest of the state space
Observe that L(x) is the composition of functions which are smooth everywhere on M and, in particular, along the trajectories of system (2.1), namely L(X(t, x)) is smooth in t. For this reason and due to L f L(W) = 0, we can always find a class-K ∞ function σ 0 such that: for all x such that |x| W ≤ 3ρ(|h(x)|). Let: It then follows from (5.21) and (5.22) that We have thus proved that L is a smooth OSS-Lyapunov function constant on invariant sets, as in Definition 3.6.

Conclusions
Classical OSS [17] results from the dualization of the ISS property by simply replacing inputs with outputs in the definition. In analogous way, dualizing the generalized ISS notion in [2] yields a notion of detectability for multistable systems, which is still called OSS but it does not require the invariant set of the system in consideration to be Lyapunov stable, while instead requiring absence of homoclinic orbits and heteroclinic cycles between the atoms of its decomposition. The resulting generalized OSS property portrays the behavior of ultimately having small distance of the states from the invariant set of the system whenever the outputs are small. This paper has provided characterizations of such property in terms of asymptotic estimates via nonlinear gain functions and in terms of smooth Lyapunov/LaSalle-like dissipation inequalities.
We bring to the attention of the reader that our converse OSS-Lyapunov theorem in Section 5 encompasses interesting results, such as attractivity of an invariant set K and absence of homoclinic orbits to K being a necessary and sufficient condition for Lyapunov stability of K on C.

Appendix A. GATTMO
This Appendix addresses the proof of Lemma 5.2.
Proof. If y = 0, the O-AG property yields lim sup t→+∞ |X(t, x)| W = 0. By definition of lim sup, for all x ∈ M and all ε > 0 there exists T x,ε > 0, such that (3.2) is satisfied for any choice of ρ ∈ K ∞ .
Consider now the case 0 < y < +∞. Select ρ as any smooth K ∞ function such that ρ(s) > η y (s) for all s > 0.
By contradiction, we assume that there exist x ∈ M and a constant ε > 0 such that, for all T x,ε > 0, there exists T ∈ [T x,ε , +∞) which verifies the following two conditions: The previous contradiction assumption can equivalently be formulated as follows: there exist x ∈ M , a constant ε > 0, and two sequences {T n } n∈N > 0 and {t n } n∈N > 0 which verify the following three conditions and for all n ∈ N. Since (A.1) holds for all n ∈ N and T n is diverging, we can take the lim sup on both sides of (A.1) so as to obtain: Moreover, since (A.2) holds for all n ∈ N andt n is diverging, we have that for someε ∈ (0, +∞]. In particular,ε ∈ (0, +∞) due to y < +∞ and O-AG.
Appendix B. GASMO with respect to two measures on A i for the forward flow We proceed to prove global asymptotic stability modulo-output with respect to two measures of the setĀ i in A i , that is existence of a class-KL function β such that: for all t ∈ [0, λ(x)) and all x ∈ A i . Throughout this Section, let Assumptions 2.2, 2.6, and 5.1, hold true.
First, we recall the following definition: Claim B.1. The set L s1,s2 satisfies the following two properties: Proof. It easily follows from the definition (B.1).
Claim B.2. For any s 1 , s 2 ∈ R with s 1 ≤ s 2 , the set L s1,s2 is closed.
Proof. Consider a sequence {x n } n∈N ∈ L s1,s2 so that lim n→+∞ x n = x. Pick any t ∈ [s 1 , s 2 ]. By continuity of solutions, we obtain ϕ t (x) = ϕ t (lim n→+∞ x n ) = lim n→+∞ ϕ t (x n ). Since the sequence ϕ t (x n ) converges to ϕ t (x) and belongs to E 0 by definition of L s1,s2 , and due to E 0 being a closed set, we conclude ϕ t (x) ∈ E 0 . Since the last statement holds for all t ∈ [s 1 , s 2 ], we conclude that x ∈ L s1,s2 .

Lemma B.3. (Existence of an arbitrary small forward invariant neighborhood modulo output on
Proof. Pick a compact neighborhood Q ofĀ i . The proof is then adapted from Lemma 6 in [11] and consists of four steps: Step 1: Consider the following definitions: Proof. Indeed, we have that x ∈ r≥0 A r if and only if Clearly, a necessary condition for x to be an element of r≥0 A r is x ∈ Q. Furthermore, only the following scenarios may arise: ). In this case, Lemma 5.3 implies that x ∈ j>i R(W j ). Therefore, since X(t, x) is attracted backward in time to W j with j > i, and due to the fact that Since a locally Lipschitz vector field f generates a flow ϕ t which maps any compact neighborhood of a backward invariant set to a compact neighborhood of the same set whenever t ≥ 0, we have that A 0 r is a compact neighborhood ofĀ i . Furthermore, recall that the setsĀ i and L −r,0 are compact (Lem. E.7 and Claim B.2). It thus follows that A r is compact. Moreover, since the sequences A 0 r r∈R ≥0 and {L −r,0 } r∈R ≥0 are monotone non-increasing in r (Claim B.1), we conclude that {A r } r∈R ≥0 is also monotone non-increasing and, in particular, is a nested sequence of compact sets converging toĀ i .
