On zero-input stability inheritance for time-varying systems with decaying-to-zero input power

On zero-input stability inheritance for time-varying systems with

decaying-to-zero input power

José L. Mancilla-Aguilar

a

_{, Hernan Haimovich}

b,

_*

a_{Departamento de Matemática, Instituto Tecnológico de Buenos Aires, Av. Eduardo Madero 399, Buenos Aires, Argentina}

b_{International French-Argentine Center for Information and Systems Science (CIFASIS), CONICET-UNR, Ocampo y Esmeralda, 2000 Rosario, Argentina}

Keywords: Nonlinear systems Time-varying systems Stability

Convergence Omega-limit sets

a b s t r a c t

Stability results for time-varying systems with inputs are relatively scarce, as opposed to the abundant literature available for time-invariant systems. This paper extends to time-varying systems existing results that ensure that if the input converges to zero in some specific sense, then the state trajectory will inherit stability properties from the corresponding zero-input system. This extension is non-trivial, in the sense that the proof technique is completely novel, and allows to recover the existing results under weaker assumptions in a unifying way.

1. Introduction

Stability properties for systems with inputs find natural ap-plication in control systems. Input-to-state stability (ISS) [1–3], integral ISS (iISS) [4,5], converging-input converging-state (CICS) [6,7], uniformly bounded-energy input bounded state (UBEBS) [8], bounded-energy-input convergent-state (BEICS) [9,10] and Lp

-input converging-state [11] are examples of such properties. Most of the existing analyses and characterizations of these prop-erties apply to time-invariant systems. Analogous results for time-varying systems are very scarce. There exist some charac-terizations of the ISS property [12–14] and a recent result by the authors characterizing the iISS property [15]. In a more general setting, some asymptotic behaviour results exist for asymptoti-cally autonomous differential equations [16,17], and some also dealing with weak invariance principles [18]. An asymptotically autonomous differential equation is one such that the functionf0 defining its dynamicsx

˙

=

f0(t

,

x) approaches a time-invariant functiong, i.e.f0(t

,

x)

→

g(x) ast

→ ∞, in some suitable sense.

A time-invariant systemx

˙

= ¯

f(x

,

u), with an inpututhat con-verges to zero can be interpreted as an asymptotically autonomous system [f0(t

,

x)

:= ¯

f(x

,

u(t))

→

g(x)

:= ¯

f(x

,

0)] under reasonable assumptions. By contrast, time-varying systems of the formx

˙

=

f(t

,

x

,

u) do not in general allow such a possibility. An interesting result in the latter case is provided in [18], where the concept of weakly asymptotically autonomous system is introduced, which,

*

Corresponding author.

E-mail addresses:[email protected](J.L. Mancilla-Aguilar),

loosely speaking, means thatx

˙

=

f0(t

,

x) approaches the differen-tial inclusion

˙

x

∈

F(x) ast

→ ∞

in some appropriate sense. The latter can be employed in the time-varying case withf0(t

,

x)

:=

f(t

,

x

,

u(t)).

An iISS system has, inter alia, the property that inputs with bounded energy, where energy is measured according to the iISS gain, produce state trajectories that asymptotically converge to zero. The latter is the BEICS property [9]. The function that weighs the input in order to measure input energy, i.e. the iISS gain in the iISS setting, is extremely important in the sense that a system may be iISS for some iISS gains but not for others. Interesting examples of some perhaps counter-intuitive facts are given in [19] and [20], where globally asymptotically stable systems (exponen-tially in [20]) are destabilized by additive inputs of arbitrarily small energy (exponentially decaying in [20]). The main point we make is that the ensuing stability or instability depends on how input energy is measured.

This work relates to the CICS and BEICS properties. Roughly speaking, these properties entail that if the system input converges to zero in some specific manner, then the state will also converge to zero. These properties are of importance in stability analy-sis for cascade systems and also in ensuring stability robustness under certain types of disturbances. We consider time-varying systems with inputs and pinpoint specific input power ‘measures’ (see Section2.3) so that solutions corresponding to inputs with decaying-to-zero power may inherit specific properties from the corresponding zero-input system. More precisely, suppose that the zero-input system has a uniformly locally asymptotically stable compactumC within an open setG contained in the ‘‘region of attraction’’ (see [21] for the latter concept in time-varying systems,

and Section2). Letxbe a forward complete solution of

˙

x

=

f(t

,

x

,

u) corresponding to an inputuhaving decaying-to-zero power. Then, one of the results that we prove is that if the

ω

-limit set ofxhas nonempty intersection withG, thenxapproachesC.

