A characterization of strong iISS for time-varying impulsive systems
Hernan Haimovich Jose L. Mancilla-Aguilar Paula Cardone
eess.SY cs.SY math.OC






A characterization of strong iISS
 for time-varying impulsive systemsE-mails: , 
    Hernan Haimovich
Centro de Ciencias de la
 Información y Sistemas (CIFASIS) 

CONICET – UNR

2000 Rosario, Argentina.
José Luis Mancilla-Aguilar
Centro de Sistemas y Control 

Instituto Tecnológico de Buenos Aires

Av. Eduardo Madero 399,
 Buenos Aires, Argentina.
Paula Cardone
CIFASIS 

CONICET – UNR

Ocampo y Esmeralda,
 2000 Rosario, Argentina.

    December 30, 2023
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  For general time-varying or switched (nonlinear) systems, converse Lyapunov theorems for stability are not available. In these cases, the integral input-to-state stability (iISS) property is not equivalent to the existence of an iISS-Lyapunov function but can still be characterized as the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded energy input-bounded state (UBEBS). For impulsive systems, asymptotic stability can be weak (when the asymptotic decay depends only on elapsed time) or strong (when such a decay depends also on the number of impulses that occurred). This paper shows that the mentioned characterization of iISS remains valid for time-varying impulsive systems, provided that stability is understood in the strong sense.



Stability, impulsive systems, time-varying systems, bounded energy, nonlinear systems.




§ INTRODUCTION


Impulsive systems are dynamical systems whose state evolves continuously most of the time but may exhibit jumps (discontinuities) at isolated time instants (see <cit.>). The continuous evolution of the state (i.e. between jumps) is governed by ordinary differential equations. The time instants when jumps occur are part of the impulsive system definition and the after-jump value of the state vector is governed by a static (i.e. not differential) equation. Uniform asymptotic stability of the origin requires that the norm of the state decays asymptotically to zero as elapsed time advances. For an impulsive system, uniform asymptotic stability can be defined in two different ways, depending on whether the decay depends only on elapsed time (we use the name weak in this case) <cit.> or also on the number of impulses that have occurred (strong) <cit.>.

Input-to-state stability (ISS) <cit.> and integral-ISS (iISS) <cit.> are arguably the most important and useful state-space based nonlinear notions of stability for systems with inputs. The iISS property gives a state bound that is the sum of a decaying-to-zero term whose amplitude depends only on the initial state, and a term depending (nonlinearly) only on an integral of a nonlinear function of the input. The latter term can be interpreted as an input energy bound. As is the case with uniform asymptotic stability, for impulsive systems the decaying-to-zero term can take impulse occurrence into account or not, giving rise to two different ways of defining iISS (strong or weak). The weak iISS property is the most usual for impulsive systems <cit.>, whereas the strong version is in agreement with iISS for hybrid systems <cit.>.





Several different sufficient conditions for weak iISS of impulsive systems involving time-invariant or time-varying flow and jump equations, with or without time delays, exist <cit.>. However, to the best of our knowledge, conditions that are both necessary and sufficient only exist for strong iISS when the impulsive system can be posed as a time-invariant hybrid system where the (time-invariant) flow map, in addition, satisfies a convexity property with respect to the input variable <cit.>. For time-invariant nonimpulsive systems, iISS was shown to be equivalent to the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded-energy input bounded state (UBEBS). This characterization of iISS was extended to time-varying and switched (nonimpulsive) systems <cit.>, and has been recently shown to remain valid for impulsive systems provided stability is understood in the weak sense and the number of jumps that occur in any given time interval is bounded in relation to the interval's length but irrespective of initial time <cit.>.

In this paper, we show that the previously derived characterization of iISS (namely, iISS = 0-GUAS + UBEBS) remains valid for impulsive systems provided stability is understood in the strong sense and without having to bound the number of jumps as in the weak case. As was the case with the previous results <cit.>, the current results apply to cases where both the ordinary differential equation defining continuous state evolution (i.e. the flow equation) and the static equation defining after-jump values (i.e. the jump equation) can be time-varying and lack time continuity. The results of <cit.> are then shown to be a particular case of the current ones. 



Notation. , , _>0 and _≥ 0 denote the natural numbers, reals, positive reals and nonnegative reals, respectively. |x| denotes the Euclidean norm of x ∈^p. 

 We write α∈ if α:_≥ 0→_≥ 0 is continuous, strictly increasing and α(0)=0, and α∈ if, in addition, α is unbounded. We write β∈ if β:_≥ 0×_≥ 0→_≥ 0, β(·,t)∈ for any t≥ 0 and, for any fixed r≥ 0, β(r,t) monotonically decreases to zero as t→∞. For every n∈ and r≥ 0, we define the closed ball B_r^n := {x∈^n : |x| ≤ r}. A function h : D ⊂×^n →^n is said to be a Carathéodory function if h(t,ξ) is measurable in t for fixed ξ, continuous in ξ for fixed t, and for every compact set K ⊂ D, there exists an integrable function m_K(t) such that |h(t,ξ)| ≤ m_K(t) for all (t,ξ)∈ K (see <cit.>).



