A superposition approach for the ISS Lyapunov-Krasovskii theorem with pointwise dissipation

A sup erp osition approac h for the ISS Ly apuno v-Kraso vskii theorem with p oin t wise dissipation Andrii Mironc henko a, ∗ , F abian Wirth b , An toine Chaillet c , Lucas Briv adis c a Dep artment of Mathematics, University of Bayreuth, Germany (email: andrii.mir onchenko@uni-b ayr euth.de) b F aculty of Computer Scienc e and Mathematics, University of Passau, Germany (email: firstname.lastname@uni-p assau.de) c Université Paris-Saclay, CNRS, CentraleSup élec, L ab or atoir e des signaux et systèmes, 91190, Gif-sur-Yvette, F r anc e (email: firstname.lastname@c entr alesup ele c.fr) Abstract W e show that the existence of a Lyapuno v-Krasovskii functional (LKF) with p oin twise dissipation (i.e. dissipation in terms of the current solution norm) suffices for ISS, pro vided that uniform global stabilit y can also be ensured using the same LKF. T o this end, w e develop a stabilit y theory , in whic h the b eha vior of solutions is not assessed through the classical norm but rather through a sp ecific LKF, which ma y pro vide significan tly tigh ter estimates. W e discuss the adv an tages of our approach b y means of an example. Keywor ds: Nonlinear con trol systems, input-to-state stability , time-dela y systems, infinite-dimensional systems. 1. In tro duction Input-to-state stability (ISS), in tro duced by E.D. Son- tag in the late 1980s [29], has b ecome a central to ol in the analysis and control of nonlinear dynamical systems [30, 17]. Originally defined in the context of ordinary dif- feren tial equations, it has b een extended more recen tly to infinite-dimensional systems [18, 9], including time-dela y systems [3]. F or time-dela y systems, ISS can be established by means of Lyapuno v-Krasovskii functionals (LKF s) [28]: ISS holds if the LKF dissipates along the system’s solutions, modulo a positive term in volving the input norm. So far, the only general conditions to ensure ISS based on LKF imp ose that the dissipation can b e expressed in terms of the LKF itself (whic h is also a necessary requirement for ISS [10]). It has b een conjectured in [4] that a p oin twise dissipa- tion, in v olving merely the norm of current v alue of the so- lution, is enough to guarantee ISS. While this conjecture has b een pro ved for sp ecific classes of systems [4, 2, 14] and for the weak er notion of integral ISS [1], it has not yet b een pro ved or dispro v ed in its full generalit y . It is w orth men tioning that, in [7], the authors emplo yed an ISS LKF in the so-called “implication form” and show ed that if a p oin t wise dissipation holds whenev er the LKF dominates the input magnitude then ISS can b e concluded. Still, this condition remains significan tly more conserv ative than the existence of an ISS LKF with p oin t wise dissipation. ⋆ This w ork has b een supp orted by Ba yF rance, pro ject FK-20- 2022. A. Mironc henko has b een supp orted by the Heisenberg pro- gram of the German Research F oundation (DF G), grant MI 1886/3-1. ∗ Corresponding author. Solving this conjecture would be interesting not only for the sak e of mathematical curiosity , but also for more prac- tical considerations, as a p oint wise dissipation is usually easier to obtain than an estimate in volving the LKF. It w ould also unify the theory with that of input-free sys- tems, since it has b een kno wn for a long time that, in the absence of inputs, a p oin twise dissipation is enough to conclude global asymptotic stabilit y [11]. Despite significan t efforts on this question, it is not ev en kno wn whether the conjecture is true if we additionally assume that solutions are globally uniformly bounded, or ev en if the origin is uniformly globally stable (UGS). In this pap er, w e partially solve this question b y sho wing that, if the LKF that dissipates p oin t wisely can also b e used to establish UGS, then the ISS prop ert y holds. F or input- free systems, our results recov er and, in fact, strengthen the Kraso vskii theorem for asymptotic stability [11]. In terestingly , our main result establishes a stronger prop ert y , whic h we call V -ISS. This prop ert y is similar to ISS, but measures the b eha vior of the system’s solutions through a given LKF V rather than through the classical sup norm of the state. W e b eliev e this notion ma y be of in terest on its own as, dep ending on the considered LKF, it may provide a tighter estimate of the solutions’ behav- ior. In particular, we prov e a V -ISS superp osition theorem (a c haracterization of V -ISS in terms of resp ectiv e attrac- tivit y and stabilit y notions, see [31, 20]) and use it for the deriv ation of Ly apunov-Kraso vskii sufficien t conditions. The paper is organized as follows. In Section 2, we re- call some basics ab out time-delay systems, introduce the V -ISS concept, and state our main result. In Section 3, w e compare this result with the other approaches existing in the literature to ensure ISS based on a point wise dissi- pation [7, 14, 25]. W e also provide an academic example illustrating the added v alue of our approach. In Section 4, we adapt some other classical stabilit y concepts, in the same spirit as V -ISS, and state a sup erp osition principle for V -ISS. W e also pro vide some LKF-based conditions to establish these prop erties, and giv e some tec hnical obser- v ations that are needed for the pro of of the main result in Section 5. In Section 7, a detailed pro of of the V -ISS sup erposition theorem is presen ted. In Section 7.3, we pro ve sufficient conditions for ISS in terms of mixed sta- bilit y notions, which may b e instrumen tal for the further study of Ly apunov-Kraso vskii functionals with p oint wise dissipation. A conference version of this pap er where the main results ha ve b een stated without detailed pro ofs, examples, and discussion, has app eared in [24]. Notation. By R + := [0 , ∞ ) we denote the set of nonneg- ativ e real n umbers. F or x ∈ R n , | x | denotes its Euclidean norm and | A | denotes the corresp onding induced matrix norm of A ∈ R n × n . Given in terv als I , Q , J ⊂ R , C ( I , J ) denotes the set of contin uous functions from I to J , and C ( I × Q , J ) the set of con tinuous maps from I × Q to J . “F or all t ∈ I a.e. ” means for all t ∈ I , except p os- sibly on a set of zero Leb esgue measure. Giv en θ > 0 , X := C ([ − θ , 0] , R ) . U denotes the space L ∞ loc ( R + , R ) of all signals u : R + → R that are Leb esgue measurable and lo cally essentially b ounded. Giv en an interv al I ⊂ R + and a lo cally essentially bounded signal u : I → R m , ∥ u ∥ := ess sup t ∈I | u ( t ) | . Giv en u ∈ U m , u I : I → R m denotes its restriction to the interv al I , in particular ∥ u I ∥ = ess sup t ∈I | u ( t ) | . Given T ∈ R + ∪ { + ∞} , θ > 0 , x ∈ C ([ − θ, T ) , R n ) and t ∈ [0 , T ) , the function x t ∈ X n is the history function defined as x t ( τ ) := x ( t + τ ) for τ ∈ [ − θ , 0] . W e also use the standard classes of compari- son functions: K := { γ ∈ C ( R + , R + ) | γ (0) = 0 , γ is strictly increasing } , K ∞ := { γ ∈ K | γ is unbounded } , L := { γ ∈ C ( R + , R + ) | γ is strictly decreasing, lim t →∞ γ ( t ) = 0 } , KL := { β ∈ C ( R + × R + , R + ) | β ( · , t ) ∈ K ∀ t ≥ 0 , β ( r, · ) ∈ L ∀ r > 0 } . 2. Preliminaries and main result 2.1. Time-delay systems W e consider retarded differen tial equations of the form ˙ x ( t ) = f ( x t , u ( t )) , (1) where x t ∈ X n , n ∈ N , denotes the history function de- fined as x t ( s ) := x ( t + s ) for all s ∈ [ − θ, 0] , and θ > 0 is the fixed maximal time-delay inv olved in the dynamics. The input u is assumed to b e in U m , m ∈ N . Our standing assumption on the v ector field f is the following. Assumption 1. The ve ctor field f : X n × R m → R n is (i) Lipschitz c ontinuous in its first ar gument on b ounde d subsets of X n × R m , i.e., for al l C > 0 , ther e exists L f ( C ) > 0 , such that for al l ϕ, φ ∈ X n with ∥ ϕ ∥ ≤ C and ∥ φ ∥ ≤ C and al l v ∈ R m with | v | ≤ C , | f ( ϕ, v ) − f ( φ, v ) | ≤ L f ( C ) ∥ ϕ − φ ∥ ; (2) (ii) jointly c ontinuous in its ar guments. Our aim in this paper is to analyze the stability of the equilibria. F or definiteness, we assume that 0 is an equi- librium of the undisturb ed system, that is: Assumption 2. Thr oughout f (0 , 0) = 0 . By [3, Theorem 2], Assumption 1 guaran tees that, for an y initial condition x 0 ∈ X n and an y input u ∈ U m , there is a unique maximal solution (in the Carathéo dory sense) of (1), which we denote b y x ( · , x 0 , u ) . Given t in the domain of existence of this solution, the corresp onding history function is denoted by x t ( x 0 , u ) ∈ X n . The triple ( X n , U m , φ ) , where φ is the flo w mapping ( x 0 , u ) and t in the maximal in terv al of existence to φ ( t, x 0 , u ) := x t ( x 0 , u ) defines an abstract control system in the sense of [20]. In view of Assumption 1, the system (1) satisfies the b oundedness-implies-con tin uation (BIC) prop ert y , i.e., ev- ery maximal solution that is b ounded on its whole domain of existence is defined on R + , see [3, Theorem 2]. 