In this context, our contribution is the following. First, we provide a convergence result for time-varying systems with inputs under very mild assumptions on the functionfdefining the system dynamics. Worthy of mention is that we do not requiref(t

,

x

,

u) to be continuous int, nor locally Lipschitz inx. As a consequence, solutions are not necessarily unique. Second, we pinpoint input power ‘measures’ for which such convergence is possible. These ‘measures’ relate to specific bounds onf. Third, we extend some of the main results in [6,9] and [11] to time-varying systems, under weaker assumptions and in a unifying way. We emphasize that these extensions are novel and nontrivial, since existing results for time-invariant systems, such as those in [9] and [11], cannot be adapted to the current setting (the corresponding proofs rely on converse Lyapunov theorems that do not remain valid).

The remainder of this paper is organized as follows. In Section2 we introduce the notation, definitions and main assumptions re-quired. Our main result and explanations of how our result sub-sumes other existing results are contained in Section3. Section4 contains some secondary technical results and conclusions are drawn in Section5.

2. Preliminaries

2.1. Notation and preliminary definitions

The reals, nonnegative reals, naturals and nonnegative integers are denotedR,R≥0,NandN0, respectively. For

ξ

∈

Rn,

|

ξ

|

denotes its Euclidean norm. For a given nonempty subsetA

⊂

_Rn,

|

ξ

|

_A denotes the distance from

ξ

∈

_Rn_{toA, that is}

_|

_ξ

_|

A

=

inf{|

ξ

−

ζ

| :

ζ

∈

A}. Givenr

≥

0,Ar

= {

ξ

∈

Rn

:

|

ξ

|

A

≤

r} and

Br(

ξ

)

= {

ξ

}

r for every

ξ

∈

Rn. Thus, if

ξ

∈

Rnandr

≥

0, the statements

ξ

∈

Ar and

|

ξ

|

A

≤

rare equivalent. Forp

≥

1 and

m

∈

_N,Lp_m,loc(Lpm) denotes the set of all the Lebesgue measurable

functions

v

:

_R≥0

→

Rm such that

|

v

|

p is integrable on each finite intervalI

⊂

_R≥0(|

v

|

pis integrable onR≥0). Whenm

=

1 we just writeLp_loc andLp_{. For a Lebesgue measurable setJ}

_⊂

R,

|J|

will denote its Lebesgue measure. Given a metric space (U

,

d) and an interval I

⊂

_R, we say that

v

:

I

→

U is piecewise constant if there exists a partitionI1

, . . . ,

ImofIsuch thatIiis an

interval for everyiand

v

is constant onIi. The functionu

:

I

→

U

is Lebesgue measurable if there exists a sequence of piecewise-constant functionsuk

:

I

→

Usuch that limk→∞uk(t)

=

u(t)

for almost allt

∈

I, that is

|{t

∈

I

:

limk→∞uk(t)

̸=

u(t)}| = 0.

WhenU is separable,u

:

I

→

U is measurable if and only if u−1_{(V) is Lebesgue measurable for every open subsetVofU(see} Remark C.1.1. in [22]). A function

ω

:

U

→

_Ris proper if for all r

∈

_Rthe sublevel set

ω

−1_((−∞

_,

_{r]) is compact. We write}

_σ

_∈

_K if

σ

:

_R≥0

→

R≥0is continuous, strictly increasing, and

σ

(0)

=

0. We write

σ

∈

K∞if

σ

∈

Kand

σ

is unbounded.

2.2. Problem statement

This work deals with time-varying control systems of the gen-eral form

˙

x

=

f(t

,

x

,

u) (1)

wheref

:

_R≥0

×

X

×

U

→

RnwithX an open subset ofRnand (U

,

d) a metric space. An input is a Lebesgue measurable function u

:

_R≥0

→

UandUis the set of all the inputs. We suppose thatUis nonempty and there exists 0

∈

U, where ‘‘0’’ is nothing but some element inU that we distinguish from the rest. For an arbitrary

µ

∈

U, we define

|

µ

| :=

d(

µ,

0), i.e.

|

µ

|

is the distance between

µ

and 0. In the case in whichU

⊂

_Rm_{, 0 denotes the origin of}

Rmand dwill be the metric induced by Euclidean norm. Thezero inputis the map0

∈

Usuch that0(t)

≡

0. With system(1)we associate the zero-input system

˙

x

=

f(t

,

x

,

0)

=:

f0(t

,

x)

.