§ PROBLEM STATEMENT




 §.§ Impulsive systems



Consider the time-varying impulsive system with inputs

  
  
    ẋ(t)    =f(t,x(t),u(t)),x(t^-)+g^-  for  t∉σ,    [0] 
    
        x(t)    =x(t^-) + g(t,x(t^-),u(t)),f  for  t∈σ,


where t_0≥ 0 is the initial time, σ={τ_k}_k=1^N, with N finite or N=∞, is a strictly increasing sequence of impulse times in _>0, the state variable x(t)∈^n, the continuous-time input variable u(t)∈^m and f (the flow map) and g (the jump map) are functions from _≥ 0×^n×^m to ^n. The ordinary differential equation (<ref>) defines the continuous evolution of the state vector x and (<ref>) defines the value of x at the impulse times. To ensure that the jumps in x caused by (<ref>) cannot occur infinitely frequently, it is assumed that τ_k→∞ when N=∞. By convention we define τ_0=0 (however, τ_0 is not considered an impulse time) and, when N is finite, we set τ_N+1:=∞. We will employ  to denote the set of all these admissible impulse time sequences, i.e.  denotes the set of all strictly increasing sequences of positive real numbers that either have a finite number of elements or are unbounded. Let 𝒰 be the set of all the functions u:_≥ 0→^m that are Lebesgue measurable and locally bounded. We will use the term “input” to refer to a pair w=(u,σ) ∈× consisting of a continuous-time input u and an admissible impulse-time sequence σ.  We assume that for each u∈ the map f_u(t,ξ):=f(t,ξ,u(t)) is a Carathéodory function and hence the (local) existence of solutions of the differential equation ẋ(t)=f(t,x(t),u(t)) is ensured (see <cit.>).




A solution to (<ref>) corresponding to an initial time t_0, an initial state x_0∈^n and an input w=(u,σ) ∈× is a right-continuous function x:[t_0,T_x)→^n such that:


  * x(t_0)=x_0; 

  * x is a Carathéodory solution of the differential equation ẋ(t)=f(t,x(t),u(t)) on [τ_k,τ_k+1)∩ [t_0,T_x) for all 0≤ k ≤ N; and 

  * for all t∈σ∩ (t_0,T_x) it happens that x(t)=x(t^-)+ g(t,x(t^-),u(t)), where x(t^-) := lim_s→ t^- x(s).

The solution x is said to be maximally defined if no other solution y satisfies y(t) = x(t) for all t∈ [t_0,T_x) and has T_y > T_x. A solution x is forward complete if T_x=∞. We will use (t_0,x_0,w) to denote the set of maximally defined solutions of (<ref>) corresponding to initial time t_0, initial state x_0, and input w. We say that (<ref>) is forward complete for a given σ∈ if for every t_0 ≥ 0, x_0 ∈^n and w=(u,σ) with u∈, any solution x∈(t_0,x_0,w) is forward complete. Given σ∈, we define n^σ_(t_0,t] to be the number of elements of σ (i.e. the number of jumps) that lie in the interval (t_0,t]:

    n^σ_(t_0,t]   := #[ σ∩ (t_0,t] ].




 §.§ Stability definitions



Stability notions for systems with inputs that are uniform with respect to initial time, such as uniform ISS and iISS, bound the state trajectory in relation to initial state, elapsed time and input. In the context of impulsive systems, the input can be interpreted as having both a continuous-time and an impulsive component. 
Given an input w = (u,σ) and ρ_1,ρ_2∈, we thus define

    w _(ρ_1,ρ_2)   := ∫_0^∞ρ_1(|u(s)|) ds + ∑_t∈σρ_2(|u(t)|).

The quantity defined in (<ref>) can be loosely interpreted as a measure of the energy content of an input that has some impulsive behaviour at the time instants t∈σ.

We are interested in determining whether some stability property holds not just for a single impulse-time sequence σ∈ but also for some family ⊂. 
We thus consider the uniform stability notions given in Definition <ref>. To simplify notation, for every interval J ⊂ [0,∞) and u∈, we define u_J via u_J(t) := u(t) if t∈ J and u_J(t) := 0 otherwise; for an input w=(u,σ), we define w_J := (u_J,σ). 

  
  Given ⊂, we say that the impulsive system (<ref>)
  is
  
  
  * strongly 0-GUAS uniformly over (the family of impulse-time sequences)  if there exists β∈ such that
    
    |x(t)|    ≤β(|x(t_0)|,t-t_0+n^σ_(t_0,t])   ∀ t≥ t_0,

    for every x∈(t_0,x_0,w_0) with t_0 ≥ 0, x_0 ∈^n
    and w_0=(0,σ) with σ∈.
  