Step 2: We claim that, for sufficiently large r and sufficiently smallT , there exists R > 0 such that, for all r ≥ R, it holds A r ⊆ N . Therefore, for all such rs and for all y ∈ A r , it takes at least d 2F time units for the flux ϕ t (y) to reach the border ðA 0 1 . It follows that ϕ T (A r ) ⊆ A 0 1 for all r ≥ R and all T ∈ [0,T ] withT := min 1, d 2F .
Step 3: We claim that, for all r ≥ R and all T ∈ [0,T ], ϕ T (A r ) ⊆ A 0 r . As a preliminary result, we note that ϕ T (A 0 r ) = T ≤t≤r+T ϕ t (Q) and this is easily proved by comparing the following definitions: Then, the following passages hold true for all r ≥ R and all T ∈ [0,T ] withT = min 1, d 2F : Step 4: By making use of the flow property 1) in Claim B.1, it holds for all T ∈ [0,T ] withT := min 1, d 2F , allT ≥ T , and all r ≥ R that and clearly A 0 r ∩ L −r,T −T ⊆ A 0 r . We are now going to show that, for all t ≥ 0 and all r ≥ R, it holds: To this end, pick any t ≥ 0 and choose an integer n ∈ N so that t n ≤T . Then, by virtue of inclusion (B.4), we obtain: Consider now any r ≥ R. Let V be defined as: SinceĀ i is generally not forward invariant, V is not a compact neighborhood ofĀ i . However, it can be proved along the lines of Section E that A i ∩ L −∞,+∞ is compact and invariant. Therefore, V qualifies as a compact neighborhood of A i ∩ L −∞,+∞ and its definition implies: By virtue of (B.6), we have ϕ t (V ∩ L 0,t ) ⊆Ā 0 r for all t ∈ [0, r]. Conversely, if t > r, inclusion (B.5) and the flow property 1) in Claim B.1 imply: with t := t − r > 0. We have thus concluded (B.2).
Lemma B.5. (Arbitrary boundedness till hitting time on A i ) For all compact neighborhoods Q ofĀ i , there exists a compact neighborhood V of A i ∩ L −∞,+∞ such that V ⊆ Q and, for all x ∈ V , it holds: Proof. Pick an arbitrary compact neighborhood Q ofĀ i . Select a compact neighborhood V of A i ∩ L −∞,+∞ according to Lemma B.3. Then, statement (B.7) follows immediately by observing that, by definition of λ(·), the property x ∈ L 0,λ(x) holds for all x ∈ M and, in particular, for all x ∈ V .
For ease of presentation, Corollary B.6. (Lyapunov stability modulo output on A i ) The following property holds true: Moreover, there exists a class-K ∞ function φ such that: We are now going to prove proposition (B.9). We first claim that for all ε > 0 there exists δ > 0 such that: and all x ∈ V 2 . Being V a compact neighborhood ofÂ i , we can select δ > 0 so that NÂ i (δ) ⊆ V and we will thus have (B.10). Being δ a function of ε, we can then select φ = δ −1 .
Lemma B.7. (Uniform attraction modulo output on A i ) The following property holds true: Moreover, it also holds that: Proof. We first prove (B.11). By contradiction, assume that there exists a compact subset K ⊆ A i and a constant ε > 0 such that, for all T K,ε ≥ 0 and some T ≥ T K,ε we have for some x andt ∈ [T K,ε , T ]. The previous contradiction assumption can equivalently be formulated as follows: there exists a compact subset K ⊆ A i , a constant ε > 0, three sequences {T n } n∈N > 0, {t n } n∈N > 0, and {x n } n∈N ∈ K such that x n ∈ L 0,Tn and n ≤t n ≤ T n and |X(t n , x n )|Ā i > ε ∀ n ∈ N. (B.13) By virtue of Corollary B.6 (Lyapunov stability modulo output in A i ), there exists δ > 0 such that, for all T ≥ 0 and all t ∈ [0, T ], |X(t, x)|Ā i ≤ ε whenever x ∈ L 0,T ∩ NÂ i (δ). By Bolzano-Weierstrass, we can consider a subsequence {x n } n∈N ∈ K such that lim n→+∞ x n =: Proof. Pick any T ∞ > 0. There exists N ∈ N such that T ∞ ≤ T n , for all n ∈ N ≥N . Recall that x n ∈ K ∩ L 0,Tn . By monotonicity (Claim B.1), x n ∈ K ∩ L 0,T∞ for all n ∈ N ≥N . By closedness of K ∩ L 0,T∞ (Claim B.2), it follows that x ∞ ∈ K ∩ L 0,T∞ . Since the latter property holds for all T ∞ > 0, we conclude that x ∞ ∈ K ∩ L 0,+∞ .