2.2. Definitions In this section, we introduce a v ariant of ISS, sp ecifically tailored to the analysis of dela y systems using Ly apunov- Kraso vskii functionals. T o this aim, w e first recall the follo wing definition [3]. Definition 2.1 (LKF candidate). A map V ∈ C ( X n , R + ) is c al le d a Lyapuno v-Krasovskii functional candidate (LKF candidate) , if ther e ar e ψ 1 , ψ 2 ∈ K ∞ so that the fol lowing sandwich b ounds hold: ψ 1 ( | ϕ (0) | ) ≤ V ( ϕ ) ≤ ψ 2 ( ∥ ϕ ∥ ) ∀ ϕ ∈ X n . (3) It is said to b e co erciv e if, for ψ 1 , ψ 2 as ab ove, the fol low- ing str onger c ondition is satisfie d: ψ 1 ( ∥ ϕ ∥ ) ≤ V ( ϕ ) ≤ ψ 2 ( ∥ ϕ ∥ ) ∀ ϕ ∈ X n . Remark 2.2. W e note that in [3], it is r e quir e d that LKF c andidates ar e also Lipschitz c ontinuous on b ounde d sets of X n . F or our r esults, however, we do not ne e d this extr a assumption. 2 Next, w e revisit the notion of ISS by estimating the system’s prop erties through an asso ciated LKF candidate, rather than through the standard norms. Definition 2.3 ( V -ISS / ISS). Given an LKF c andi- date V : X n → R + , the system (1) is c al le d V -input- to-state stable ( V -ISS) if ther e exist β ∈ KL and γ ∈ K ∞ such that, for al l x 0 ∈ X n and al l u ∈ U m , V ( x t ( x 0 , u )) ≤ β ( V ( x 0 ) , t ) + γ ( ∥ u ∥ ) , ∀ t ≥ 0 . (4) System (1) is c al le d input-to-state stable (ISS) , if it is V - ISS for V ( ϕ ) = ∥ ϕ ∥ . Remark 2.4. In the ab ove definition, we tacitly assume that al l solutions ar e wel l-define d on R + . Similar c onven- tions ar e taken for other definitions in this work. If (4) wer e to hold only on the domain of definition of x ( · , x 0 , u ) , then (4) would guar ante e that the solution in b ounde d on this interval, and the BIC pr op erty of (1) guar ante e d by our r e gularity Assumption 1 would imply that the domain of definition is, in fact, R + , and thus (4) is valid on R + . It is worth noting that V -ISS may pro vide a tighter es- timate on the solutions’ norm. F or instance, consider the follo wing widely-used class of quadratic LKF candidates V ( ϕ ) := ϕ (0) ⊤ P ϕ (0) + Z 0 − θ ϕ ( τ ) ⊤ Qϕ ( τ ) dτ , where P , Q ∈ R n × n denote symmetric, p ositiv e definite matrices. F or suc h LKF candidates, V -ISS ensures an upp er b ound on the solution’s norm in terms of V ( x 0 ) , and, since P , Q are p ositiv e definite, in terms of | x 0 (0) | 2 + R 0 − θ | x 0 ( τ ) | 2 dτ , whereas the classical ISS would upp er- b ound them in terms of ∥ x 0 ∥ , whic h ma y be significantly larger for some particular initial states. The follo wing statement clarifies the prop erties induced b y V -ISS and relates it to the class ical definition of ISS. Prop osition 2.5 ( V -ISS ⇒ ISS). Given an LKF c an- didate V : X n → R + , c onsider the fol lowing statements: i) System (1) is V -ISS. ii) Ther e exist β ∈ KL and γ ∈ K ∞ such that, for al l x 0 ∈ X n and al l u ∈ U m , the flow of (1) satisfies V ( x t ( x 0 , u )) ≤ β ( ∥ x 0 ∥ , t ) + γ ( ∥ u ∥ ) , ∀ t ≥ 0 . (5) iii) Ther e exist β ∈ KL and γ ∈ K ∞ such that, for al l x 0 ∈ X n and al l u ∈ U m , the flow of (1) satisfies | x ( t, x 0 , u ) | ≤ β ( ∥ x 0 ∥ , t ) + γ ( ∥ u ∥ ) , ∀ t ≥ 0 . (6) iv) System (1) is ISS. Then the fol lowing r elations hold: i) ⇒ ii) ⇔ iii) ⇔ iv). If V is c o er cive, then al l four statements ar e e quivalent. Pro of. i) ⇒ ii) ⇒ iii) follows easily from the fact that b y definition V satisfies a sandwich condition as in (3). T o see iii) ⇒ iv), note that from (6) we can get (4) with V ( · ) = ∥ · ∥ , b y using ˜ β ( r, t ) := β ( r, t − θ ) , r ≥ 0 , t ≥ θ and extending this to a K L -function that is sufficiently large on R + × [0 , θ ] . The fact that iv) ⇒ ii) is also straigh tforward b y noticing that b y (3) we hav e V ( ϕ ) ≤ ψ 2 ( ∥ ϕ ∥ ) for a certain ψ 2 ∈ K ∞ and for all ϕ ∈ X n . It remains to pro ve that iii) ⇒ iv). T o this end, notice that iii) guarantees that, for all t ≥ θ , ∥ x t ( x 0 , u ) ∥ = max τ ∈ [ − θ , 0] | x ( t + τ , x 0 , u ) | ≤ max τ ∈ [ − θ , 0] β ( ∥ x 0 ∥ , t + τ ) + γ ( ∥ u ∥ ) = β ( ∥ x 0 ∥ , t − θ ) + γ ( ∥ u ∥ ) . (7) On the other hand, for t ∈ [0 , θ ] , it holds that ∥ x t ( x 0 , u ) ∥ = max  max s ∈ [ t − θ, 0] | x ( s, x 0 , u ) | , max s ∈ [0 ,t ] | x ( s, x 0 , u ) |  ≤ max {∥ x 0 ∥ , β ( ∥ x 0 ∥ , 0) + γ ( ∥ u ∥ ) } . Since β ( r, 0) ≥ r for all r ≥ 0 (as can b e seen from (6) with u ≡ 0 ), w e obtain that ∥ x t ( x 0 , u ) ∥ ≤ β ( ∥ x 0 ∥ , 0) + γ ( ∥ u ∥ ) , ∀ t ∈ [0 , θ ] . (8) Com bining (7) and (8), w e conclude that ∥ x t ( x 0 , u ) ∥ ≤ ˜ β ( ∥ x 0 ∥ , t ) + γ ( ∥ u ∥ ) , ∀ t ≥ 0 , where ˜ β is the function defined for all r, t ≥ 0 as ˜ β ( r, t ) =  ( θ − t ) r + β ( r , 0) , if t ∈ [0 , θ ] , β ( r, t − θ ) , if t > θ. The conclusion follo ws by noticing that ˜ β ∈ KL . □ The equiv alence of the items iii) and iv) can be found, e.g., in [16, Proposition 1.4.2]. W e mention it here merely for the sak e of completeness. T o analyze the V -ISS prop erty using ISS Ly apunov- Kraso vskii functionals, w e use the following notions of ISS LKF. They all rely on the upp er right-hand Dini derivative of the map V along the solutions of system (1), defined for all ϕ ∈ X n and u ∈ U m as ˙ V u ( ϕ ) := lim sup h → 0 + V ( x h ( ϕ, u )) − V ( ϕ ) h . (9) Definition 2.6 (p oin twise/LKF-wise ISS LKF). Consider system (1) . F or this system an LKF V : X n → R + is c al le d: 3 (i) an ISS LKF in implication form with LKF-wise dis- sipation if ther e exist χ ∈ K ∞ and α ∈ P such that, for al l ϕ ∈ X n and al l u ∈ U m , V ( ϕ ) ≥ χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ − α ( V ( ϕ )) . (10) (ii) an ISS LKF in implication form with p oin twise dis- sipation if ther e exist χ ∈ K ∞ and α ∈ P such that, for al l ϕ ∈ X n and al l u ∈ U m , | ϕ (0) | ≥ χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ − α ( | ϕ (0) | ) . (11) (iii) an ISS LKF in dissipative form with LKF-wise dis- sipation if ther e exist α, χ ∈ K ∞ such that, for al l ϕ ∈ X n and al l u ∈ U m , ˙ V u ( ϕ ) ≤ − α ( V ( ϕ )) + χ ( ∥ u ∥ ) . (12) (iv) an ISS LKF in dissipativ e form with p oin twise dis- sipation if ther e exist α, χ ∈ K ∞ such that, for al l ϕ ∈ X n and al l u ∈ U m , ˙ V u ( ϕ ) ≤ − α ( | ϕ (0) | ) + χ ( ∥ u ∥ ) . (13) In al l c ases the function χ is c al le d the gain r elate d to V . Remark 2.7. In Definition 2.1 we imp ose d little r e gular- ity on V – in p art b e c ause for many ar guments only c on- tinuity is r e quir e d. On the other hand this makes the ver- ific ation of the c onditions app e aring in Definition 2.6 te- dious, b e c ause the expr ession in (9) r e quir es evaluation for al l u ∈ U m . F r e quently, it is assume d that V is lo c al ly Lipschitz c ontinuous, se e e.g. [3]. In this c ase, the Driver derivative [5, 3] is useful. Given ϕ ∈ X n , u ∈ R m , define pseudotr aje ctories for h ∈ [0 , θ ) by ϕ h, u ( τ ) :=  ϕ ( τ + h ) τ ∈ [ − θ , − h ] , ϕ (0) + τ f ( ϕ, u ) τ ∈ [ − h, 0] . (14) The Driver derivative of V at ϕ in dir e ction f ( ϕ, u ) is then D + V ( ϕ, u ) := lim sup h → 0 + 1 h ( V ( ϕ h, u ) − V ( ϕ )) . (15) If V is lo c al ly Lipschitz c ontinuous, then for any input u ∈ U m the Dini derivative and the p ointwise Driver derivative c oincide almost everywher e along a tr aje ctory, [26, The or em 2]. In addition, by [27] it is sufficient to have de c ay estimates along the absolutely c ontinuous so- lutions t 7→ x t ( ϕ, u ) c orr esp onding to initial functions ϕ ∈ C 1 ([ − θ , 0] , R n ) . With these to ols it is not har d to se e that it is sufficient to formulate the c onditions of Def- inition 2.6 in terms of Driver derivatives, e.g. (10) c an b e expr esse d as V ( ϕ ) ≥ χ ( | u | ) ⇒ D + V ( ϕ, u ) ≤ − α ( V ( ϕ )) , (10’) and similarly for (11) – (13) . In the dissipative form of the ISS LKF, w e hav e to re- quire the deca y rate α to b e a K ∞ -function. A t the same time, for ISS LKF s in implication form, it suffices that α is merely a P -function. As the following prop osition sho ws (motiv ated by [12, Remark 4.1], [17, Prop osition 2.17]), by a suitable scaling of an ISS Lya- puno v functional (in implication form), we can alwa ys ob- tain a K ∞ deca y rate. Recall that a nonline ar sc