  
  
  
  
  
  
  * UBEBS uniformly over  if there exist α,ρ_1,ρ_2∈ and c≥ 0 such that
    
    α(|x(t)|)    ≤ |x(t_0)| +  w_(t_0,t]_(ρ_1,ρ_2) + c   ∀ t≥ t_0,

    for every x∈(t_0,x_0,w) with t_0 ≥ 0, x_0 ∈^n and w ∈×. The pair (ρ_1,ρ_2) will be referred to as an UBEBS gain.
  
  * strongly iISS uniformly over  if there exist β∈ and α,ρ_1,ρ_2 ∈ such that
    
    α(|x(t)|) ≤β(|x(t_0)|,t-t_0+n^σ_(t_0,t])
    
          +  w_(t_0,t]_(ρ_1,ρ_2)

    for all t≥ t_0, for every x∈(t_0,x_0,w) with t_0 ≥ 0, x_0 ∈^n and w ∈×. The pair (ρ_1,ρ_2) will be referred to as an iISS gain.
  


Due to the blanket assumption we have made on f, any of the conditions (<ref>), (<ref>) or (<ref>) implies that the solution x is forward complete. Suppose that x is a solution satisfying (<ref>), (<ref>) or (<ref>) and that its maximal interval of definition is [t_0,T) or [t_0,T] with T<∞. If x(T) is defined, then the initial value problem ż(t)=f(t,z(t),u(t)), z(T)=x(T) has a solution z which is defined on some interval [T,T+δ) with δ>0. In consequence, x admits a prolongation defined on some interval [t_0,T+δ') with δ'>0 small enough, which is absurd. If x(T) is not defined, then, due to standard results on ordinary differential equations, |x(t)|→∞ as t→ T^-, but this is impossible since x is bounded on [t_0,T). 


The weak versions of 0-GUAS or iISS are obtained by replacing the second argument of the function β in (<ref>) or (<ref>) by just t-t_0 (i.e. the number of jumps n^σ_(t_0,t] does not appear).

If (<ref>) is (weakly or strongly) 0-GUAS uniformly over , then under u≡ 0 the state converges asymptotically to the origin. In the weak case, the convergence warranty depends on the elapsed time t-t_0 but is insensitive to the occurrence of jumps. In addition, this convergence is uniform over initial times and over impulse time sequences within the family . The uniform-over- UBEBS property just imposes a bound on the state trajectory without necessarily guaranteeing convergence. The bound is uniform over initial times and over all σ∈, and depends on the initial state norm and the input energy. The uniform-over- (weak or strong) iISS property imposes a bound that is also uniform over initial times and over all σ∈. This bound is formed by a term similar to the 0-GUAS property and another term equal to the input energy.











§ CHARACTERIZATION OF STRONG IISS





 §.§ Main result




We require the following definitions, as employed in <cit.>.

  
  A function h : _≥ 0×^n ×^m →^n is said to belong to class , written h∈, if the following items hold:
  
  
  *  there exist ν_h ∈ and a nondecreasing function N_h:_≥ 0→_>0 such that |h(t,ξ,μ)|≤ N_h(|ξ|)(1+ν_h(|μ|)) for all t≥ 0, all ξ∈^n and all μ∈^m;
  
  * for every r>0 and ε >0 there exists δ>0 such that for all t≥ 0, |h(t,ξ,μ) - h(t,ξ,0)|<ε if |ξ|≤ r and |μ| ≤δ.
  
  
  
  
  *  h(t,ξ,0) is locally Lipschitz in ξ,     uniformly in t, i.e. for every ξ∈^n there are an open ball B containing ξ and a constant L≥ 0 so that for every ξ_1,ξ_2 ∈ B and t≥ 0 it happens that |h(t,ξ_1,0) - h(t,ξ_2,0)| ≤ L |ξ_1-ξ_2|.
  





Our main result is the following.

  
  Consider the impulsive system (<ref>), suppose that f,g∈ and let ⊂. Then, (<ref>) is strongly iISS uniformly over  if and only if it is strongly 0-GUAS and UBEBS, both uniformly over .

The proof of Theorem <ref> will be developed along Sections <ref> and <ref>.



 §.§ Intermediate results



The proof of Theorem <ref> follows the same steps as that of the proof of Theorem 3.2 of <cit.> but suitably modified for the strong case. For the sake of conciseness and to clarify the current contribution, we will emphasize the main differences and remove the parts that are identical or very similar.






The integral expression for the solution of (<ref>) is given by:

    x(t) = x(t_0) + ∫_t_0^t f(s,x(s),u(s)) ds +[0]
    
      +∑_τ∈σ∩(t_0,t] g(τ,x(τ^-),u(τ)).

The proof of our main result requires the generalization of Gronwall inequality for continuous functions with isolated jumps given as Lemma 3.1 in <cit.>. We copy the corresponding statement here for simplicity.