. By Lipschitz continuity of trajectories with initial condition in K and local Lipschitz continuity of | · |Â i , there exists an index Q 1 ∈ N such that, for all q ∈ N ≥Q1 , it holds: By Lyapunov stability modulo output, for all such qs, all T ≥ 0, and all t ∈ [0, T ], +T . However, we can always select Q 2 ∈ N so as to have Q 2 ≥ Q 1 and q ≥ T x∞, δ 2 for all q ∈ N ≥Q2 . Since T q ≥ q and x q ∈ L 0,Tq for all q ≥ Q 2 , we have |X(t, x q )|Ā i ≤ ε for all t ∈ [q, T q ], which contradicts (B.13).
(B. 16) By virtue of (B.11) with K =K and ε =ε, it holds that: Pickn ∈ N such thatn ≥ TK ,ε . By the definitions of λ(·) and L ·,· , it holds for all w ∈ M that w ∈ L 0,λ(w) . In particular, we have that xn ∈K ∩ L 0,λ(xn) . Since TK ,ε ≤n ≤ λ(xn), it holds from (B. 1. for each fixed r > 0, T r : R >0 → R >0 is continuous and is strictly decreasing; 2. for each fixed ε > 0, T r (ε) is strictly increasing as r increases and lim r→∞ T r (ε) = ∞; such that: Furthermore, there exists a class-KL function β such that Proof. For all r > 0 we can identify a compact set NÂ i (r) ∩ A i which plays the role ofK in Lemma B.7. Therefore, Now consider the following definitions:T The proof of Lemma 3.1 in [8] shows that T r,ε indeed satisfies properties (i) and (ii) and, moreover, T r,ε ≥T r,ε ≥ T r,ε . It is then clear from the definition of A r,ε that |X(t, x)|Ā i < ε whenever x ∈ NÂ i (r) ∩ A i and t ∈ [T r,ε , λ(x)]. The equivalence between existence of family of mappings {T r } r>0 and the existence of a class-KL function satistying (B.18) follows along the lines of Proposition 2.5 in [8] and for this reason we omit the proof here.
Appendix C. GASMO with respect to two measures on sector B i for the backward flow We proceed to prove global asymptotic stability modulo-output with respect to two measures of the setB i in B i , that is existence of a class-KL function β such that: for all t ∈ [0, λ (x)) and all x ∈ B i . Throughout this Section, let Assumptions 2.2, 2.6, and 5.1, hold true.
For ease of presentation, we consider the backward flow: with f as in (2.1). Let X (t, x) and ϕ t (x) equivalently denote the uniquely defined solution of (C.1) for all t ∈ R with initial condition x ∈ M . Clearly, it holds X (t, x) = X(−t, x) and ϕ t (x) = ϕ −t (x). In a similar way, we can define: with Λ being a closed set, and then observe that Proof. Pick a compact neighborhood Q ofB i . The proof is then adapted from Lemma 6 in [11] and consists of four steps: Step 1: Consider the following definitions: Proof. Indeed, we have that x ∈ r≥0 A r if and only if Clearly, a necessary condition for x to be an element of r≥0 A r is x ∈ Q. Furthermore, only the following scenarios may arise: ). In this case, Lemma 5.3 implies that x ∈ j≤i R (W j ). Therefore, since X(t, x) is attracted forward in time to W j with j ≤ i, and due to the fact that Q ∩ j≤i W j ⊆ B i ∩ j≤i W j = ∅ with Q being a closed set, there existst > 0 such that ϕ −t (x) / ∈ Q, which in turn contradicts condition (C.5) for r = t =t. -x ∈ Q \ L 0,+∞ . In this case, there existst ≤ 0 such that ϕ t (x) / ∈ E 0 , which contradicts condition (C.5) by selecting r = |t|. -x ∈ Q ∩ L 0,+∞ ∩ B i . Then, x ∈B i and condition (C.5) is trivially satisfied. -x ∈ Q ∩ L 0,+∞ ∩ R (Z ∞ ). Then, due to Lemma 5.5 and since lim t→+∞ |X(t, x)| Z ∞ = 0, there existst > 0 such that X(t, x) / ∈ E 0 for all t >t, and thus x / ∈ L 0,+∞ .
Since a locally Lipschitz vector field f generates a flow ϕ t which maps any compact neighborhood of a forward invariant set to a compact neighborhood of the same set whenever t ≤ 0, we have that A 0 r is a compact neighborhood ofB i . Furthermore, recall that the setsB i and L 0,r are compact (Lem. E.7 and Claim B.2). It thus follows that A r is compact. Moreover, since the sequences A 0 r r∈R ≥0 and {L 0,r } r∈R ≥0 are monotone non-increasing in r (Claim B.1), we conclude that {A r } r∈R ≥0 is also monotone non-increasing and, in particular, is a nested sequence of compact sets converging toB i .
Step 2: We claim that, for sufficiently large r and sufficiently smallT , ϕ T (A r ) ⊆ A 0 1 , for all T ∈ [0,T ]. Indeed, let D denote the minimum distance of the boundary ðA 0 1 fromB i , that is D := inf a∈ðA 0 1 δ(a,B i ). Due to A 0 1 being a compact neighborhood ofB i , we have that D > 0. Let F denote the maximum velocity of the flux ϕ t (·) in A 0 1 , namely F := sup x∈A 0 1 |f (x)| g . By virtue of Lemma G.2, for the neighborhood N : there exists R > 0 such that, for all r ≥ R, it holds A r ⊆ N . Therefore, for all such rs and for all y ∈ A r , it takes at least d 2F time units for the flux ϕ t (y) to reach the boundary ðA 0 1 . It follows that ϕ T (A r ) ⊆ A 0 1 for all r ≥ R and all T ∈ [0,T ] withT := min 1, d 2F .