aling is a function µ ∈ K ∞ , con tinuously differentiable on (0 , + ∞ ) , satisfying µ ′ ( s ) > 0 for all s > 0 and such that lim s → 0 µ ′ ( s ) exists, and is finite (that is, it b elongs to R + ). Prop osition 2.8. L et V ∈ C ( X n , R + ) b e an ISS LKF in implic ation form with p ointwise dissip ation with a de- c ay r ate α ∈ P . Then ther e is a nonline ar sc aling ξ ∈ K ∞ , which b elongs to C 1 ( R + , R + ) , is infinitely differ entiable on (0 , + ∞ ) , and such that ξ ◦ V ( · ) is an ISS LKF in impli- c ation form with p ointwise dissip ation with a de c ay r ate b elonging to K ∞ . Pro of. Let V ∈ C ( X n , R + ) b e an ISS LKF in impli- cation form with point wise dissipation with the sandwich b ounds ψ 1 , ψ 2 ∈ K ∞ , a decay rate α ∈ P and a Lyapuno v gain χ ∈ K ∞ . According to [17, Prop osition A.13], there are ω ∈ K ∞ and σ ∈ L suc h that α ( r ) ≥ ω ( r ) σ ( r ) , r ≥ 0 . Without loss of generalit y , w e can assume that σ and ψ 1 are infinitely differentiable on (0 , + ∞ ) (otherwise take 1 2 σ , 1 2 ψ 1 and smo oth them). F rom (11), it follo ws that for all ϕ ∈ X n and u ∈ U m | ϕ (0) | ≥ χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ − ω ( | ϕ (0) | ) σ ( | ϕ (0) | ) . As σ ∈ L , − σ is an increasing function, and by the sand- wic h inequality (3), we ha ve that | ϕ (0) | ≥ χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ − ω ( | ϕ (0) | ) σ ◦ ψ − 1 1 ( V ( ϕ )) . Th us, for all ϕ ∈ X n and all u ∈ U m , w e hav e | ϕ (0) | ≥ χ ( ∥ u ∥ ) ⇒ 1 σ ◦ ψ − 1 1 ( V ( ϕ )) ˙ V u ( ϕ ) ≤ − ω ( | ϕ (0) | ) . (16) Define ξ ∈ C 1 ( R + , R + ) b y ξ ( r ) := Z r 0 1 σ ◦ ψ − 1 1 ( s ) ds, r ≥ 0 . (17) As σ is infinitely differen tiable and nev er zero, ξ is well- defined on R + and infinitely differen tiable. As σ can also b e chosen small enough so that ξ ∈ K ∞ , it is easy to see that ξ is a nonlinear scaling. Define W := ξ ◦ V and observe 4 b y exploiting the c hain rule for Dini deriv atives 1 , where w e use that ξ has a p ositiv e deriv ative, that ˙ W u ( ϕ ) = 1 σ ( V ( ϕ )) ˙ V u ( ϕ ) and th us (16) can b e transformed in to | ϕ (0) | ≥ χ ( ∥ u ∥ ) ⇒ ˙ W u ( ϕ ) ≤ − ω ◦ ξ − 1 ( | ϕ (0) | ) , (18) and ω ◦ ξ − 1 ∈ K ∞ as a comp osition of K ∞ -functions. As ξ ◦ ψ 1 ( | ϕ (0) | ) ≤ W ( x ) ≤ ξ ◦ ψ 2 ( ∥ ϕ ∥ ) ∀ x ∈ X n , (19) w e see that W is an ISS LKF in implication form with p oin t wise dissipation and K ∞ -deca y rate. □ Remark 2.9. A c ounterp art of Pr op osition 2.8 c an b e shown also for ISS LKF in implic ation for with LKF-wise dissip ation. The following result states that any ISS LKF in dissipa- tiv e form with p oint wise dissipation is also an ISS LKF in implication form with p oin t wise dissipation. Prop osition 2.10. ( Dissip ative form ⇒ implic ation form ) F or system (1) , if V is an ISS LKF in dissip ative form with p ointwise dissip ation, then it is also an ISS LKF in implic ation form with p ointwise dissip ation. Pro of. If V is an ISS LKF in dissipativ e form with p oin t wise dissipation, then (13) holds with some α , χ ∈ K ∞ . Thus, | ϕ (0) | ≥ α − 1 ◦ 2 χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ − 1 2 α ( | ϕ (0) | ) , and the claim follo ws. □ Our main result also exploits the following relaxation of the concept of ISS LKF in implication form so that the deca y estimate ensures uniform global stability . Definition 2.11 (UGS LKF). An LKF c andidate V : X n → R + is c al le d a UGS LKF for (1) if ther e exists χ ∈ K ∞ , such that for al l ϕ ∈ X n and al l u ∈ U m , V ( ϕ ) ≥ χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ 0 . (20) As will b e formalized in Section 4.2, the existence of a UGS LKF ensures uniform global stabilit y of the origin. 1 A goo d reference for the particular prop ert y we need seems elu- sive, so we provide a brief direct argument: By definition ˙ W u ( ϕ ) = lim sup h → 0 + 1 h ( ξ ◦ V ( x h ( ϕ, u )) − ξ ◦ V ( ϕ )) . As ξ is smo oth, we ma y apply the mean v alue theorem to obtain ξ ◦ V ( x h ( ϕ, u )) − ξ ◦ V ( ϕ ) = ξ ′ ( r h )( V ( x h ( ϕ, u )) − V ( ϕ )) , where r h is some p oint in b etw een V ( ϕ ) and V ( x h ( ϕ, u )) . As h → 0 + we ha ve ξ ′ ( r h ) → ξ ′ ( V ( ϕ )) > 0 . Mul- tiplication by a p ositiv ely conv ergent sequence do es not change the limit superior and we obtain ˙ W u ( ϕ ) = ξ ′ ( V ( ϕ )) ˙ V u ( ϕ ) . 2.3. Main r esult In [4], it has b een conjectured that the existence of an ISS LKF in dissipativ e form with p oin twise dissipation is enough to ensure ISS. In ligh t of Prop osition 2.10, this conjecture would b e solved if we managed to show that the existence of an ISS LKF V in implication form with p oin t wise dissipation is enough to ensure ISS. T o date, this conjecture remains op en, but our main result states that ISS (and, actually , V -ISS) indeed holds if V is also a UGS LKF. Theorem 2.12 (ISS with p oin t wise dissipation). L et Assumptions 1 and 2 hold. If ther e exists an LKF c andidate V : X n → R + which is simultane ously an ISS LKF with p ointwise dissip ation (in either implic ation or dissip ative form) and a UGS LKF for (1) , then (1) is V -ISS and, in p articular, ISS. The pro of of this result requires the introduction of fur- ther notions related to V -stabilit y and corresponding LKF to ols. It is therefore p ostp oned to Section 5. 3. Discussion Let us briefly discuss the nov elty of Theorem 2.12 by comparing it to other approac hes in the literature. 3.1. R elations to existing r esults A sufficient condition for ISS in terms of p oin twise dis- sipation has b een obtained in [7]. In the terminology of Definition 2.6, it can b e expressed as V ( ϕ ) ≥ χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ − α ( | ϕ (0) | ) . (21) This condition lies halfwa y b et ween (10) and (11), in the sense that the dissipation is requested in a p oin twise manner but it needs to hold whenev er the LKF itself qual- itativ ely dominates the input norm. It has b een shown in [7, Theorem 2] that (21) is sufficien t to ensure ISS. This result can be seen as a corollary of Theorem 2.12 since (21) implies that V is b oth an ISS LKF with p oint wise dissipation in implication form, and a UGS LKF. Our re- sult therefore strengthens [7, Theorem 2] in three different w ays. • Our requiremen ts on V are w eaker than those in [7, Theorem 2], as V is requested to decay only when | ϕ (0) | ≥ γ ( ∥ u ∥ ) . • Theorem 2.12 do es not merely ensure ISS but also V -ISS, which is a p oten tially stronger prop ert y . • Our requiremen ts on the nonlinearit y f (Assump- tion 1) are w eaker than those in [7, Theorem 2]. Namely , we do not assume Lipsc hitz contin uit y of f with respect to its second argument (the input u ), whic h was imp ortan t in [7]. 5 In a different direction, [4, 2, 14] assume growth restric- tion on the LKF candidate or on the vector field in order to establish ISS based on p oin twise dissipation. In The- orem 2.12, no suc h gro wth constraints are needed, at the price of a uniform global stability requirement (through the same LKF V ). An extension of results using growth restrictions is the so-called exponential tric k studied in [25] and [13, Sec- tion 4.3]. This metho d considers an LKF candidate of the form W ( ϕ ) = w 1 ( ϕ (0)) + Z 0 − θ w 2 ( ϕ ( s )) ds. (22) Under suitable conditions on W and the system (1), it is sho wn that for certain c, κ > 0 the mo dified LKF W c,κ ( ϕ ) = κw 1 ( ϕ (0)) + Z 0 − θ e cs w 2 ( ϕ ( s )) ds. (23) yields an LKF with LKF-wise dissipation (hence, ISS), if p oin t wise dissipation holds for W . An example in [14] sho ws that this is by no means a univ ersal method, but ma y or may not work dep ending on the system. A further metho d is prop osed in [14], where it is assumed that V : X n → R + is Lipschitz contin uous on b ounded sets LKF, and the follo wing p oin twise decay estimate holds: ˙ V u ≡ u ( ϕ ) ≤ − α ( Q ( ϕ (0))) + γ ( | u | ) , ϕ ∈ X n , u ∈ R m , (24) where α, γ ∈ K ∞ and Q : R n → R + is C 1 , p ositiv e definite, and radially unbounded. [14, Theorem 1] states th at if there exists a σ ∈ K ∞ suc h that for all ϕ ∈ X n , u ∈ R m ∇ Q ( ϕ (0)) f ( ϕ, u ) ≤ σ  max τ ∈ [ − θ , 0] Q ( ϕ ( τ ))  + γ ( | u | ) , (25) and lim s →∞ α ( s ) σ ( se θ ) > 0 , (26) then the system is ISS. In [14, Prop osition 2], an example of a dela y system is presen ted, for whic h one can construct the functions V and Q as ab ov e to conclude ISS via [14, Theorem 1]. It w as also sho wn that the constructed functional V do es not satisfy the assumptions of Theorem 2.12. W e note that even though the author of [13, Example 4.11] claims that their particular example shows that all systems cov ered by Theorem 2.12 can b e treated using an approac