[<cit.>]
  
  Let 0 ≤ t_0 < T ≤∞ and let y:[t_0,T) → be a right-continuous function having a finite left-limit at every discontinuity instant. Suppose that the points of discontinuity of y can be arranged into a sequence σ∈. Let p∈ and q_1,q_2 ≥ 0. If y satisfies
  
    y(t)    ≤ p + q_1 ∫_t_0^t y(s) ds + q_2 ∑_s∈σ∩ (t_0,t] y(s^-)

  for all t ∈ [t_0,T), then in the same time interval y also satisfies
  
    y(t)    ≤ p (1+q_2)^n^σ_(t_0,t]· e^q_1 (t-t_0).




We will also require Lemma 3.2 of <cit.> (which is a generalization of Lemma 3 in <cit.>) suitably modified for the strong 0-GUAS case. The proof is a very minor modification of the corresponding proof in <cit.> and hence omitted.
[cf. Lemma 3.2 in <cit.>]
  
  Let ⊂, let the impulsive system (<ref>) be strongly 0-GUAS uniformly over  and let β∈ characterize the strong 0-GUAS property. Suppose that f,g ∈ and let ν_f and ν_g be, respectively, the functions corresponding to f and g as per item <ref>) of Definition <ref>. Let χ_f,χ_g ∈ satisfy χ_f ≥ν_f and χ_g ≥ν_g. Then, for every r>0 and every η>0, there exist L=L(r) and κ = κ(r,η) such that if x∈(t_0,x_0,w) with t_0 ≥ 0, x_0∈^n, w=(u,σ) ∈× satisfies |x(t)| ≤ r for all t≥ t_0, then also
  
    |x(t)| ≤β(|x_0|,t-t_0+n^σ_(t_0,t]) +
        [ (t-t_0+n^σ_(t_0,t])η. 
    .+ κw_(t_0,t]_(χ_f,χ_g)]   (1+L)^n^σ_(t_0,t]· e^L(t-t_0).


The only difference with respect to the corresponding bound in Lemma 3.2 of <cit.> is the inclusion of the number of jumps n^σ_(t_0,t] within the second argument of β in (<ref>). The corresponding proof is almost identical. 


The proof of our main result also requires a suitably modified version of Lemma 3.3 of <cit.>. In this case, the removal of the assumption on the boundedness of the number of jumps in a given interval, given by the uniform incremental boundedness (UIB) property in <cit.>, makes the corresponding proof sufficiently different so as to include it here. 

   
  Consider the impulsive system (<ref>), suppose that f,g∈ and let ⊂. If (<ref>) is strongly 0-GUAS and UBEBS, both uniformly over , then there exist α̃,ρ̃_1,ρ̃_2∈ for which the estimate (<ref>) holds for every x ∈(t_0,x_0,w) with t_0≥ 0, x_0∈^n and w∈×.
  
    α̃(|x(t)|) ≤ |x(t_0)| + w_(t_0,t]_(ρ̃_1,ρ̃_2)  ∀ t≥ t_0.



  Let α, ρ_1, ρ_2 and c be as in the estimate (<ref>). Let ρ̃_1 := max{ρ_1,ν_f} and ρ̃_2 := max{ρ_2,ν_g}. For r≥ 0 define
  
    α̅(r)    := sup_x∈𝒯(t_0,x_0,w),  t≥ t_0≥ 0,  |x_0|≤ r,   w∈×,  w≤ r |x(t)|

where w:=w_(ρ̃_1,ρ̃_2). From this definition, it follows that α̅ is nondecreasing and from (<ref>) that it is finite for all r≥ 0. 

Let β∈ be the function which characterizes the uniform-over- strong 0-GUAS property of (<ref>). From the latter property, it follows that α̅(0) = 0. Next, we show that lim_r→ 0^+α̅(r)=0. Let r^*=α^-1(2+c) and L=L(r^*)>0 be given by Lemma <ref>. Let ε>0 be arbitrary. Pick 0<δ_1<1 such that δ_1 ≤β(δ_1,0)<ε/2 and T̃>0 such that β(δ_1,T̃)<δ_1/2. Define η = δ_1/4 (T̃+1) e^-L̃(T̃+1), with L̃ = max{L, log(1+L) } and let κ=κ(r^*,η)>0 be given by Lemma <ref>. Last, pick 0<δ_2<1 such that

  δ_2 < δ_1/4 κ (T̃ + 1) e^-L̃(T̃ + 1).

For every j∈_0, define 

    t_j+1 := inf{ t>t_j : t-t_j+n^σ_(t_j,t]≥T̃}

and consider the intervals I_j = [t_j,t_j+1). Note that t_j+1 > t_j for every j∈_0. 




By definition of t_j+1 and since t↦ n_(t_j,t]^σ is right-continuous, it follows that






for all j ∈_0,

    T̃≤ t_j+1-t_j+n^σ_(t_j,t_j+1]   ≤T̃+1,  and
    
      t-t_j+n^σ_(t_j,t]   ≤T̃+1   ∀  t ∈ I_j.