Step 3: We claim that, for all r ≥ R and all T ∈ [0,T ], ϕ T (A r ) ⊆ A 0 r . As a preliminary result, we note that ϕ T (A 0 r ) = T ≤t≤r+T ϕ t (Q) and this is easily proved by comparing the following definitions: Then, the following passages hold true for all r ≥ R and all T ∈ [0,T ] withT = min 1, d 2F : Step 4: By making use of the flow property 1) in Claim B.1, it holds for all T ∈ [0,T ] withT := min 1, d 2F , allT ≥ T , and all r ≥ R that and clearly A 0 r ∩ L −T +T,r ⊆ A 0 r . We are now going to show that, for all t ≥ 0 and all r ≥ R, it holds: ϕ t (A 0 r ∩ L −t,r ) ⊆ A 0 r . To this end, pick any t ≥ 0 and choose an integer n ∈ N so that t n ≤T . Then, by virtue of inclusion (B.4), we obtain: Consider now any r ≥ R. Let V be defined as: SinceB i is generally not backward invariant, V is not a compact neighborhood ofB i . However, it can be proved along the lines of Section E that that B i ∩ L −∞,+∞ is compact and invariant. Therefore, V qualifies as a compact neighborhood of B i ∩ L −∞,+∞ and its definition implies: Conversely, if t > r, inclusion (C.7) and the flow property 1) in Claim B.1 imply: Moreover, there exists a class-K ∞ function φ such that: Proof. It follows along the lines of the proof of Corollary B.6.
Lemma C.5. (Uniform attraction modulo output in B i ) The following property holds true: Moreover, it also holds that: Proof. It follows along the lines of the proof of Lemma B.7.
Corollary C.6. (Existence of a family of mappings {T r } r>0 on B i ) Lyapunov stability modulo-output (Cor. C.4) and uniform attraction modulo-output (Lem. C.5) imply the existence of a family of mappings {T r } r>0 with: 1. for each fixed r > 0, T r : R >0 → R >0 is continuous and is strictly decreasing; 2. for each fixed ε > 0, T r (ε) is strictly increasing as r increases and lim r→+∞ T r (ε) = +∞; such that: Furthermore, there exists a class-KL function β such that (C.14) for all t ∈ [0, λ (x)) and all x ∈ B i .
Proof. It follows along the lines of the proof of Corollary B.10. whenever |h(ξ)| ≥ 1. Denote by Z(t, z) the uniquely defined solution of (D.1) at time t fulfilling Z(0, z) = z. It has been proven in Section 4.1 of [6] that each solution Z(t, z) of (D.1) satisfies the following property:
In order for the sought Lyapunov function to yield desired dissipation properties over M \ (D ∪ B), we first define φ : M \ W → [0, 1] as any smooth function such that: Then, we introduce another system on the state space M \ W: with f 0 (z) := |f (z)| g and controls v : Denote by Z(t, z; v) the uniquely defined solution of (D.1) at time t with initial condition Z(0, z) = z and input v ∈ V.
Here we denote with V the set of so-called auxiliary controls, that is measurable functions of time, taking values in V equipped with the weak convergence topology ( [6]., Sects. 4.2 and 4.3) SystemΣ satisfies the following properties: -Σ is affine in v on M \ W; -since |f (z, v)| g ≤ 5 for all z ∈ M \ W and all v ∈ V , systemΣ is complete; -f (z, v) =f (z) for all z / ∈ D ∪ B and all v ∈ V ; then, due to the GATTMO property, for any z / ∈ D ∪ B and any v ∈ V, there exists a t 0 > 0 such that Z(t, z; v) / ∈ D ∪ B and Z(t, z; v) = Z(t, z) for all t ∈ [0, t 0 ].

D.2 Continuity
Define for any ξ ∈ E and any v ∈ V: Lemma D.2. For every neighborhood U ofĀ i , there is a constant c such that for any ξ ∈ U \Ā i , any v ∈ V, and any t ∈ [0, θ(ξ, v)], we have a lower bound Proof. By virtue of Lemma I.1, for any ξ ∈ U ∩ E \Ā i and v ∈ V , it holds From this point onwards, the proof follows along the lines of Lemma 4.1 in [6] and we recall it here. Fix ξ ∈ U ∩ E \Ā i and v ∈ V. Note that inequality (D.8) holds for t = 0, and thus it holds for all small enough t > 0. Suppose that (D.8) fails at some t 2 ∈ (0, θ(ξ, v)], so that: Then there exists a t 1 < t 2 such that |Z(t 1 , ξ; v)|Ā i = |ξ|Ā i and |Z(t, ξ; v)|Ā i ≤ |ξ|Ā i for all t ∈ [t 1 , t 2 ]. Let Then, for almost all t ∈ [t 1 , t 2 ], it holds that: It then follows thatẇ + 2cw(t) ≥ 0 for almost all t ∈ [t 1 , t 2 ]. In particular, for all such ts, we have 0 ≤ e 2ct (ẇ(t) + 2cw(t)) = d(e 2ct w(t)) dt , implying that e 2ct w(t) ≥ e 2ct1 w(t 1 ) for all t ∈ [t 1 , t 2 ]. Thus, Proof. It directly follows from Lemma D.2.