h as describ ed in (24)–(26), no argument is pro- vided wh y this should b e the case. 3.2. Criterion of uniform glob al asymptotic stability F rom the pro of of Theorem 2.12, we will see that if the gain χ in (11) is identically zero, then the gain γ in the V -ISS estimate (4) can also be c hosen to be zero. F or the case V = ∥ · ∥ , a system is called uniformly globally asymptotically stable (UGAS), if the estimate (4) holds with γ = 0 , see, e.g., [12]. W e therefore adopt the name V -UGAS, if (4) holds for a given LKF candidate V with γ = 0 . This yields the following: Corollary 3.1 ( V -U GAS). If (1) admits an ISS LKF V with p ointwise dissip ation (in implic ation or dissip ative form) with gain χ ≡ 0 (in (11) or (13) ), then it is V - UGAS. Pro of. Checking the pro of of Theorem 2.12 (and, in particular, the pro of of Prop osition 4.7 that w e use there), w e see that the system (1) satisfies the V -UGS property with 0 gain and V -ULIM prop ert y with 0 gain. This im- plies by arguments similar to those in [21, Theorem 2] that (1) is V -UGAS. □ V -UGAS has its ro ots in the ISS literature on finite- dimensional systems and was instrumental for the deriv a- tion of conv erse Lyapuno v results, [12]. Corollary 3.1 ex- tends the classical global asymptotic result by Krasovskii [6, Chapter 5, Theorem 2.1, p. 132] from input-free sys- tems to systems with inputs, and w e even obtain V -UGAS, in contrast to merely UGAS obtained in [6, Chapter 5, Theorem 2.1, p. 132]. 3.3. Example W e exhibit below an example of an ISS system for which Theorem 2.12 can b e used to prov e ISS, while none the approac hes describ ed in Section 3.1 can b e applied for the family of quadratic LKF candidates under consideration. Consider the system ˙ x ( t ) = − x ( t ) + x ( t − 1) − ψ ( x 2 ( t ) − u 2 ( t )) x ( t ) , x (0) = x 0 ∈ C ([ − 1 , 0] , R ) =: X , (27) where ψ : R → R is a smo oth function such that ψ ( s ) = 0 for all s ≤ 0 , ψ ( s ) > 0 for all s > 0 , and ψ ( s ) s → 0 as s → + ∞ . Prop osition 3.2. Consider system (27) . F or al l c on- stants c ≥ 0 , κ > 0 the functional define d by V c,κ ( ϕ ) := κϕ 2 (0) + Z 0 − 1 e cs ϕ 2 ( s ) ds, ϕ ∈ X , (28) is an LKF c andidate that is Lipschitz c ontinuous on b ounde d sets. In addition, (i) the system is V 0 , 1 -UGS with zer o gain and V 0 , 1 -ISS. (ii) the system is ISS. (iii) for every ( c, κ )  = (0 , 1) ther e exists a ϕ ∈ X such that | ϕ (0) | > 0 and ˙ V c,κ,u ≡ 0 ( δ ϕ ) > 0 for al l δ ∈ (0 , 1] . In p articular, V c,κ do es not c ertify lo c al stability with zer o input, nor ISS. (iv) ther e is no choic e of c ≥ 0 , κ > 0 such that V := V c,κ admits an LKF-wise dissip ation estimate of the form (10) or (12) . (v) ther e is no choic e of c ≥ 0 , κ > 0 for which ther e exist α, χ ∈ K ∞ such that (10’) is satisfie d for V := V c,κ . 6 (vi) ther e is no choic e of c ≥ 0 , κ > 0 for which ther e exist α, γ ∈ K ∞ , and a Q : R n → R + that is C 1 , p os- itive definite and r adial ly unb ounde d, such that (24) is satisfie d for V := V c,κ . Remark 3.3. The c onse quenc e of the pr evious Pr op osi- tion 3.2 for the other analysis metho ds mentione d at the b e ginning of this se ction ar e the fol lowing: (i) F or system (27) and the pr op ose d class of LKF c andi- dates the r esults of [8] ar e not applic able. By the analysis, the only p ossible c andidate is V 0 , 1 . However, even in this c ase (21) c annot b e satisfie d. (ii) As far as the exp onential trick is c onc erne d, we have se en that amongst the family of LKF c andidates the only viable LKF is inde e d V 0 , 1 and an applic ation of the exp o- nential trick c an only deterior ate the pr op erties of the LKF c andidate. Incidental ly, in this way we have pr esente d a new example of a situation wher e the exp onential trick is not applic able. Ar guably, the example is c onc eptual ly sim- pler than the one pr esente d in [14]. (iii) Also the metho d describ e d in (24) – (26) fails for sys- tem (27) b e c ause alr e ady the b asic assumption c annot b e met. On the other hand, examples in [13] show that this metho d is applic able in situations wher e the metho ds of the pr esent p ap er ar e not. So none of the appr o aches is strictly sup erior to the other. Pro of. (of Prop osition 3.2) It is easy to see that for ev ery c ≥ 0 , κ > 0 the conditions of Definition 2.1 are satisfied, meaning that V c,κ is an LKF candidate. In view of [3, Example 1], V c,κ is Lipschitz con tinuous on b ounded sets. F ollowing Remark 2.7, it is sufficient to consider the Driv er deriv ativ e in the remainder of the pro of. This has the added b enefit that our considerations are directly com- parable to the metho ds discussed in Section 3.1. (i), (ii). Let c ≥ 0 and κ > 0 . F or the Driver deriv a- tiv e of V c,κ in ϕ ∈ X we obtain (following the detailed calculations pro vided in [3, Example 1]) D + V c,κ ( ϕ, u ) = − 2 κϕ 2 (0) + 2 κϕ (0) ϕ ( − 1) − 2 κψ ( ϕ 2 (0) − u 2 ) ϕ 2 (0) + ϕ 2 (0) − e − c ϕ 2 ( − 1) − c Z 0 − 1 e cs ϕ 2 ( s ) ds = −  2 κψ ( ϕ 2 (0) − u 2 ) − e c κ 2 − 1 + 2 κ  ϕ 2 (0) −  e c/ 2 κϕ (0) − e − c/ 2 ϕ ( − 1)  2 − c Z 0 − 1 e cs ϕ 2 ( s ) ds. (29) In particular, for c = 0 and κ = 1 we obtain D + V 0 , 1 ( ϕ, u ) = − 2 ψ ( ϕ 2 (0) − u 2 ) ϕ 2 (0) − ( ϕ (0) − ϕ ( − 1)) 2 ≤ 0 . (30) This shows that V 0 , 1 is a UGS Lyapuno v function with zero Ly apunov gain and thus system (27) is V 0 , 1 -UGS with zero gain. In addition, observe that | ϕ (0) | ≥ 2 | u | ⇒ ψ ( ϕ 2 (0) − u 2 ) ≥ ˜ ψ ( | ϕ (0) | ) , where ˜ ψ ( s ) := min | r |≤ s/ 2 ψ ( s 2 − r 2 ) = min { ψ ([ 3 4 s 2 , s 2 ]) } , for all s ≥ 0 . Clearly , ˜ ψ ∈ P , and we get from (30) that | ϕ (0) | ≥ 2 | u | ⇒ D + V 0 , 1 ( ϕ, u ) ≤ − 2 ˜ ψ ( | ϕ (0) | ) ϕ 2 (0) . (31) Th us, b y Remark 2.7, V 0 , 1 is an ISS LKF in implication form (Definition 2.6) and by Theorem 2.12 system (27) is V 0 , 1 -ISS and ISS, whic h prov es items (i) and (ii). (iii). Fix a pair ( c, κ )  = (0 , 1) . As ψ is C ∞ and takes the constan t v alue zero for s ∈ ( −∞ , 0] , all deriv atives of ψ v anish in 0 and there are constants M , C > 0 , such that | s | ≤ M implies 0 ≤ ψ ( s ) ≤ C s 2 . Introducing the constan t K := e c κ 2 + 1 − 2 κ = ( e c − 1) κ 2 + ( κ − 1) 2 > 0 , where w e use ( c, κ )  = (0 , 1) , the expression (29) can be written as D + V c,κ ( ϕ, u ) = − 2 κψ ( ϕ 2 (0) − u 2 ) ϕ 2 (0) + K ϕ 2 (0) −  e c/ 2 κϕ (0) − e − c/ 2 ϕ ( − 1)  2 − c Z 0 − 1 e cs ϕ 2 ( s ) ds. (32) Consider ϕ ∈ X such that the following conditions are satisfied: (a) 0 < ϕ 2 (0) ≤ M , (b) 6 κC ϕ 4 (0) ≤ K , (c) ϕ ( − 1) = e c κϕ (0) , (d) 3 c R 0 − 1 e cs ϕ 2 ( s ) ds ≤ K ϕ 2 (0) . Then (32) for u = 0 yields D + V c,κ ( ϕ, 0) ( c ) = − 2 κψ ( ϕ 2 (0)) ϕ 2 (0) + K ϕ 2 (0) − c Z 0 − 1 e cs ϕ 2 ( s ) ds. ( a ) ≥ − 2 κC ϕ 6 (0) + K ϕ 2 (0) − c Z 0 − 1 e cs ϕ 2 ( s ) ds ( d ) >  K − 2 κC ϕ 4 (0) − K 3  ϕ 2 (0) ( b ) > 0 . With the choice of ϕ satisfying (a)–(d), it is clear that for all δ ∈ (0 , 1] also δ ϕ satisfies (a)–(d). W e thus obtain that D + V c,κ ( δ ϕ, 0) > 0 . This shows the claim. (iv) By (iii) the only p ossibilit y to obtain (10) or (12) is to choose c = 0 , κ = 1 . Ho wev er, in this case the equalit y (30) holds. Cho osing 0  = ϕ ∈ X with ϕ ( − 1) = ϕ (0) = 0 w e obtain D + V 0 , 1 ( ϕ, u ) = 0 for all u ∈ R . How ev er, b oth the conditions (10) or (12) require that D + V 0 , 1 ( ϕ, 0) < 0 . It is therefore imp ossible to satisfy (10) or (12) for all ϕ, u . (v) Again, by (iii) w e only need to consider the case c = 0 and κ = 1 . Fix an arbitrary χ ∈ K ∞ . By c ho osing ϕ ∈ X with 0 < ϕ (0) = ϕ ( − 1) and u = ϕ (0) , we hav e from (30) that D + V 0 , 1 ( ϕ, u ) = 0 . No w using the integral term in (28), w e can ensure that V ( ϕ ) ≥ χ ( | u | ) , so (v) holds. (vi) By (iii), w e only need to consider the case c = 0 , κ = 1 . Cho osing u = 0 , and ϕ ∈ X with 0 < ϕ (0) = ϕ ( − 1) , we ha ve by (30) that D + V 0 , 1 ( ϕ, 0) = − 2 ψ ( ϕ 2 (0)) ϕ 2 (0) . Since ψ ( s ) s → 0 as s → ∞ , w e see that D + V 0 , 1 ( r ϕ, 0) → 0 as r → ∞ . Th us for ev ery α ∈ K ∞ , and Q : R n → R + p ositiv e definite and radially unbounded the condition (24) do es not hold, as this w ould require that D + V 0 , 1 ( r ϕ, 0) → −∞ as r → ∞ . □ 7 4. V -stability theory On our wa y to proving Theorem 2.12, we dev elop the theory of V -stabilit y , which studies the long-term b eha vior of solutions not in terms of the classical ∥ · ∥ -norm, but rather through a particular LKF candidate V . This section aims to in tro duce several V -stability notions, to provide LKF conditions to establish them in practice, and, more imp ortan tly , to give a sup erposition theorem for V -ISS in this new setup, characterizing V -ISS in terms of these w eaker prop erties. 