We claim that 
lim_j→∞t_j = ∞.  For a contradiction, suppose that lim_j→∞t_j = M < ∞. As every convergent sequence is a Cauchy sequence, for every ρ>0 there exists N = N(ρ) ∈ such that |t_j+1-t_j|<ρ for all j ≥ N. But t_j+1-t_j+n^σ_(t_j,t_j+1]≥T̃>0 and hence 0 < T̃≤ρ+n^σ_(t_j,t_j+1] and T̃-ρ<n^σ_(t_j,t_j+1]. Taking ρ<T̃ we have that 0<T̃-ρ < n^σ_(t_j,t_j+1] and thus n^σ_(t_j,t_j+1]≥ 1 for all j≥ N. Then, n^σ_(t_N,M) = ∑_j=N^∞ n^σ_(t_j,t_j+1]≥∑_j=N^∞ 1 = ∞, contradicting the assumption that σ has no finite accumulation points. Therefore, lim_j→∞t_j = ∞. 

For every x∈(t_0,x_0,w), with t_0≥ 0, |x_0|≤δ_1, w ∈× and w≤δ_2, we also have x ∈(t_j,x(t_j),w) for all j∈. By induction, we will show that |x(t)| ≤ε for all t∈ I_j = [t_j,t_j+1) and that |x(t_j+1)| < δ_1. For j=0 and applying Lemma <ref>, it follows that for all t ∈ I_0, we have

    |x(t)|    ≤β(|x_0|,t-t_0+n^σ_(t_0,t])
       +[(t-t_0+n^σ_(t_0,t])η + κw] e^L̃(t-t_0+n^σ_(t_0,t])
       ≤β(δ_1,0) + [(T̃+1)η + κδ_2] e^L̃(T̃+1) < ε/2 + δ_1/2 < ε,

and that

    |x(t_1)|    ≤β(|x_0|, t_1-t_0+n^σ_(t_0,t_1])
       +[(t_1-t_0+n^σ_(t_0,t_1])η + κw] e^L̃(t_1-t_0+n^σ_(t_0,t_1])
       ≤β(δ_1,T̃)+[(T̃+1)η + κδ_2] e^L̃(T̃+1) < δ_1.

So our induction assumption holds for j=0. Next, suppose that it holds for arbitrary j∈ N_0. Applying Lemma <ref>, then for t∈ I_j+1 we have that

    |x(t)|    ≤β(|x(t_j+1)|,0) + [(T̃+1)η + κδ_2] e^L̃(T̃+1)
    
         ≤β(δ_1,0) + [(T̃+1)η + κδ_2] e^L̃(T̃+1) < ε/2 + δ_1/2 < ε,

where we have used the fact that |x(t_j+1)| ≤δ_1, and that

    |x(t_j+2)|    ≤β(|x(t_j+1)|, t_j+2-t_j+1+n^σ_(t_j+1,t_j+2])[0]
       +[(T̃+1)η + κδ_2] e^L̃(T̃+1)
    
         ≤β(δ_1,T̃)+[(T̃+1)η + κδ_2] e^L̃(T̃+1) < δ_1.

Hence our induction assumption holds for j+1. As a consequence, |x(t)| ≤ε must hold for all t ≥ t_0.



Thus, if δ=min{δ_1,δ_2}, for all
x∈(t_0,x_0,w), with t_0≥ 0, |x_0|≤δ, w∈× with
w≤δ, we have |x(t)|≤ε for all t≥ t_0. Therefore,
α̅(r)≤α̅(δ)<ε for all
0<r<δ and lim_r→ 0^+α̅(r)=0.
 
Since α̅ is nondecreasing and
lim_r→ 0^+α̅(r)=0 there exists α̂∈ such
that α̂(r)≥α̅(r) for all r≥ 0. Let
x ∈(t_0,x_0,w) with t_0≥ 0, x_0 ∈^n and w∈×. Let t≥ t_0. 
Due to causality, there exists
x^*∈(t_0,x_0,w_(t_0,t]) such that x^*(τ)=x(τ) for all
τ∈ [t_0,t]. By using the definition of α̅ and the
fact that α̂(r)≥α̅(r), we then have

  |x(t)| = |x^*(t)| ≤α̂(|x_0|) + α̂(w_(t_0,t]).

Define α̃∈ via α̃(s) = α̂^-1(s)/2. Applying α̃ to both sides of the preceding inequality and using the fact that α̃(a+b) ≤α̃(2a) +α̃(2b), we reach

  α̃(|x(t)|) ≤ |x_0| + w_(t_0,t],

which establishes the result.








 §.§ Proof of Theorem <ref>



The proof of our main result requires the following ϵ-δ characterization of the uniform-over- strong iISS property. The statement follows from suitable modification of that of Theorem 3.1 of <cit.>. Whether this characterization holds or not under such a modification is a nontrivial question. We hence provide the proof in the Appendix.