Proof. The proof follows along the lines of Lemma 4.3 of [6] and we adapt it here to the manifold case. Assume without loss of generality that ξ k ∈ U with Observe that, due to |f (·, ·)| g ≤ 5 and Lemma D.3, the reachable set R ≤T (U ) is a subset of

Assume by induction that for some
We are now going to prove that, for any ε > 0, there exists an index K ∈ N such that, for any k ∈ N ≥K and any t ∈ [0, T i+1 − T i ], it holds d[Z(T i + t, ξ k ; v k ), Z(T i + t, ξ; v)] ≤ ε, and thus lim k→+∞ d[Z(T i+1 , ξ k ; v k ), Z(T i+1 , ξ; v)] = 0. Assume without loss of generality that the orbit {Z(T i + t, ξ k , v k ), t ∈ [0, T i+1 − T i ]} enters a single coordinate chart H i for all ks large enough. Denote Z(t) := Z(T i + t, ξ; v) and Z k (t) := Z(T i + t, ξ k ; v k ). Then, in local coordinates, we have: Due to the weak convergence of v k to v, the first two integrals tend to 0, namely for any ε > 0 there exists a K 1 ∈ N with K 1 > K 0 such that, for all k ∈ N ≥K1 , Then, for all such ks and all t ∈ [0, T i+1 − T i ], we have, by the Gronwall inequality, The Lemma is thus proved by letting ε → 0.
By virtue of Proposition 7 in [14], there exists two class-K ∞ functions Ξ, µ 2 such that the following upper bound holds true for the class-KL functionβ of Claim D.1: We are now going to use function Ξ in the definition of a provisional candidate Lyapunov function forΣ. Define, We also define for ξ ∈ (A i ∩ E) ∪Ā i : Clearly, it holds that V (ξ) ≤ V (ξ, 0) where we have denoted with 0 the auxiliary control identically equal to 0. Furthermore, since j≤i W j ⊆ A i ∩ L −∞,+∞ , observe that Proof. The proof follows along the lines of Lemma 4.6 in [6] and we recall it here. Pick two sequences {ξ k } k∈N ∈ A i ∩ E \Ā i and {v k } k∈N ∈ V such that ξ k → ξ and v k → v for some ξ ∈ A i ∩ E \Ā i and v ∈ V. We need to show that: Case V (ξ, v) < +∞. The GATTMO property (D.6) guarantees that, for any ε > 0, there exists some T ∈ (0, θ(ξ, v)) such that Without loss of generality we can assume that the ξ k are within the unit distance from ξ. Recall that the reachable set from th eunit ball around ξ is bounded. By virtue of Lemma D.4, Z(t, ξ k ; v k ) converges to Z(t, ξ; v) uniformly on [0, T ]. Since Ξ is uniformly continuous on compacts, it then follows that there exists some K ∈ N such that This implies that: for all k ∈ N ≥K . By Lemma D.6, there exists some K 1 ∈ N such that K 1 ≥ K and θ(ξ , v k ) > T for all k ∈ N ≥K1 . Thus, for all k ≥ K 1 , As ε → 0, we obtain V (ξ, v) ≤ lim inf k→+∞ V (ξ k , v k ).
Case V (ξ, v) = +∞. It easily follows that θ(ξ, v) = +∞. For any integer k ≥ 0, there exists a time T k ≥ 0 such that Repeating the same argument used above, one sees that Consequently, for all such hs, Since k > 0 can be picked arbitrarily, it follows that Proof. The proof follows along the lines of Lemma 4.7 in [6] and we recall it here. Let {v k } k∈N be a sequence of open loop auxiliary controls such that V (ξ, v k ) → V (ξ) as k → +∞. Without loss of generality we are assuming that all these controls are defined for all positive t. Extract from {v k } k∈N a subsequence such that, without relabeling, the v k s converge to somev ∈ V. By Lemma D.7, Corollary D.9. For any ξ ∈ A i ∩ E \Ā i , v ∈ V, and T ∈ [0, θ(ξ, v)), it holds that Proof. The proof follows along the lines of Corollary 4.8 in [6] and we recall it here. Assume by contradiction that there exists ξ ∈ A i ∩ E \Ā i , v ∈ V, and T ∈ [0, θ(ξ, v)) such that (D.19) fails. By Lemma D.8, there exists a control v 1 ∈ V such that V (Z(T, ξ; v)) = V (Z(T, ξ; v), v 1 ). Definev to be the concatenation of v and v 1 . Then, letting θ := θ(Z(T, ξ; v), v 1 ) and noticing that θ(ξ,v) = θ + T , we get, by out assumption, Claim D.11. For any closed set U and any Proof. By closedness of U, we can find w 2 := arg inf a∈U d [a, x 2 ] ∈ M . Then, the minimizing property of geodesics Proof. Assume by contradiction that: ∃ε > 0 ∀r > 0 ∃ξ r ∈ A i : |ξ r |Ā i ≤ r and V (ξ r ) > ε.