(2)

Assumption 1. The functionf

:

_R≥0

×

X

×

U

→

Rnsatisfies the following conditions.

(A1) (Carathéodory) f(·

, ξ, µ

) is Lebesgue measurable for all (

ξ, µ

)

∈

X

×

Uandf(t

,

·

,

·) is continuous for every

t

≥

0. (A2) (Zero-input Lipschitzianity)f0(t

, ξ

) is locally Lipschitz in

ξ

uniformly in t in the following sense: for every compact subsetK

⊂

Xthere exists a nonnegative functionLK

∈

L1loc such that sup_t≥0

∫

t+T

t LK(s)ds

<

∞

for allT

>

0 and

|f

0(t

, ξ

)

−

f0(t

, ξ

′

)| ≤LK(t)|

ξ

−

ξ

′

| ∀t

≥

0

,

∀

ξ, ξ

′

∈

K

.

In view ofAssumption 1, for eacht0

≥

0 and

ξ

∈

Xthere is a unique maximally defined (forward) solutionx(t)

=

ϕ

(t

,

t0

, ξ

) of(2)which verifiesx(t0)

=

ξ

. We will denote by

[t

0

,

tt0,ξ) its maximal interval of definition. It is well-known that in the case in which

ϕ

(t

,

t0

, ξ

) belongs to a fixed compact subset ofXfor all t

∈ [t

0

,

tt0,ξ), thentt0,ξ

= ∞.

Let C

⊂

G

⊂

X be such thatC is nonempty and compact andG is open. In what follows we assume thatC is uniformly asymptotically stable with respect to(2)and thatGis contained in the region of attraction ofC. These statements are made precise in the following assumption.

Assumption 2(Zero-input stability). There exist a nonempty com-pact setCand an open setGsuch thatC

⊂

G

⊂

Xand

(B1) (uniform Lyapunov stability) for every

ε >

0 there exists

δ >

0 such that for allt0

≥

0 and

ξ

∈

Cδ,

ϕ

(t

,

t0

, ξ

)

∈

Cε for allt

≥

t0;

(B2) (uniform boundedness of solutions) for every compact set K

⊂

Gthere exists a compact setΓ

⊂

X such that for all t0

≥

0 and

ξ

∈

Kwe have that

ϕ

(t

,

t0

, ξ

)

∈

Γfor allt

≥

t0; (B3) (uniform attractiveness) for every compact setK

⊂

Gand

every

ε >

0 there existsT

=

T(K

, ε

)

≥

0 such that for allt0

≥

0 and

ξ

∈

Kwe have that

ϕ

(t

,

t0

, ξ

)

∈

Cε for all t

≥

t0

+

T.

Note that under the uniform Lyapunov stability in (B1) above, it follows thatCis forward invariant under(2), i.e. for allt0

≥

0 and

ξ

∈

C,

ϕ

(t

,

t0

, ξ

)

∈

Cfor allt

≥

t0. WhenG

=

X

=

Rn, thenCis globally uniformly asymptotically stable with respect to(2).

Remark 1. When the zero-input system (2) is time-invariant, i.e.f0(t

, ξ

)

≡

f

∗

0(

ξ

),Assumption 2is satisfied with any compact setC

⊂

Xwhich is asymptotically stable with respect to(2)[that is (i)Cis stable: for every

ε >

0 there exists

δ >

0 such that for all

ξ

∈

C_δ,

ϕ

(t

,

0

, ξ

)

∈

C_εfor allt

≥

0 and (ii)Cis attractive: there exists

δ

0

>

0 such that

|

ϕ

(t

,

0

, ξ

)|C

→

0 for all

ξ

∈

Cδ0] and with G

=

A, whereA

= {

ξ

∈

X

: |

ϕ

(t

,

0

, ξ

)|_C

→

0}is the region of attraction ofC. □

The problem we address in this paper is the following: Give conditions under which the property of convergence to C that applies to solutions of the zero-input system(2)is inherited by (i.e. also applies to) solutions of(1).

2.3. Admissible inputs and further assumptions

One of the conditions that we will give towards solving the considered problem is that the inputushould converge to 0 in some appropriate sense. To make this notion more precise, we require the following assumption.