  
  Let ρ_1,ρ_2∈ and ⊂. Consider the notation w = w_(ρ_1,ρ_2) and for r ≥ 0, B_r^ := {w∈× :  w ≤ r}. Then, system (<ref>) is strongly iISS uniformly over  with iISS gain (ρ_1,ρ_2) if and only if 
  the following conditions hold:
  
  
  * For every T≥ 0, r≥ 0, s≥ 0, there exists C>0 such that every x∈(t_0,x_0,w) with t_0 ≥ 0, x_0 ∈ B_r^n and w ∈ B_s^ satisfies |x(t)| ≤ C for all t ≥ t_0 such that t + n^σ_(t_0,t]≤ t_0+T.
  
  * For each ϵ > 0, there exists δ > 0 such that every x∈(t_0,x_0,w) with t_0 ≥ 0, x_0 ∈ B_δ^n and w ∈ B_δ^ satisfies |x(t)| ≤ϵ for all t≥ t_0.
  
  * There exists α̃∈ such that for every r,ϵ > 0 there exists T>0 so that for every x∈(t_0,x_0,w) with t_0 ≥ 0, x_0∈ B_r^n and w ∈×, then

      α̃(|x(t)|) ≤ϵ + w

    for all t≥ t_0 such that t + n^σ_(t_0,t]≥ t_0+T.
  


We may finally provide a proof to our main result.




[Proof of Theorem <ref>]

(⇒) Considering w=(u,γ) with u=0, the estimate (<ref>) reduces to α(|x(t)|) ≤β(|x(t_0)|,t-t_0+n^σ_(t_0,t]) and hence |x(t)| ≤α^-1(β(|x(t_0)|,t-t_0+n^σ_(t_0,t])). The function β̃:= α^-1β satisfies β̃∈, and hence (<ref>) follows with β replaced by β̃. Therefore, clearly strongly iISS implies strongly 0-GUAS, both uniformly over .

Consider β∈ from (<ref>), define β_0 ∈ via β_0(r) = β(r,0). Then, |x(t)| ≤α^-1[ β_0(|x(t_0)|) + w_(t_0,t]_(ρ_1,ρ_2)]. 
Define ψ∈ via ψ(r) = min{β_0^-1(α(r)/2),  α(r)/2  }. Applying ψ to each side of the latter inequality and using the fact that ϕ(a+b) ≤ϕ(2a) + ϕ(2b) for every ϕ∈ and a,b≥ 0, yields

    ψ(|x(t)|)    ≤ψα^-1[ β_0(|x(t_0)|) + w_(t_0,t]_(ρ_1,ρ_2)]
       ≤ψα^-1 [ 2β_0(|x(t_0)|) ] + ψα^-1 [2w_(t_0,t]_(ρ_1,ρ_2) ]
       ≤ |x(t_0)| + w_(t_0,t]_(ρ_1,ρ_2),

and hence (<ref>) follows with α replaced by ψ. We have shown that strong iISS implies UBEBS, both uniformly over .

(⇐)
Let α̃,ρ̃_1,ρ̃_2∈ be given by Lemma <ref>, so that (<ref>) is satisfied. We will prove that (<ref>) is strongly iISS uniformly over  with iISS gain (ρ̃_1,ρ̃_2) by establishing each of the items of Theorem <ref>.

<ref>) Let T≥ 0, r≥ 0 and s≥ 0. Let x ∈(t_0,x_0,w) with t_0≥ 0, x_0∈ B_r^n, w∈ B^_s. From (<ref>) we have: α̃(|x(t)|) ≤ |x(t)| + w_(t_0,t]_(ρ̃_1,ρ̃_2)≤ r+s, and hence |x(t)| ≤α̃^-1(r+s) =: C for all t≥ t_0. This establishes item <ref>) of Theorem <ref>.

<ref>) Let ϵ > 0. Let δ = α̃(ϵ)/2. Then, if x ∈(t_0,x_0,w) with t_0≥ 0, x_0∈ B_δ^n and w∈ B_δ^, from (<ref>) then α̃(|x(t)|) ≤ |x_0| + w_(t_0,t]_(ρ̃_1,ρ̃_2)≤δ + δ = 2 δ. It follows that |x(t)| ≤α̃^-1(2δ) = ϵ for all t≥ t_0. This establishes item <ref>) of Theorem <ref>.

<ref>) Let α = α̃/2 ∈. Let r,ϵ > 0 and let x ∈(t_0,x_0,w) with t_0≥ 0, x_0∈ B_r^n and w∈×. 
We distinguish two cases:


  * w≥ r,

  * w < r.


In case (a), from (<ref>) we have α̃(|x(t)|) ≤ |x_0| + w_(t_0,t]≤ r + w≤ 2w, hence α(|x(t)|) = α̃(|x(t)|)/2≤w≤ϵ + w for all t≥ t_0.