4.1. V -ISS sup erp osition the or em In the same w ay as the classical concept of ISS can b e extended to V -ISS, w e can consider the follo wing notions. Definition 4.1 ( V -stabilit y). Given an LKF c andidate V : X n → R + , the system (1) is (i) c al le d V -uniformly lo cally stable ( V -ULS) , if ther e ex- ist σ, γ ∈ K ∞ and r > 0 such that, for al l x 0 ∈ X n with V ( x 0 ) ≤ r and al l u ∈ U m with ∥ u ∥ ≤ r , V ( x t ( x 0 , u )) ≤ σ ( V ( x 0 )) + γ ( ∥ u ∥ ) ∀ t ≥ 0 . (33) (ii) c al le d V -uniformly globally stable ( V -UGS) , if ther e exist σ, γ ∈ K ∞ such that for al l x 0 ∈ X n , u ∈ U m the estimate (33) holds. (iii) c al le d V -uniformly globally b ounded ( V -UGB) , if ther e exist σ, γ ∈ K ∞ and c ≥ 0 such that for al l x 0 ∈ X n , u ∈ U m the fol lowing holds: V ( x t ( x 0 , u )) ≤ σ ( V ( x 0 )) + γ ( ∥ u ∥ ) + c ∀ t ≥ 0 . (34) (iv) said to have the V -uniform limit prop ert y ( V -ULIM) , if ther e exists γ ∈ K ∞ ∪ { 0 } so that for every ε > 0 and for every r > 0 ther e exists a τ = τ ( ε, r ) ≥ 0 such that, for al l x 0 ∈ X n with V ( x 0 ) ≤ r and al l u ∈ U m with ∥ u ∥ ≤ r , ther e is a t ∈ [0 , τ ] such that V ( x t ( x 0 , u )) ≤ ε + γ ( ∥ u ∥ ) . (35) (v) said to have a V -global uniform asymptotic gain ( V - GUA G) , if ther e exists a γ ∈ K ∞ ∪ { 0 } such that for al l ε, r > 0 ther e is a τ = τ ( ε, r ) ≥ 0 such that for al l u ∈ U m and al l x 0 ∈ X n with V ( x 0 ) ≤ r , it holds that V ( x t ( x 0 , u )) ≤ ε + γ ( ∥ u ∥ ) ∀ t ≥ τ . (36) (vi) said to have a V -uniform asymptotic gain ( V -UAG) , if ther e exists a γ ∈ K ∞ ∪ { 0 } such that for al l ε, r > 0 ther e is a τ = τ ( ε, r ) ≥ 0 such that for al l u ∈ U m with ∥ u ∥ ≤ r and al l x 0 ∈ X n with V ( x 0 ) ≤ r , the ine quality (36) holds. The system (1) is c al le d UGS , if it is V -UGS with V ( ϕ ) = ∥ ϕ ∥ , and similarly for the other stability notions. The UGS prop ert y has already b een used in the lit- erature on delay systems [19, 23]. The ULIM notion shares some similarities with the more classical LIM prop- ert y [31, 18], with the difference that here the maximal time needed for (35) to hold is required to be uniform on b ounded balls of b oth initial states and inputs. Note that in V -UAG prop ert y we are chec king the prop ert y (36) for inputs of a b ounded magnitude, whereas for V - GUA G property we require the v alidit y of (36) for all in- puts. The follo wing result can b e shown analogously to Prop osition 2.5. Prop osition 4.2 ( V -UGS ⇒ UGS). Given an LKF c andidate V : X n → R + , if (1) is V -UGS, then it is UGS. Recall that b y Assumption 2, 0 ∈ X n is an equilibrium of the undisturbed system. T wo more concepts will be imp ortan t for the c haracterization of V -ISS. Definition 4.3 ( V -CEP). Given an LKF c andidate V : X n → R + , we say that ϕ is V -contin uous at the equilib- rium of (1) if for every ε, h > 0 ther e exists a δ = δ ( ε, h ) > 0 , such that for al l x 0 ∈ X n with V ( x 0 ) ≤ δ and al l u ∈ U m with ∥ u ∥ ≤ δ the solution t 7→ x t ( x 0 , u ) is define d at le ast over [0 , h ] , and t ∈ [0 , h ] , V ( x 0 ) ≤ δ, ∥ u ∥ ≤ δ ⇒ V  x t ( x 0 , u )  ≤ ε. (37) In this c ase, we wil l also say that the system has the V - CEP prop ert y . Definition 4.4 ( V -BRS). Given an LKF c andidate V : X n → R + , we say that (1) has V -b ounded reachabilit y sets (is V -BRS) , if for any C > 0 and any τ > 0 it holds that sup  V ( x t ( x 0 , u )) : V ( x 0 ) ≤ C, ∥ u ∥ ≤ C , t ∈ [0 , τ ]  < ∞ . Here again, V -BRS constitutes the natural extension of the BRS prop ert y . It is worth recalling that, unlik e for ordinary differential equations [12], BRS is not implied b y forw ard completeness for delay systems [15]. Remark 4.5. If V is a c o er cive LKF c andidate, then ther e is no distinction b etwe en the various stability c on- c epts in the V -version and the classic al variants define d using the norm on X n . Mor e gener al ly, for LKF c an- didates V 1 , V 2 we have that V 1 -pr op erties ar e e quivalent to V 2 -pr op erties, if ther e ar e ψ 1 , ψ 2 ∈ K ∞ such that ψ 1 ◦ V 1 ≤ V 2 ≤ ψ 2 ◦ V 1 . On the other hand, if V is not c o er cive, then we have alr e ady se en in Pr op osition 2.5 that V -ISS implies ISS and similarly, it is not har d to show that the V -pr op erties imply the standar d norm pr op erties, but the c onverse dir e ction is usual ly false. The main result of this section is the characterization of V -ISS as a combination of the ab ov e stability and attrac- tivit y prop erties. 8 Theorem 4.6 ( V -ISS sup erp osition theorem). Given an LKF c andidate V : X n → R + , the fol lowing statements ar e e quivalent for system (1) (i) V -ISS. (ii) V -GUAG and V -UGS. (iii) V -UAG and V -UGS. (iv) V -UAG, V -CEP and V -BRS. (v) V -ULIM, V -ULS and V -BRS. (vi) V -ULIM and V -UGS. It is worth noting that, for the particular case that V ( ϕ ) = ∥ ϕ ∥ for all ϕ ∈ X n , Theorem 4.6 reduces to the ISS sup erposition theorem for general con trol systems pro ved in [20]. The pro of is based on a series of technical lemmas, whic h w e state and prov e in Section 7. Pro of. (i) ⇒ (ii). Since β ( V ( x 0 ) , t ) ≤ β ( V ( x 0 ) , 0) for all x 0 ∈ X n and all t ≥ 0 , we see that V -ISS implies V -UGS. The V -GUAG prop ert y follows from Lemma 7.5. (ii) ⇒ (iii) ⇒ (iv). These are immediate consequences of the definitions. (iv) ⇒ (v). It is immediate from the definition that V -UAG implies V -ULIM. The combination V -UAG ∧ V - CEP implies V -ULS by Lemma 7.6. (v) ⇒ (vi). By Prop osition 7.7, V -ULIM ∧ V -BRS implies V -UGB. V -UGB ∧ V -ULS implies V -UGS b y Lemma 7.4. (vi) ⇒ (i). V -ULIM ∧ V -UGS implies V -UA G b y Lemma 7.8. V -UAG ∧ V -UGS implies V -GUAG by Lemma 7.9. V -GUAG ∧ V -UGS implies V -ISS by Lemma 7.10. □ 4.2. Lyapunov-Kr asovskii c ondition for V -UGS The next result states that the existence of a UGS LKF V , as introduced in Definition 2.11, guarantees V -UGS. Prop osition 4.7 (LKF condition for V -UGS). If (1) admits a UGS LKF V then (1) is V -UGS (and thus UGS). Pro of. By assumption (20), there exists χ ∈ K ∞ suc h that V ( ϕ ) ≥ χ ( ∥ u ∥ ) ⇒ ˙ V u ( ϕ ) ≤ 0 . (38) for all ϕ ∈ X n and all u ∈ U m . Fix x 0 ∈ X n and u ∈ U m . Then the maximal solution x ( · , x 0 , u ) of (1) exists on some in terv al [ − θ , t m ( x 0 , u )) with t m ( x 0 , u ) ∈ (0 , + ∞ ] . W e consider tw o cases, whether or not V ( x 0 ) ≤ χ ( ∥ u ∥ ) . First let V ( x 0 ) ≤ χ ( ∥ u ∥ ) . Seeking a con tradiction, as- sume that there is a time t 2 ∈ (0 , t m ( x 0 , u )) such that V ( x t 2 ( x 0 , u )) > χ ( ∥ u ∥ ) . Let t 1 b e the maximal time t ∈ [0 , t 2 ) suc h that V ( x t ( x 0 , u )) = χ ( ∥ u ∥ ) , whic h exists b y contin uity of solutions. Due to the con tinuit y of solu- tions, V ( x t ( x 0 , u )) > χ ( ∥ u ∥ ) for all t ∈ ( t 1 , t 2 ) , and hence it holds from (38) that ˙ V u ( t + · ) ( x t ( x 0 , u )) ≤ 0 , ∀ t ∈ ( t 1 , t 2 ) , (39) and thus V ( x t ( x 0 , u )) ≤ V ( x t 1 ( x 0 , u )) = χ ( ∥ u ∥ ) for all t ∈ ( t 1 , t 2 ) , a contradiction. W e conclude for the case V ( x 0 ) ≤ χ ( ∥ u ∥ ) that V ( x t ( x 0 , u )) ≤ χ ( ∥ u ∥ ) , ∀ t ∈ [0 , t m ( x 0 , u )) . (40) W e now pro ceed to the second case, namely when V ( x 0 ) > χ ( ∥ u ∥ ) . Then either V ( x t ( x 0 , u )) > χ ( ∥ u ∥ ) for all t ∈ [0 , t m ( x 0 , u )) , or there is some minimal time t 3 > 0 so that V ( x t 3 ( x 0 , u )) = χ ( ∥ u ∥ ) ≤ V ( x 0 ) . Arguing as in (39), we see that for t ∈ [0 , t m ( x 0 , u )) resp. t ∈ [0 , t 3 ) V ( x t ( x 0 , u )) ≤ V ( x 0 ) . (41) In the second case we hav e for t > t 3 b y the co cycle prop ert y and the argumen ts leading to (40) that, for all t ∈ [ t 3 , t m ( x 0 , u )) , V ( x t ( x 0 , u )) = V ( x t − t 3 ( x t 3 ( x 0 , u ) , u ( t 3 + · ))) ≤ χ ( ∥ u ∥ ) . W e conclude from (40) and (41) that, in all cases, V ( x t ( x 0 , u )) ≤ max { V ( x 0 ) , χ ( ∥ u ∥ ) } , ∀ t ∈ [0 , t m ( x 0 , u )) . Since (1) satisfies the BIC prop erty (see, e.g., [3, Theorem 2]), the ab o ve inequality ensures that t m ( x 0 , u ) = + ∞ , and thus this estimate holds for all t ≥ 0 , and V -UGS follo ws. UGS is then a consequence of Prop osition 4.8. □ 4.3. Bounds on solutions W e finally present some tec hnical results pro viding b ounds on the solutions (and on their deriv ative) of a V - UGS system. While the definition of V -UGS provides an upp er bound on V ( x t ) , a b ound on the whole history norm can b e ob- tained after one full dela y p erio d. Prop osition 4.8 (Bounds on history). Given an LKF c andidate V : X n → R + , if (1) is V -UGS then ther e ar e σ, γ ∈ K ∞ such that, for al l x 0 ∈ X n and al l u ∈ U m , ∥ x t ( x 0 , u ) ∥ ≤ σ ( V ( x 0 )) + γ ( ∥ u ∥ ) ∀ t ≥ θ . (42) Pro of. Since V is an LKF candidate and (1) is V - UGS, w e may apply (33) and (3) to conclude that there are ψ 1 , ˜ σ , ˜ γ ∈ K ∞ so that, for all x 0 ∈ X n and u ∈ U m , ψ 1 ( | x ( t, x 0 , u ) | ) ≤ ˜ σ ( V ( x 0 )) + ˜ γ ( ∥ u ∥ ) ∀ t ≥ 0 . As ψ − 1 1 ( a + b ) ≤ ψ − 1 1 (2 a ) + ψ − 1 1 (2 b ) , a, b ≥ 0 , w e hav e | x ( t, x 0 , u ) | ≤ ψ − 1 1  2 ˜ σ ( V ( x 0 ))  + ψ − 1 1  2 ˜ γ ( ∥ u ∥ )  . Consequen tly , for all t ≥ θ , ∥ x t ( x 0 , u ) ∥ = max τ ∈ [ − θ , 0] | x ( t + τ , x 0 , u ) | ≤ σ ( V ( x 0 )) + γ ( ∥ u ∥ ) , 9 with σ := ψ − 1 1 ◦ 2 ˜ σ and γ := ψ − 1 1 ◦ 2 ˜ γ . □ V -UGS also provides a b ound on the solutions’ deriv a- tiv e after a full delay p eriod. T o establish this fact, we first mak e the following observ ation. Prop osition 4.9 (Bounds on the v ector field). If Assumptions 1 and 2 hold, then f is K -bounded , i.e., ther e exist ξ 1 , ξ 2 ∈ K such that | f ( ϕ, v ) | ≤ ξ 1 ( ∥ ϕ ∥ ) + ξ 2 ( | v | ) ∀ ϕ ∈ X n , v ∈ R m . (43) Pro of. Fix ϕ ∈ X n and v ∈ R m . Due to Lipschitz con tinuit y of f on b ounded balls w.r.t. the first argument, there is a strictly increasing con tinuous function L (char- acterizing a Lipsc hitz constant of f ), so that | f ( ϕ, v ) | ≤ | f (0 , v ) | + | f ( ϕ, v ) − f (0 , v ) | ≤ ξ ( | v | ) + L (max {∥ ϕ ∥ , | v |} ) ∥ ϕ ∥ , where ξ ( s ) := max | v |≤ s | f (0 , v ) | for s ≥ 0 . As f (0 , 0) = 0 b y Assumption 2, it holds that ξ (0) = 0 . Since f is con tinuous in its second argumen t, it also holds that ξ is a con tinuous nondecreasing function: see [17, Lemma A.27], and hence it can be upp er-b ounded b y a K ∞ -function. F urthermore, w e hav e | f ( ϕ, v ) | ≤ ξ ( | v | ) + L (max {∥ ϕ ∥ , | v |} ) max {∥ ϕ ∥ , | v |} ≤ ξ ( | v | ) + L ( ∥ ϕ ∥ ) ∥ ϕ ∥ + L ( | v | ) | v | . The function L ( · ) can be ma jorized by a contin uous in- creasing function, and thus r 7→ L ( r ) r can be ma jorized by a K ∞ function. These considerations establish the claim. □ Based on Prop osition 4.9, we hav e the following. Lemma 4.10 (Bounds on solution deriv ativ es). Given an LKF c andidate V : X n → R + , assume that (1) is V -UGS, and let Assumptions 1 and 2 hold. Then ther e exist µ 1 , µ 2 ∈ K ∞ such that, for al l x 0 ∈ X n and al l u ∈ U m , | ˙ x ( t, x 0 , u ) | ≤ µ 1 ( V ( x 0 )) + µ 2 ( ∥ u ∥ ) a.e. t ≥ θ. (44) Pro of. By Proposition 4.8, there exist σ 1 , σ 2 ∈ K ∞ suc h that, for all x 0 ∈ X n and all u ∈ U m , ∥ x t ( x 0 , u ) ∥ ≤ σ 1 ( V ( x 0 )) + σ 2 ( ∥ u ∥ ) , ∀ t ≥ θ . Let ξ 1 , ξ 2 ∈ K ∞ satisfy the assertion of Prop osition 4.9. Fix x 0 ∈ X n and u ∈ U m . The it follows from (43) that, for almost all t ≥ θ , | ˙ x ( t, x 0 , u ) | =   f ( x t ( x 0 , u ) , u ( t ))   ≤ ξ 1 ( ∥ x t ( x 0 , u ) ∥ ) + ξ 2 ( | u ( t ) | ) ≤ ξ 1  σ 1 ( V ( x 0 )) + σ 2 ( ∥ u ∥ )  + ξ 2 ( ∥ u ∥ ) ≤ ξ 1  2 σ 1 ( V ( x 0 ))  + ξ 1  2 σ 2 ( ∥ u ∥ )  + ξ 2 ( ∥ u ∥ ) and the claim follows with µ 1 := ξ 1 ◦ 2 σ 1 and µ 2 := ξ 1 ◦ 2 σ 2 + ξ 2 . □ 5. Pro of of Theorem 2.12 No w we can establish our main result (Theorem 2.12). W e restate it here for the sak e of readability . Theorem 2.12 L et Assumption 1 and 2 hold. If ther e exists an LKF c andidate V : X n → R + which is simulta- ne ously an ISS LKF with p ointwise dissip ation (in either implic ation or sum form) and a UGS LKF for (1) , then (1) is V -ISS and, in p articular, ISS. Pro of. In view of Prop osition 2.10, it is enough to assume that V is b oth an ISS LKF in implication form with p oin twise dissipation and a UGS LKF. F urthermore, b y Prop osition 2.8, w e can assume that the decay rate of V is a K ∞ -function (note that rescaled function constructed in Prop osition 2.8 will also b e a UGS-LKF for (1)). The pro of uses the V -ISS sup erposition theorem (Theo- rem 4.6) and pro ceeds to sho w that (1) is both V -UGS and V -ULIM. The former is a direct consequence of Prop osi- tion 4.7. F or the latter, we first fix χ ∈ K ∞ as a Lyapuno v gain as in Definitions 2.11 and 2.6 (if the Lyapuno v gains are differen t, we can define χ as the maxim um of b oth). Also let ψ 2 ∈ K ∞ b e an upp er b ound on V as in the sand- wic h condition (3). Now, seeking a contradiction, assume that the system (1) do es not satisfy the V -ULIM condi- tion (35) with the gain γ := ψ 2 ◦ 2 χ and some function τ = τ ( ε, r ) to be defined later. Hence, there are some r , ε > 0 , certain x 0 ∈ X n with V ( x 0 ) ≤ r , and some u ∈ U m with ∥ u ∥ ≤ r such that V ( x t ( x 0 , u )) ≥ ε + γ ( ∥ u ∥ ) , ∀ t ∈ [0 , τ ( r, ε )] . (45) By (3), it follows that ψ 2 ( ∥ x t ( x 0 , u ) ∥ ) ≥ ε + ψ 2 ◦ 2 χ ( ∥ u ∥ ) , ∀ t ∈ [0 , τ ( r , ε )] , (46) whic h in turn implies that ∥ x t ( x 0 , u ) ∥ ≥ max  ψ − 1 2 ( ε ) , 2 χ ( ∥ u ∥ )  , ∀ t ∈ [0 , τ ( r, ε )] . Hence, there exists an increasing finite sequence of time in- stan ts t k ∈ [0 , τ ( r, ε )] , k ∈ { 0 , 1 , . . . , K } , K ∈ N , satisfying t k − t k − 1 ≤ θ for all such k , such that | x ( t k , x 0 , u ) | ≥ max  ψ − 1 2 ( ε ) , 2 χ ( ∥ u ∥ )  . (47) Note that K ≥ τ ( ε, r ) θ − 1 . (48) By Lemma 4.10, there exist µ 1 , µ 2 ∈ K ∞ suc h that | ˙ x ( t, x 0 , u ) | ≤ µ 1 ( V ( x 0 )) + µ 2 ( ∥ u ∥ ) ≤ µ ( r ) , t ≥ θ a.e. , (49) where µ := µ 1 + µ 2 . F or each k , consider the interv al I k :=  t k − ψ − 1 2 ( ε ) 2 µ ( r ) , t k + ψ − 1 2 ( ε ) 2 µ ( r )  . As ψ 2 is an upper b ound for V , it can be c hosen arbitrarily large. Thus, we can assume that ψ 2 ( s ) ≥ s for all s ≥ 0 10 and that ψ − 1 2 ( ε ) µ ( r ) ≤ θ , so that the abov e in terv als do not o verlap. In view of (49), for all k ∈ { 0 , . . . , K } , w e ha ve for all t ∈ I k that | x ( t, x 0 , u ) | ≥ max  ψ − 1 2 ( ε ) 2 , 2 χ ( ∥ u ∥ ) − ψ − 1 2 ( ε ) 2  . Note that if c > max { a, 2 b − a } for some a, b, c ≥ 0 , then c > max { a, b } (consider a > b and a ≤ b ). It follows that | x ( t, x 0 , u ) | ≥ max  1 2 ψ − 1 2 ( ε ) , χ ( ∥ u ∥ )  , ∀ t ∈ I k . (50) Since ψ 2 ( s ) ≥ s for all s ≥ 0 , (45) ensures that V ( x t ( x 0 , u )) ≥ χ ( ∥ u ∥ ) , ∀ t ∈ [0 , τ ( r , ε )] . Consequen tly , we get from (20) that ˙ V u ( t + · ) ( x t ( x 0 , u )) ≤ 0 , ∀ t ∈ [0 , τ ( r , ε )] . (51) Using first [22, Lemma 3.4], and then (51), it follows that V ( x τ ( r,ε ) ( x 0 , u )) − V ( x 0 ) ≤ Z τ ( r,ε ) 0 ˙ V u ( t + · ) ( x t ( x 0 , u )) dt ≤ K X k =0 Z I k ˙ V u ( t + · ) ( x t ( x 0 , u )) dt. Using (11) and (50) on the interv als I k , w e get that V ( x τ ( r,ε ) ( x 0 , u )) − V ( x 0 ) ≤ − K X k =0 Z I k α ( | x ( t, x 0 , u ) | ) dt ≤ − K X k =0 Z I k α ◦ 1 2 ψ − 1 2 ( ε ) dt ≤ − ( K + 1) ψ − 1 2 ( ε ) µ ( r ) α ◦ 1 2 ψ − 1 2 ( ε ) ≤ − τ ( r, ε ) ψ − 1 2 ( ε ) θ µ ( r ) α ◦ 1 2 ψ − 1 2 ( ε ) , where the last inequality results from (48). This implies that r ≥ V ( x 0 ) ≥ τ ( r, ε ) ψ − 1 2 ( ε ) θ µ ( r ) α ◦ 1 2 ψ − 1 2 ( ε ) . (52) F or the particular c hoice τ ( r, ε ) := 4 r θµ ( r ) ψ − 1 2 ( ε ) α ◦ 1 2 ψ − 1 2 ( ε ) , (53) (52) yields a con tradiction. Thus, (1) satisfies the V -ULIM estimate (35) with the function τ given in (53) and the gain γ = ψ 2 ◦ 2 χ , whic h concludes the pro of. Thus the system is V -UGS and V -ULIM and therefore V -ISS by Theorem 4.6 and th us ISS by Prop osition 2.5. □ 6. Conclusion and p ersp ectiv es It has b een shown that ISS can b e inferred from an ISS LKF with p oin t wise dissipation, provided that the same LKF can b e used to establish UGS. F or this, w e hav e relied on a sup erposition principle for a v arian t of ISS, in whic h solutions are estimated through the LKF rather than through the classical ∥ · ∥ -norm of the state. As evi- denced in the example section, the result applies in situa- tions where previously kno wn sufficien t conditions do not suffice. While our result relaxes the ISS conditions imp osed in [7], the original question p osed in [4], namely whether a p oin t wise dissipation is enough to guaran tee ISS, remains op en. A p oten tial next step in that direction w ould b e to sho w that ISS indeed holds under a p oin twise dissipation if the system is assumed to b e UGS, thus without assuming a common LKF for b oth ISS and UGS. 7. App endix: ISS sup erp osition theorems 7.1. Criteria for V -ULS and V -UGS In this section, we provide analytic criteria for some of the V -stability properties that will be useful in the proof of the main results. Lemma 7.1. L et V b e an LKF c andidate for system (1) . Then (1) is V -ULS if and only if for every ε > 0 ther e exists a δ > 0 such that, for al l x 0 ∈ X n and al l u ∈ U m , V ( x 0 ) ≤ δ, ∥ u ∥ ≤ δ, t ≥ 0 ⇒ V  x t ( x 0 , u )  ≤ ε. (54) Pro of. " ⇒ ". Let (1) b e V -ULS for the giv en LKF candidate V . L et σ, γ ∈ K ∞ and r > 0 be such that (33) holds for these functions and