In order to prove the continuity of V (·), we also need the following result.
Lemma D.14. Function V : Proof. The proof follows along the lines of Proposition 4.11 in [6] and we adapt it here with minor modifications and some additions. Lemma D.12 together with V (Ā i ) = 0 proves continuity atĀ i . Fix ξ ∈ A i ∩ E \Ā i . Suppose For each k, Lemma D.8 provides the auxiliary control v k ∈ V such that V (ξ k ) = V (ξ k , v k ). Notice that lim j→+∞ V (ξ kj , v kj ) exists because of (D.23). By sequential compactness of V, there exists a subsequence of v kj converging to somev ∈ V. Without relabeling, we assume that v kj →v. It then follows from Lemma D.7 that Consequently, by definition of V (ξ), it follows that V (ξ) ≤ lim inf k→+∞ V (ξ k ). To complete the proof, we will show that (D.24) To this end, pick ε > 0. By making use of Lemma D.8 again, let v ∈ V be a control such that V (x) = V (ξ, v).
Inequality (D.30) shows that V is proper in E 1 , whereas inequality (D.31) shows that V is positive on E \Ā i . We are now going to use (D.30) and (D.31) to prove existence of lower bound (D.28) on E 1 . Lemma D.14 has proved that V (·) is continuous, and thus it attains its minimum on any compact set. For each positive l define r l := K 0 + 2 l and m l := inf V (z) : z ∈ E 1 , r l ≤ |z|Ā i ≤ r 1 .
Note that Ξ is a class-K ∞ function and that the sequence {m l } l∈R ≥1 is nonincreasing and positive. Therefore, we can find a class-K ∞ function α such that α(s) < m l for all s ∈ [r l , r l−1 ] for all l ≥ 2 and α(s) < Ξ(s − 1) for all s > K 0 + 2.
Lemma D.18. There exists a K ∞ functionᾱ such that Proof. The Lemma follows from continuity of V in Lemma D.14 and bound (D.17).
We have proven that the OSS dissipation inequality holds for V (·) along the trajectories of the slower system Σ in (D.1) which are contained in A i \ (D ∪ B). It follows immediately that the same estimate holds along the trajectories of the original system Σ.
Corollary D.20. For any x ∈ A i \ (D ∪ B), and any t 0 such that X(t, x) / ∈ (D ∪ B), the following dissipation inequality holds for the trajectories of system (2.1):

D.5 Extension and smoothing
In this Section, we extend the definition of V to the whole sector A i and we show how some classical results concerning the smoothing of Lyapunov functions can be generalized for systems evolving on manifolds. It turns out that most proofs cannot be adapted to the manifold case in a straightforward fashion, unless few assumptions and facts are established beforehand, as in the following.
The (possible empty) set of all proximal subgradients of V at x is called the proximal subdifferential and is denoted d P V (x).
Lemma D.22. Consider any system of type (2.1), some x ∈ M , and some function V : M → R. Then, if there exist some continuous α x (t) : R ≥0 → R and ε > 0 such that the following inequality holds for all τ < ε, then for any ζ ∈ d P V (x) the proximal form of (D.34) holds: with some continuous function Θ : O → R; -two positive, continuous function Υ 1 and Υ 2 on O.
Then there exists a functionṼ : O → R, locally Lipschitz con O, such that such that for almost all x ∈ O. Then, there exists a smooth function Ψ : O → R such that Lemma D.25. Under the assumptions of Lemma D.23, there also exists a smooth functionV on O, satisfying inequalities: We are now ready to state our main result.
Corollary D.26. There exist a smooth function V 3 : A i → R >0 and class-K ∞ functions α 1 , α 2 , α 3 satisfying the following two properties:  1] be any smooth function with the property and φ is nonzero elsewhere. Let V 2 : A i \Ā i → R ≥0 be defined as: It is then easy to show that:α Proof. Consider a sequence {x n } n∈N ∈ L −∞,0 such that lim n→+∞ x n = x for some x ∈ M . We shall prove that x ∈ L −∞,0 . Specifically, we prove that X(t, x) ∈ E 0 for all t ≤ 0. To this end, select any t ≤ 0. It holds: where the first inequality holds by definition of x, the second one holds by continuity of trajectories, and the third one by definition of X n := X(t, x n ). By definition of L −∞,0 , it holds X n ∈ E 0 . Since E 0 is closed and the sequence {X n } n∈N is converging to X(t, x), it follows that X(t, x) ∈ E 0 .
Proof. The proof follows along the lines of [11].