Next, consider case (b). From (<ref>), we have  α̃(|x(t)|) ≤ r + w < 2r =: r̃ for all t≥ t_0. Let β∈ characterize uniform-over- strong 0-GUAS property, so that (<ref>) is satisfied under zero input, and let L = L(r̃) > 0 be given by Lemma <ref>. Define L̃ := max{L,log(1+L)}, let ϵ̃= ϵ and T̃>0 satisfy β(r̃,T̃) < ϵ̃/2. Define η=ϵ̃/4 (T̃+1) e^-L̃ (T̃+1). Let κ = κ(r̃,η) > 0 be given by Lemma <ref>. Let δ = ϵ̃/4κe^-L̃ (T̃ + 1). Define N := ⌈r/δ⌉ and  T:= N (T̃+1), where ⌈ s⌉ denotes the least integer not less than s∈.

Let s_0 := t_0 and for i=1 to N, define

    s_i := inf{ t ≥ s_i-1 : t - s_i-1 + n^σ_(s_i-1,t]≥T̃}.

Then, for i=1,…,N we have s_i-1 < s_i < ∞ and

    T̃≤ s_i - s_i-1 + n^σ_(s_i-1,s_i]≤T̃ + 1.

Consider the intervals I_i=[s_i-1, s_i], with i=1,…, N. We claim that there exists j≤ N-1 for which w_(s_j,s_j+1]≤δ. For a contradiction, suppose that w_(s_j,s_j+1] > δ for all 0 ≤ j ≤ N-1. Then, w≥w_(s_0,s_N] = ∑_j=0^N-1w_(s_j,s_j+1] > Nδ≥ r, contradicting case (b). Therefore, let 0≤ j≤ N-1 be such that w_(s_j,s_j+1]≤δ.

Since x∈(s_j,x(s_j),w) and |x(t)|≤r̃ for all t≥ s_j, from Lemma <ref> and using the bounds (<ref>), it follows that
  
    |x(s_j+1)| ≤β(|x(s_j)|, T̃)+
    [ (T̃ + 1) η + κw_(s_j,s_j+1]] e^L̃(T̃+1)
    ≤β(r̃,T̃) + [(T̃ + 1) η + κδ] e^L̃ (T̃+1)≤ϵ̃.

Therefore, using (<ref>) with t_0 replaced by s_j+1, we reach

    α̃(|x(t)|) ≤ |x(s_j+1)| +  w_(s_j+1,t]≤ϵ̃+ w

for all  t≥ s_j+1 and hence also for all t≥ s_N. Since ∑_i=1^N s_i - s_i-1 + n^σ_(s_i-1,s_i] = s_N - s_0 + n^σ_(s_0,s_N]≤ N(T̃+1) = T, then t + n^σ_(t_0,t]≥ t_0 + T implies that t≥ s_N. Therefore, (<ref>) holds for all t≥ t_0 for which t + n^σ_(t_0,t]≥ t_0 + T. Since α =α̃/2 ≤α̃, it follows that item <ref>) of Theorem <ref> also is satisfied.




 §.§ Previous results as a particular case


In this section we will show that the main result in <cit.>, namely Theorem 3.2 in <cit.>, is a Corollary of Theorem <ref>. We recall that a subset ⊂ℐ is uniformly incrementally bounded (UIB) if there exists a nondecreasing function ϕ : _> 0→_≥ 0 so that n^σ_(t_0,t]≤ϕ(t-t_0) for every σ∈ and all t>t_0≥ 0 (see Definition 3.2 in <cit.>).

(<cit.>)  Consider the impulsive system (<ref>) and suppose that f,g∈. Let ⊂ be a UIB set of impulse time sequences. Then, (<ref>) is weakly iISS uniformly over  if and only if it is weakly 0-GUAS and UBEBS, both uniformly over .


The proof of the only if part is straightforward and does not require the UIB hypothesis. As for the if part, assume that (<ref>) is weakly 0-GUAS and UBEBS and that  is UIB. Let β∈ be the function that characterizes the weak 0-GUAS stability property of the system (<ref>). Let ϕ be the function appearing in the definition of the UIB property. Due to Lemma 6.1 in <cit.>, there exists β̂∈ such that

    β(r,s)≤β̂(r,s+ϕ(s)),  ∀ (r,s)∈_≥ 0^2.

Then, for every x∈(t_0,x_0,w_0) with t_0 ≥ 0, x_0 ∈^n and w_0=(0,σ) with σ∈ we have that for all t≥ t_0

    |x(t)|    ≤β(|x(t_0)|,t-t_0) 
          ≤β̂(|x(t_0)|,t-t_0+ϕ(t-t_0)) 
       ≤β̂(|x(t_0)|,t-t_0+n^σ_(t_0,t]).

So (<ref>) is strongly 0-GUAS uniformly over . Applying Theorem <ref> it follows that (<ref>) is then strongly iISS and therefore weakly iISS, both uniformly over .  




§ CONCLUSIONS



We have addressed the characterization of the integral input-to-state stability property in terms of global uniform asymptotic stability under zero input and a uniformly bounded-energy input bounded state property. We have shown that this characterization remains valid for impulsive systems with time-varying flow and jump maps if both global uniform stability and integral input-to-state stability are understood in the strong sense. This characterization was established under a partial Lipschitz continuity assumption on the jump map [see item <ref>) of Definition <ref>]. Future work is aimed at removing this assumption and establishing relationships between the ISS and iISS properties for impulsive systems.









































 §.§ Proof of Theorem <ref>


  Necessity is straightforward, so we just establish sufficiency. 