the neighborho od specified b y r . Let ε > 0 b e arbitrary and choose δ = δ ( ε ) := min n σ − 1  ε 2  , γ − 1  ε 2  , r o . With this c hoice, (54) follows from (33). " ⇐ " Let (54) hold. F or ε ≥ 0 define δ ( ε ) := sup { s ≥ 0 : V ( x 0 ) ≤ s ∧ ∥ u ∥ ≤ s ⇒ sup t ≥ 0 V  x t ( x 0 , u )  ≤ ε } . Clearly , (54) implies that δ ( · ) is w ell defined, increasing and contin uous in 0 . By [17, Prop osition A.16], there exists ˆ δ ∈ K with ˆ δ ≤ δ . Set r := sup s ≥ 0 ˆ δ ( s ) ∈ R + ∪ {∞} and define γ := ˆ δ − 1 : [0 , r ) → R , and extend it in an arbitrary w ay to a K ∞ -function. Then for V ( x 0 ) < r and ∥ u ∥ < r , w e hav e V  x t ( x 0 , u )  ≤ γ (max { V ( x 0 ) , ∥ u ∥} ) ≤ γ ( V ( x 0 )) + γ ( ∥ u ∥ ) , whic h shows V -ULS. □ It is useful to hav e a quantitativ e restatement of the V - BRS prop ert y using estimates of comparison t yp e. The 11 pro of of the following result is inspired b y a similar pro of in [22, Lemma 2.12]. In the statemen t, we call a function h : R 3 + → R + increasing, if ( r 1 , r 2 , r 3 ) ≤ ( R 1 , R 2 , R 3 ) im- plies that h ( r 1 , r 2 , r 3 ) ≤ h ( R 1 , R 2 , R 3 ) , where we use the comp onen t-wise partial order on R 3 + . W e call h strictly increasing if ( r 1 , r 2 , r 3 ) ≤ ( R 1 , R 2 , R 3 ) and ( r 1 , r 2 , r 3 )  = ( R 1 , R 2 , R 3 ) imply h ( r 1 , r 2 , r 3 ) < h ( R 1 , R 2 , R 3 ) . Lemma 7.2. Consider an LKF c andidate V for (1) . The fol lowing statements ar e e quivalent: (i) (1) has V -b ounde d r e achability sets. (ii) Ther e exists a c ontinuous, incr e asing function µ : R 3 + → R + , such that for al l x 0 ∈ X n , u ∈ U m and al l t ≥ 0 we have V  x t ( x 0 , u )  ≤ µ ( V ( x 0 ) , ∥ u ∥ , t ) . (55) (iii) Ther e exists a c ontinuous function µ : R 3 + → R + such that for al l x 0 ∈ X n , u ∈ U m and al l t ≥ 0 the in- e quality (55) holds. Pro of. (i) ⇒ (ii). Define ˜ µ : R 3 + → R + b y ˜ µ ( C 1 , C 2 , τ ) := sup V ( x 0 ) ≤ C 1 , ∥ u ∥≤ C 2 , t ∈ [0 ,τ ] V  x t ( x 0 , u )  , (56) whic h is well-defined in view of the item (i). Clearly , ˜ µ is increasing by definition. In particular, it is lo cally inte- grable. Define ˆ µ : (0 , + ∞ ) 3 → R + b y setting for C 1 , C 2 , τ > 0 ˆ µ ( C 1 , C 2 , τ ) := 1 C 1 C 2 τ Z 2 C 1 C 1 Z 2 C 2 C 2 Z 2 τ τ ˜ µ ( r 1 , r 2 , s ) dsdr 2 dr 1 + C 1 C 2 τ . By construction, ˆ µ is strictly increasing and contin uous on (0 , + ∞ ) 3 . W e can enlarge the domain of definition of ˆ µ to all of R 3 + using monotonicity . T o this end w e define for C 2 , τ > 0 : ˆ µ (0 , C 2 , τ ) := lim C 1 → +0 ˆ µ ( C 1 , C 2 , τ ) , for C 1 ≥ 0 , τ > 0 : ˆ µ ( C 1 , 0 , τ ) := lim C 2 → +0 ˆ µ ( C 1 , C 2 , τ ) and for C 1 , C 2 ≥ 0 w e define ˆ µ ( C 1 , C 2 , 0) := lim τ → +0 ˆ µ ( C 1 , C 2 , τ ) . All these limits are well-defined as ˆ µ is increasing on (0 , + ∞ ) 3 , and w e obtain that the resulting function is increasing on R 3 + . Note that the construction do es not guarantee that ˆ µ is con tinuous. T o obtain contin uit y , choose a contin uous strictly increasing function ν : R + → R + with ν ( r ) > max { ˆ µ (0 , 0 , r ) , ˆ µ (0 , r, 0) , ˆ µ ( r , 0 , 0) } , r ≥ 0 , and define for ( C 1 , C 2 , τ ) ≥ (0 , 0 , 0) µ ( C 1 , C 2 , τ ) := max  ν  max { C 1 , C 2 , τ }  , ˆ µ ( C 1 ,C 2 , τ )  + C 1 C 2 τ . The function µ is con tinuous as µ ( C 1 , C 2 , τ ) = ν  max { C 1 , C 2 , τ }  + C 1 C 2 τ whenev er C 1 , C 2 or τ is small enough. At the same time w e hav e for C 1 > 0 , C 2 > 0 and τ > 0 that µ ( C 1 , C 2 , τ ) ≥ ˆ µ ( C 1 , C 2 , τ ) ≥ 1 C 1 C 2 τ Z 2 C 1 C 1 Z 2 C 2 C 2 Z 2 τ τ 1 dsdr 2 dr 1 ˜ µ ( C 1 , C 2 , τ ) + C 1 C 2 τ ≥ ˜ µ ( C 1 , C 2 , τ ) . This implies that (ii) holds with this µ . (ii) ⇒ (iii) is eviden t. (iii) ⇒ (i) follo ws due to contin uity of µ . □ The next result states that a similar c haracterization can b e derived for the V -UGS prop erty , in which the upp er b ound is time-indep enden t. Prop osition 7.3. Consider system (1) and an LKF c an- didate V . System (1) is V -UGB if and only if the c ondition of L emma 7.2 (ii) holds with a µ that do es not dep end on t . Pro of. It is immediate from the definitions that if (1) is V -UGB, then (1) is V -BRS and µ in Lemma 7.2 can b e c hosen as µ ( V ( x 0 ) , ∥ u ∥ ) := σ ( V ( x 0 )) + γ ( ∥ u ∥ ) + c with σ, γ ∈ K ∞ and c ≥ 0 . Conv ersely , let Lemma 7.2 (ii) hold with a con tinuous increasing µ = µ ( V ( x 0 ) , ∥ u ∥ ) . Then for an y t ≥ 0 , x 0 ∈ X n , u ∈ U m it holds that V  x t ( x 0 , u )  ≤ max  µ ( V ( x 0 ) , V ( x 0 )) , µ ( ∥ u ∥ , ∥ u ∥ )  . By assumption, ˜ σ : r 7→ µ ( r, r ) is a contin uous increasing function. Define σ ( r ) := ˜ σ ( r ) − ˜ σ (0) . Then σ ∈ K ∞ and w e hav e for any t ≥ 0 , x 0 ∈ X n , u ∈ U m : V  x t ( x 0 , u )  ≤ σ ( V ( x 0 )) + σ ( ∥ u ∥ ) + ˜ σ (0) , whic h shows that (1) is V -UGB. □ Finally , we can characterize uniform global stabilit y in a similar w ay . Lemma 7.4. Consider system (1) and an LKF c andidate V . Then (1) is V -UGS if and only if it is V -ULS and V -UGB. Pro of. It is immediate from the definitions that V - UGS implies b oth V -ULS and V -UGB. F or the con verse direction, assume that (1) is V -ULS and V -UGB. This means, that there exist σ 1 , γ 1 , σ 2 , γ 2 ∈ K ∞ and r, c > 0 suc h that, for all x 0 with V ( x 0 ) ≤ r and all u with ∥ u ∥ ≤ r , V  x t ( x 0 , u )  ≤ σ 1 ( V ( x 0 )) + γ 1 ( ∥ u ∥ ) ∀ t ≥ 0 , and suc h that for all x 0 ∈ X n and all u ∈ U m the follo wing estimate holds: V  x t ( x 0 , u )  ≤ σ 2 ( V ( x 0 )) + γ 2 ( ∥ u ∥ ) + c ∀ t ≥ 0 . 12 Assume without restriction that σ 2 ( s ) ≥ σ 1 ( s ) and γ 2 ( s ) ≥ γ 1 ( s ) for all s ≥ 0 . Pic k k 1 , k 2 > 0 so that c = k 1 σ 2 ( r ) and c = k 2 γ 2 ( r ) . Then for all ( x 0 , u ) ∈ X n × U m with V ( x 0 ) ≥ r or ∥ u ∥ ≥ r w e hav e c ≤ k 1 σ 2 ( V ( x 0 )) + k 2 γ 2 ( ∥ u ∥ ) . Th us for all x 0 ∈ X n and all u ∈ U m it holds that V  x t ( x 0 , u )  ≤ (1 + k 1 ) σ 2 ( V ( x 0 )) + (1 + k 2 ) γ 2 ( ∥ u ∥ ) . This sho ws V -UGS of (1). □ 7.2. A uxiliary lemmas for the V -ISS sup erp osition the o- r em W e pro ceed with a sequence of lemmas needed to sho w Theorem 4.6. Lemma 7.5. L et V ∈ C ( X n , R + ) b e an LKF c andidate. If (1) is V -ISS, then (1) is V -GUAG. Pro of. Let (1) b e V -ISS with the corresp onding β ∈ K L and γ ∈ K ∞ . T ake arbitrary ε, r > 0 . Define τ = τ ( ε, r ) as the solution of the equation β ( r , τ ) = ε (if it exists, then it is unique b ecause of the monotonicit y of β in the second argumen t, if it does not exist, w e set τ ( ε, r ) := 0 ). Then for all t ≥ τ , all x 0 ∈ X n with V ( x 0 ) ≤ r and all u ∈ U m w e hav e V ( x t ( x 0 , u )) ≤ β ( V ( x 0 ) , t ) + γ ( ∥ u ∥ ) ≤ β ( V ( x 0 ) , τ ) + γ ( ∥ u ∥ ) ≤ ε + γ ( ∥ u ∥ ) , and the implication (36) holds. □ Lemma 7.6. L et V ∈ C ( X n , R + ) b e an LKF c andidate. If (1) is V -UAG and V -CEP, then it is V -ULS. Pro of. W e will show that (54) holds so that the claim follo ws from Lemma 7.1. Let τ and γ b e the functions from Definition 4.1 (vi). Let ε > 0 and τ := τ ( ε/ 2 , 1) . Pic k any δ 1 ∈ (0 , 1] such that γ ( δ 1 ) < ε/ 2 . Then for all x 0 ∈ X n with V ( x 0 ) ≤ 1 and all u ∈ U m with ∥ u ∥ ≤ δ 1 w e hav e sup t ≥ τ V ( x t ( x 0 , u )) ≤ ε 2 + γ ( ∥ u ∥ ) < ε. (57) Since (1) is V -CEP , there is some δ 2 = δ 2 ( ε, τ ) > 0 so that V ( η ) ≤ δ 2 ∧ ∥ u ∥ ≤ δ 2 ⇒ sup t ∈ [0 ,τ ] V ( x t ( η , u )) ≤ ε. T ogether with (57), this prov es (54) with δ := min { 1 , δ 1 , δ 2 } . □ Prop osition 7.7. L et V ∈ C ( X n , R + ) b e an LKF c an- didate. If (1) is V -BRS and has the V -ULIM pr op erty. Then (1) is V -UGB. Pro of. Let γ ∈ K ∞ ∪ { 0 } and τ b e given by the V -ULIM prop erty according to Definition 4.1 (iv). Pick an y r > 0 and set ε := r 2 . Since (1) has the V - ULIM prop ert y , there exists a τ = τ ( ϵ, r ) (more precisely , τ = τ ( r 2 , max { r, γ − 1 ( r 4 ) } ) from the V -ULIM prop ert y), suc h that V ( x 0 ) ≤ r , ∥ u ∥ ≤ γ − 1 ( r 4 ) ⇒ ∃ t ≤ τ : V  x t ( x 0 , u )  ≤ r 2 + γ ( ∥ u ∥ ) ≤ 3 r 4 . (58) Without loss of generality , w e can assume that τ is increas- ing in r . In particular, it is lo cally integrable. Defining ¯ τ ( r ) := 1 r R 2 r r τ ( s ) ds for r > 0 , we see that ¯ τ ( r ) ≥ τ ( r ) and ¯ τ is contin uous. F or any r 2 > r 1 > 0 via the c hange of v ariables s = r 2 r 1 w , we hav e also that ¯ τ ( r 2 ) = 1 r 2 Z 2 r 2 r 2 τ ( s ) ds = 1 r 2 Z 2 r 1 r 1 τ  r 2 r 1 w  r 2 r 1 dw > 1 r 1 Z 2 r 1 r 1 τ ( w ) dw = ¯ τ ( r 1 ) , and thus ¯ τ is increasing. W e define further ¯ τ (0) := lim r → +0 ¯ τ ( r ) (the limit exists as ¯ τ is increasing). Since (1) is V -BRS, Lemma 7.2 implies that there exists a contin uous, increasing function µ : R 3 + → R + , such that for all x 0 ∈ X n , u ∈ U m and all t ≥ 0 the estimate (55) holds. F rom this w e conclude that we hav e the implication V ( x 0 ) ≤ r , ∥ u ∥ ≤ γ − 1 ( r 4 ) , t ≤ ¯ τ ( r ) ⇒ V  x t ( x 0 , u )  ≤ ˜ σ ( r ) , (59) where ˜ σ : r 7→ µ  r , γ − 1 ( r 4 ) , ¯ τ ( r )  , r ≥ 0 . Note that ˜ σ is con tinuous and increasing, since µ, γ , ¯ τ are contin uous, increasing functions. W e see from (59) that ˜ σ ( r ) ≥ V ( x 0 ) whenever V ( x 0 ) ≤ r , and thus ˜ σ ( r ) ≥ r > 3 r 4 for an y r > 0 . Assume that there exist x 0 with V ( x 0 ) ≤ r , u ∈ U m with ∥ u ∥ ≤ γ − 1 ( r 4 ) and t ≥ 0 suc h that V  x t ( x 0 , u )  > ˜ σ ( r ) . Define t m := sup { s ∈ [0 , t ] : V  x s ( x 0 , u )  ≤ r } ≥ 0 . The quantit y t m is well-defined, since V ( x 0 ) ≤ r . In view of the co cycle prop ert y , it holds that x t ( x 0 , u ) = x t − t m ( x t m ( x 0 , u ) , u ( · + t m )) , where clearly u ( · + t m ) ∈ U m . Assume that t − t m ≤ τ ( r ) . Since V ( x t m ( x 0 , u )) ≤ r , (59) implies that V ( x t ( x 0 , u )) ≤ ˜ σ ( r ) for all t ∈ [ t m , t ] . Otherwise, if t − t m > τ ( r ) , then due to (58) there exists t ∗ < τ ( r ) , so that V  x t ∗  x t m ( x 0 , u ) , u ( · + t m )  = V ( x t ∗ + t m ( x 0 , u )) ≤ 3 r 4 , whic h con tradicts the definition of t m , since t m + t ∗ < t . Hence V ( x 0 ) ≤ r , ∥ u ∥ ≤ γ − 1 ( r 4 ) , t ≥ 0 ⇒ V ( x t ( x 0 , u )) ≤ ˜ σ ( r ) . (60) 13 Denote σ ( r ) := ˜ σ ( r ) − ˜ σ (0) , for any r ≥ 0 . Clearly , σ ∈ K ∞ . F or eac h x 0 ∈ X n , u ∈ U m define r := max { V ( x 0 ) , 4 γ ( ∥ u ∥ ) } . Then (60) immediately shows for all x 0 ∈ X n , u ∈ U m , t ≥ 0 that V ( x t ( x 0 , u )) ≤ σ  max { V ( x 0 ) , 4 γ ( ∥ u ∥ ) }  + ˜ σ (0) ≤ σ ( V ( x 0 )) + σ  4 γ ( ∥ u ∥ )  + ˜ σ (0) , whic h shows that (1) is V -UGB. □ Lemma 7.8. L et V ∈ C ( X n , R + ) b e an LKF c andidate. If (1) is V -ULIM and V -UGS, then (1) is V -UAG. Pro of. Let γ 1 , σ ∈ K ∞ b e the functions asso ciated to V -UGS, see Definition 4.1 (ii) and let γ 2 ∈ K ∞ b e the gain asso ciated to V -ULIM, see Definition 4.1 (iv). Consider γ : s 7→ max { γ 1 ( s ) , γ 2 ( s ) } , s ≥ 0 , and note that γ can b e used in the estimates b oth for V -ULIM and V -UGS. Define ˜ γ ( s ) = σ (2 γ ( s )) + γ ( s ) , s ≥ 0 . W e claim that with this gain the V -UAG estimates hold. T o this end, fix ε > 0 , r > 0 and let ˜ ε := 1 2 σ − 1 ( ε ) > 0 . By the V -ULIM property , there exists τ = τ ( ˜ ε, r ) suc h that, for all x 0 with V ( x 0 ) ≤ r and all u ∈ U m with ∥ u ∥ ≤ r , there is a t ≤ τ such that V ( x t ( x 0 , u )) ≤ ˜ ε + γ ( ∥ u ∥ ) . (61) In view of the cocycle property and as the system is V -UGS, we hav e for the ab o ve x 0 , u, t and any s ≥ 0 V ( x s + t ( x 0 , u )) = V  x s ( x t ( x 0 , u ) , u ( t + · ))  ≤ σ  V  x t ( x 0 , u )  + γ ( ∥ u ∥ ) ≤ σ  ˜ ε + γ ( ∥ u ∥ )  + γ ( ∥ u ∥ ) . Using the inequality σ ( a + b ) ≤ σ (2 a ) + σ (2 b ) , v alid for an y a, b ≥ 0 , we obtain that V ( x s + t ( x 0 , u )) ≤ ε + ˜ γ ( ∥ u ∥ ) , s ≥ 0 . Ov erall, the system is V -UAG with gain ˜ γ and threshold ˜ τ = ˜ τ ( ε, r ) := τ ( ˜ ε, r ) . □ The following lemma shows how the V -UAG prop ert y can be “upgraded” to the V -GUAG prop ert y if V -UGS holds. Lemma 7.9. L et V ∈ C ( X n , R + ) b e an LKF c andidate. If (1) is V -UAG and V -UGS, then (1) is V -GUAG. Pro of. Pick any ε > 0 , r > 0 , and let τ and γ b e as in the form ulation of the V -UAG prop erty . Let x 0 ∈ X n with V ( x 0 ) ≤ r , and u ∈ U m b e arbitrary . If ∥ u ∥ ≤ r , then (36) is the desired implication. Let ∥ u ∥ > r . Hence it holds that ∥ u ∥ > V ( x 0 ) . Due to V -UGS of (1), it holds for all t, x 0 , u that V ( x t ( x 0 , u )) ≤ σ ( V ( x 0 )) + γ ( ∥ u ∥ ) , where w e assume that γ is the same as in the definition of a V -UAG prop ert y (otherwise pic k the maxim um of both). F or ∥ u ∥ > V ( x 0 ) w e obtain that V ( x t ( x 0 , u )) ≤ σ ( ∥ u ∥ ) + γ ( ∥ u ∥ ) , and th us for all x 0 ∈ X n , u ∈ U m it holds that t ≥ τ ⇒ V ( x t ( x 0 , u )) ≤ ε + γ ( ∥ u ∥ ) + σ ( ∥ u ∥ ) , whic h sho ws the V -GUAG prop erty with the asymptotic gain γ + σ . □ The final tec hnical lemma of this section is the following. Lemma 7.10. L et V ∈ C ( X n , R + ) b e an LKF c andidate. If (1) is V -GUAG and V -UGS, then (1) is V -ISS. Pro of. Assume that (1) is V -GUA G and V -UGS and that γ in (33) and (36) are the same (otherwise, let γ b e the maxim um of the tw o). Fix an arbitrary r ∈ R + . W e are going to construct a function β ∈ K L so that (4) holds. F rom V -UGS, there exist γ , σ ∈ K ∞ suc h that for all t ≥ 0 , all x 0 ∈ X n with V ( x 0 ) ≤ r , and all u ∈ U m , V ( x t ( x 0 , u )) ≤ σ ( r ) + γ ( ∥ u ∥ ) . (62) Define ε n := 2 − n σ ( r ) , for n ∈ N . The V -GUAG prop- ert y implies that there exists a sequence of times τ n := τ ( ε n , r ) , which we ma y without loss of generality assume to b e strictly increasing, such that for all x 0 ∈ X n with V ( x 0 ) ≤ r , and all u ∈ U m V ( x t ( x 0 , u )) ≤ ε n + γ ( ∥ u ∥ ) , ∀ t ≥ τ n . F rom (62), we see that w e may set τ 0 := 0 . Define ω ( r , τ n ) := ε n − 1 , for n ∈ N and ω ( r, 0) := 2 ε 0 = 2 σ ( r ) . No w extend the definition of ω to a function ω ( r, · ) ∈ L . W e obtain for t ∈ ( τ n , τ n +1 ) , n = 0 , 1 , . . . that whenever V ( x 0 ) ≤ r and u ∈ U m , it holds that V ( x t ( x 0 , u )) ≤ ε n + γ ( ∥ u ∥ ) < ω ( r , t ) + γ ( ∥ u ∥ ) . Doing this for all r ∈ R + , w e obtain the definition of the function ω . No w define ˆ β ( r, t ) := sup 0 ≤ s ≤ r ω ( s, t ) ≥ ω ( r, t ) for ( r , t ) ∈ R 2 + . F rom this definition, it follows that, for eac h t ≥ 0 , ˆ β ( · , t ) is increasing in r and ˆ β ( r, · ) is decreasing in t for each r > 0 as every ω ( r , · ) ∈ L . Moreo ver, for each fixed t ≥ 0 , ˆ β ( r, t ) ≤ sup 0 ≤ s ≤ r ω ( s, 0) = 2 σ ( r ) . This im- plies that ˆ β is con tinuous in the first argument at r = 0 for an y fixed t ≥ 0 . By [21, Prop osition 9], ˆ β can b e upp er b ounded b y cer- tain ˜ β ∈ KL , and the estimate (4) is satisfied with such a β . □ 7.3. ISS sup erp osition r esult via mixe d stability notions Our pro of technique for the main results of this work is based on the use of Theorem 4.6 ( V -ISS sup erposition the- orem for delay systems) characterizing V -ISS using com- binations of nominally weak er notions of stabilit y . Here 14 w e emplo y v ariations of the stability concepts form ulated through an LKF candidate V to show a further ISS sup er- p osition result. The main difference to previous concepts is that the requirement on the initial condition x 0 for a dynamic prop ert y to hold is form ulated in terms of the norm ∥ · ∥ as opp osed to using the LKF candidate V . Definition 7.11. L et V ∈ C ( X n , R + ) b e an LKF c andi- date as in Definition 2.1. System (1) is said to have the mixed V -uniform limit prop ert y (mixed V -ULIM) , if ther e exists γ ∈ K ∪ { 0 } such that for every ε > 0 and for every r > 0 ther e exists a τ = τ ( ε, r ) such that for al l x 0 with ∥ x 0 ∥ ≤ r and al l u ∈ U m with ∥ u ∥ ≤ r ther e is a t ≤ τ such that V ( x t ( x 0 , u )) ≤ ε + γ ( ∥ u ∥ ) . (63) Remark 7.12. L et V b e a fixe d LKF c andidate. Then if (1) is V -ULIM, it is also mixe d V -ULIM. Inde e d, let (1) b e V -ULIM. If x 0 satisfies ∥ x 0 ∥ ≤ r then V ( x 0 ) ≤ ψ 2 ( r ) , se e (3) . Henc e the V -ULIM pr op erty imme diately yields the mixe d V -ULIM pr op erty. Prop osition 7.13. L et V ∈ C ( X n , R + ) b e an LKF c an- didate. If (1) is mixe d V -ULIM and V -UGS, then (1) is ISS. Pro of. First of all, by Prop osition 4.8, V -UGS implies that there are ¯ σ , ¯ γ ∈ K ∞ suc h that for all t ≥ θ , x 0 ∈ X n and u ∈ U m w e hav e ∥ x t ( x 0 , u ) ∥ ≤ ¯ σ ( V ( x 0 )) + ¯ γ ( ∥ u ∥ ) . (64) Fix ε > 0 and r > 0 . By the mixed V -ULIM property , there exists γ ∈ K ∞ , independent of ε and r , and τ = τ ( ε, r ) so that for all x 0 with ∥ x 0 ∥ ≤ r and all u ∈ U m with ∥ u ∥ ≤ r there is a t ≤ τ such that V ( x t ( x 0 , u )) ≤ ε + γ ( ∥ u ∥ ) . (65) In view of the co cycle prop ert y , and the estimate (64), w e hav e for the ab o ve x 0 , u, t and any s ≥ θ ∥ x s + t ( x 0 , u ) ∥ =   x s ( x t ( x 0 , u ) , u ( t + · ))   ≤ ¯ σ  V  x t ( x 0 , u )  + ¯ γ ( ∥ u ∥ ) ≤ ¯ σ  ε + γ ( ∥ u ∥ )  + ¯ γ ( ∥ u ∥ ) . No w let ˜ ε := ¯ σ (2 ε ) > 0 . Using again σ ( a + b ) ≤ σ (2 a ) + σ (2 b ) , we pro ceed to ∥ x s + t ( x 0 , u ) ∥ ≤ ˜ ε + ˜ γ ( ∥ u ∥ ) , where ˜ γ ( r ) = ¯ σ (2 γ ( r )) + ¯ γ ( r ) , r ≥ 0 . Ov erall, for an y ˜ ε > 0 and an y r > 0 there exists ˜ τ = ˜ τ ( ˜ ε, r ) = τ ( 1 2 σ − 1 ( ˜ ε ) , r ) + θ , so that ∥ x 0 ∥ ≤ r ∧ ∥ u ∥ ≤ r ∧ t ≥ ˜ τ ⇒ ∥ x t ( x 0 , u ) ∥ ≤ ˜ ε + ˜ γ ( ∥ u ∥ ) , whic h shows the UAG prop ert y of (1). 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