Proof. For any set E, if x ∈ clos E then there exists a sequence {x n } n∈N ∈ E such that x n → x as n → +∞. For this reason, if we select x ∈ clos [R(Z i ) ∩ L −∞,0 ], there exists a sequence {x n } n∈N ∈ R(Z i ) ∩ L −∞,0 converging to x. Backward invariance of R(Z i ) ∩ L −∞,0 implies X(τ, x n ) ∈ R(Z i ) ∩ L −∞,0 for any τ ≤ 0. Pick any t ≤ 0. By continuity of trajectories, it holds: Since X(t, x n ) ∈ R(Z i ) ∩ L −∞,0 for all n ∈ N and X(t, x n ) → X(t, x) for any n → +∞, it follows that X(t, x n ) ∈ clos R(Z i ) ∩ L −∞,0 . Proof. Since the vector field f is locally Lipschitz continuous and due to the Z i s being compact and disjoint, we can select open neighborhoods U i of Z i such that for all t, s ∈ [−1, 1] and all i 1 = i 2 . Define the sets L j := ϕ −1 (clos U j \ U j ). By definition, L j is compact. Consider now y ∈ clos [R(Z i ) ∩ L −∞,0 ] ∩ Z j . There exists a sequence {y n } n∈N ∈ R(Z i ) ∩ L −∞,0 converging to y. Without loss of generality, we may assume y n ∈ U j . We can then pick T n < 0 so that X(T n , y n ) / ∈ U j but X(p, y n ) ∈ U j for all p ∈ (T n , 0]. It holds that X n := X(T n , y n ) ∈ L j for all n ∈ N. By compactness of L j , X n →X ∈ L j . By backward invariance and closedness of clos [R(Z i ) ∩ L −∞,0 ], we have thatX ∈ clos [R(Z i ) ∩ L −∞,0 ]. Now, let t > 0 and consider X(t,X). By continuity of trajectories, it holds: For all t ≥ 0, we can always find N so that ∀n > N , we obtain T n + t < 0, and thus X(T n + t, y n ) ∈ U j , which in turn implies X(t,X) ∈ clos U j for all t > 0. Ultimate boundedness in clos U j (Lem. 5.3) and invariance of Z j imply X(t,X) → Z j as t → +∞, and thusX ∈ A(Z j ). Moreover, definition ofX implies thatX ∈ L j , thereforē Proof. By virtue of Lemma E.4, pick By virtue of Lemmas 5.3 and 5.5, x 1 ∈ R(W l1 ) for some l 1 ∈ {1, . . . , K}. Since x 1 ∈ R(W l1 ) ∩ A(W j ) and due to the existence of a filtration ordering modulo-infinity (Assumption 2.6), we have j < l 1 . Since Due to the existence of a filtration ordering modulo-infinity, we have l 1 < l 2 . By iterating this procedure, we obtain an infinite sequence of indexes l k , which contradicts finiteness of the W k s.
Proof. Select x ∈ clos [R(W i ) ∩ L −∞,0 ]. By virtue of Lemmas 5.3 and 5.5, α(x) ⊆ W j . By virtue of Lemma E.5, j ≤ i, and thus x ∈ R(W j ) for some j ≤ i.
Proof. We are going to show that clos {A i ∩ L −∞,0 } ⊆ A i ∩ L −∞,0 . Indeed, by making use of Lemma E.6, we obtain: Appendix F. OSS-Lyapunov function L i defined on M This Appendix Section focuses on the proof of Lemma 5.16 and provides a list of additional interesting properties for function L i defined in the Lemma. The underlying assumptions in all subsequent results of this Section is the fulfillment of the hypotheses in Lemmas 5.14 and 5.15.

Appendix G. Limit of sequences of nested sets
In this section, we recall basic results on sequences of nested sets, e.g. Cantor's intersection theorem ( [12], Chap. 2).
Lemma G.1 (Cantor's intersection theorem). Let {A k } k∈N be a sequence of compact subsets of R n such that A k ⊇ A k+1 for all k ∈ N and K = k∈N A k . Then, if A k is non-empty for all k ∈ N , K is non-empty.
Lemma G.2 (Compact case). Let {A k } k∈N be a sequence of compact subsets of R n such that A k ⊇ A k+1 for all k ∈ N and K = k∈N A k . Then, for all open neighborhoods U of K, there exists k 0 ∈ N such that, for all k ≥ k 0 , we have A k ⊆ U .
Proof. Being U a neighborhood of K, it holds: where last equality follows be recalling that . Since U is an open set and A k is a compact set for all k ∈ N, it follows that A k \ U is a compact set for all k ∈ N. Therefore, expression (G.1) shows that {A k \ U } k∈N is a non-increasing sequence of compact subsets of R n whose intersection is empty. By virtue of Lemma G.1, it follows that A k0 \ U = ∅ for some k 0 ∈ N. By monotonicity of {A k \ U } k∈N , we have A k \ U = ∅ for all k ≥ k 0 , and thus A k ⊆ U for all such ks.
curve γ : R → M from the frame at γ(t 1 ) to the frame at γ(t 2 ). Let L(T x M ) denote the vector space of linear endomorphisms on a tangent space T x M . Let V be a compact subset of R n+1 .