Let α̃∈ and T>0 be given by item <ref>), the latter in correspondence with r>0 and ϵ = 1. Let C be given by item <ref>) in correspondence with s=r and T. From items <ref>) and <ref>), we then have, whenever t_0 ≥ 0, x_0∈ B_r^n and w∈ B_r^, 

    |x(t)| ≤ C,      ∀ t≥ t_0, t + n^σ_(t_0,t]≤ t_0+T,
    α̃(|x(t)|) ≤ 1 + w,      ∀ t≥ t_0, t + n^σ_(t_0,t] > t_0 + T.

It follows that α̃(|x(t)|) ≤α̃(C) + 1 + w for all t≥ t_0. 

Let ϕ(r) := inf{C̃≥ 0 : α̃(|x(t)|) ≤C̃, ∀ x ∈(t_0,x_0,w), ∀ t ≥ t_0 ≥ 0, ∀ x_0 ∈ B_r^n, ∀ w∈ B_r^}. By the previous analysis, then ϕ(r) ≤α̃(C) + 1 + r < ∞ for all r≥ 0. Also, ϕ is nondecreasing and α̃(|x(t)|) ≤ϕ(|x(t_0)|) + ϕ(w) for all t≥ t_0 whenever x ∈(t_0,x_0,w) with t_0 ≥ 0, x_0 ∈^n and w ∈×. From item <ref>), it follows that lim_r↘ 0ϕ(r) = 0. There thus exists η∈ such that ϕ≤η and then

    α̃(|x(t)|) ≤η(|x(t_0)|) + η(w)   for all t≥ t_0,

whenever x ∈(t_0,x_0,w) with t_0 ≥ 0, x_0 ∈^n and w ∈×. Let ψ,α∈ be defined via ψ(s) = η^-1(s/2) and α = min{α̃,ψα̃}. Then, applying ψ to (<ref>) and using the inequality ψ(a+b) ≤ψ(2a) + ψ(2b), it follows that

    α(|x(t)|) ≤ |x(t_0)|+w  for all t≥ t_0,

whenever x ∈(t_0,x_0,w) with t_0 ≥ 0, x_0 ∈^n and w ∈×.

Define

    T_r,ϵ := inf{τ≥ 0 : α(|x(t)|) ≤ϵ + w,
    ∀ t≥ t_0, t + n^σ_(t_0,t]≥ t_0 + τ,
    ∀ x∈(t_0,x_0,w), ∀ t_0 ≥ 0,  ∀ x_0 ∈ B_r^n, ∀ w ∈×}.

By item <ref>) and since α≤α̃, then T_r,ϵ < ∞ for every r,ϵ > 0. Moreover, T_r,ϵ is nondecreasing in r for fixed ϵ>0 and nonincreasing in ϵ for fixed r>0. By (<ref>), then T_r,ϵ→ 0 as ϵ→∞ for fixed r>0.

  T_r,ϵ can be strictly upper bounded by T̅_r,ϵ with the following properties:
  
  
  * For each fixed r>0, T̅_r,· : _>0→_>0 is continuous, strictly decreasing, and onto, so that lim_ϵ↘ 0T̅_r,ϵ = ∞ and lim_ϵ→∞T̅_r,ϵ = 0.
  
  * For each fixed ϵ > 0, T̅_·,ϵ is strictly increasing and lim_r→∞T̅_r,ϵ = ∞.
  

Let ψ_r denote the inverse function of T̅_r,ϵ considered as a function of ϵ for fixed r> 0. For every r>0, then ψ_r is continuous on _>0 and lim_s↘ 0ψ_r(s) = ∞. By definition of T_r,ϵ and since T̅_r,ϵ > T_r,ϵ, we have that

    t_0 ≥ 0, x_0 ∈ B_r^n, w∈×, x∈(t_0,x_0,w), t ≥ t_0,  and
    
     t + n^σ_(t_0,t]≥ t_0 + T̅_r,ϵ  ⇒  α(|x(t)|) ≤ϵ + w

Note that t - t_0 + n^σ_(t_0,t] = T̅_r,ϵ is equivalent to ϵ = ψ_r(t-t_0+n^σ_(t_0,t]). Hence, from the implication (<ref>) at t≥ t_0 such that t - t_0 + n^σ_(t_0,t] = T̅_r,ϵ, it follows that

    t> t_0 ≥ 0, x_0 ∈ B_r^n, w∈×, x∈(t_0,x_0,w)
    ⇒  α(|x(t)|) ≤ψ_r(t-t_0+n^σ_(t_0,t]) + w

The proof concludes following exactly the same steps as for the proof of Lemma 2.7 in <cit.>.
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