Definition H.1. A vector field F : M → T M is said to be locally Lipschitz continuous uniformly on V if, for all compact sets K ⊂ M , there exists a constant C K such that, for all x ∈ K, there exists a neighborhood U ⊂ T x M of 0 such that exp x : T x M → M is bijective on U and for all r ∈ U it holds: T exp x (tv),1,0 F (exp x (r)) − F (x) g ≤ C K |r| g . (H.1) The constant C K is said to be the Lipschitz constant of F in C K .
This definition of uniform local Lipschitz continuity implies uniform local Lipschitz continuity in local coordinates, therefore we can conclude that the set of points x ∈ Ω(F (·)) where F (·) is not differentiable has zero measure. For any x ∈ M \ Ω(F (·)), and any r ∈ T x M , the covariant derivative of F (x) along r, can be defined by∇ for a smooth curve γ : R → M with γ(0) = x, γ (0) = r. Observe that, for any fixed x ∈ M \ Ω(F ), we obtain a well defined map∇F (x) : T x M → T x M given by r →∇ r F (x). To make the latter statement more precise, we bring to the attention of the reader that∇ r F (x) ∈ T x M while∇F (x) ∈ L(T x M ). We would like to extend this definition to arbitrary points x ∈ M , therefore we introduce a set-valued generalization of the covariant derivative for nonsmooth vector fields: The following Lemmas provide a local Gronwall estimate for dynamical systems evolving on Riemannian manifolds. Consider the following dynamical system: (H. 5) with states x ∈ M , and F being a locally Lipschitz continuous vector field. Let X(t, x) denote the solution trajectory of (H.5) at time t with initial condition at x. Proof. The proof follows along the lines of Proposition 1.1 in [7] with minor modifications. Without loss of generality, let τ → c(0, τ ) be parameterized by arc length, τ ∈ [0, l(0)]. For ease of presentation, we denote ∂ τ := ∂/∂ τ and ∂ t := ∂/∂ t . From the theory of ODEs, due to F being locally Lipschitz continuous on M , it follows that t → c(t, τ ) is absolutely continuous and has a Lipschitz continuous first derivative with respect to τ , i.e. F (c(t, τ )) is continuous in τ for any fixed t and |∇ ∂τ F (c(t, τ ))| g ≤ C U,T for all (t, τ ) ∈ [0, T ] × [a, b]. On the other hand, note that F (c(t, τ )) may fail to be continuous and ∇ ∂t F (c(t, τ )) g may fail to exist on some subsets of [0, T ] with zero measure. However, one may observe that |∇ ∂t F (c(t, τ ))| g is bounded from above for almost all t ∈ [0, T ]. Under the very same conditions, Theorem 9 in [10] has proven equality of mixed partial derivatives, namely that:∇ ∂t ∂ τ c(τ, t) =∇ ∂τ ∂ t c(τ, t), Furthermore, if p(t), q(t) evolve on a bounded set U ⊂ M for all t ≥ 0, there exists a constant C U such that (H.7) holds with T = +∞.
Corollary H.6 (upper bound). Consider the nonautonomous system (H.5) and the flow X(t, x; v) at time t ∈ R with initial condition x ∈ M . Assume that Λ is a compact invariant set for (H.5). Then, for all neighborhoods U of Λ and all T > 0, there is a constantĈ U,T > 0 such that, for all x ∈ U , and all t ∈ [0, T ], the following upper bound holds: Furthermore, if Λ is robustly stable on a compact neighborhood U in the sense of [8, Definition 2.2], then inequality (H.8) holds for all t ≥ 0, all x ∈ U , and with a exponential constantĈ U which does not depend upon T .
Proof. We first highlight that Proposition 5.1 in [8] can also be applied to systems evolving on manifolds. Indeed, the proof in [8] makes use of the following two facts: 1. closed and bounded sets are compact, and this is indeed true for geodesically complete manifolds; 2. Gronwall inequality holds (Corollary H.5).
It then follows that the reachable set R ≤T (U ) =: Q is bounded and we can find the Lipschitz constantĈ U,T of Corollary H.5. By definition of | · | Λ and compactness of Λ, it follows that there exists some x 0 ∈ Λ such that |x| Λ = d [x, x 0 ]. Moreover, by invariance of Λ under the flow of (2.1), we have that X(t, x 0 ) ∈ Λ for all t ∈ R. Therefore it holds, for all t ≥ 0, that: where the first inequality follows by definition of | · | Λ .
Corollary H.9 (Exponential lower bound for systems with locally Lipschitz vector fields). Assume Λ is a compact invariant set for (H.5). Then, for all neighborhoods U of Λ, there is a constant C U > 0 such that, for all x ∈ U \ Λ, and all t ≥ 0, the following lower bound holds: Proof. By definition of | · | Λ and compactness of Λ, there existsx ∈ Λ such that |X(t, x)| Λ = d [X(t, x) ,x]. By invariance of Λ under the flow of (H.5), we can then select x 0 ∈ Λ such that X(t, x 0 ) =x. Then, by virtue of Lemma H.7, we obtain: for some c which only depends upon U . The previous inequalities follow from local Lipschitz continuity of functions |·|Ā i andf (·), where the definition of local Lipschitz continuity for real-valued functions on M is adapted from Definition H.1. Finally, injecting (I.3) and (I.9) in (I.2) proves the Lemma.