Assumption 3. The functionf in(1)satisfies the following two conditions:

(C1) There exists a continuous function

γ

:

U

→

_R≥0so that: (i)

γ

(0)

=

0 and

γ

r

:=

inf|µ|≥r

γ

(

µ

)

>

0 for allr

>

0, and

(ii) for every compact setK

⊂

Xthere existsM

=

M(K)

≥

0 such that

|f

(t

, ξ, µ

)| ≤ M(1

+

γ

(

µ

)) for allt

≥

0, all

ξ

∈

K and all

µ

∈

U.

(C2) For every compact setK

⊂

X and

ε >

0 there exists

δ

=

δ

(K

, ε

)

>

0 such that for allt

≥

0,

|f

(t

, ξ, µ

)

−

f(t

, ξ,

0)|

< ε

if

ξ

∈

Kand

|

µ

| ≤

δ

.

Condition (C1) gives a specific bound on the growth of

|f

(t

, ξ, µ

)|which is uniform over allt

≥

0 and over

ξ

in compact sets. Condition (C2) requires thatf(t

, ξ,

·) be continuous at (t

, ξ,

0), uniformly overt

≥

0 and over

ξ

in compact sets.

For some results we will consider the following somewhat weaker conditions onf, which are equivalent to those of Assump-tion 3whenUis locally compact and separable (seeLemma 2.1).

Assumption 4. The functionfin(1)satisfies (C2) and:

(D1) fis bounded onR≥0

×

K

×

Bfor every pair of compact sets K

⊂

XandB

⊂

U.

The proof of the following lemma is given in Section4.

Lemma 2.1. Suppose that f satisfies condition (D1) and that U is a separable and locally compact metric space. Then, f satisfies condition (C1).

Remark 3. Several characterizations of stability properties for time-invariant systems with inputs, of the form x

˙

=

f

¯

(x

,

u), are made possible by employing (i) knowledge of the stability properties of the zero-input system, and (ii) some local Lipschitz continuity assumption onf

¯

(see e.g. [2,5]). For example, the proof of the characterization of the integral input-to-state stability (iISS) property in Theorem 1 of [5] employs the fact that the 0-input system

˙

x

= ¯

f(x

,

0) is globally asymptotically stable (Proposition II.5 and Lemma IV.10 of [5]) and the fact thatf

¯

is locally Lipschitz continuous. In this work, we do not require any additional Lipschitz continuity assumption other than that inAssumption 1. □

Associated with the function

γ

in (C1), we define the set of admissibleinputsU_γ

:= {u

∈

U

:

γ

◦

u

∈

L1_loc

}. From (A1)

and (C1), well-known results of the theory of ordinary differential equations (e.g. Theorem I.5.1 in [23]) ensure that for everyt0

≥

0,

ξ

∈

X andu

∈

U_γ there exists at least one maximally defined (forward) solutionx

: [t

0

,

tx)

→

Xto(1)which verifiesx(t0)

=

ξ

and thattx

= ∞

ifx(t) belongs to some fixed compact subset of Xfor allt

∈ [t

0

,

tx). We emphasize thatAssumptions 1and3, and

the factu

∈

U_γ are not sufficient to ensure the uniqueness of the corresponding solutions of(1), and that uniqueness of solutions is not required along this paper.

For a givenT

>

0, consider the positive semidefinite functional

∥ · ∥

_T

:

U_γ

→ [0

,

∞], defined by

∥u∥

T

:=

sup t≥0

∫

t+T

t

γ

(u(s))ds

.

(3)

Givenu

∈

U_γandT

>

0, the quantity

∥u∥

Tcan be interpreted as a

measure of the maximum energy thatucontains in any interval

of length T. Hence,

∥u∥

T

/

T is a measure of maximum average

power. For the sake of simplicity, and to distinguish

∥u∥

T from

other measures of input energy, we will refer to

∥u∥

Tas thepower

of the admissible inputu.

Proposition 2.2.Let T1

,

T2

>

0. Then, there exists k

=

k(T1

,

T2)such that

∥u∥

T2

≤

k∥u∥T1for all u

∈

Uγ.

Proof. Letkdenote the least integer not less thanT2

T1. Then,k

≥

1 andT2

≤

kT1. By direct application of(3)and simple properties of integrals and suprema, it can be shown that

∥u∥

T2

≤

k∥u∥T1. ■

Definition 2.3. We say thatu

∈

U_γconverges to 0, and writeu

→

0, if lim_τ→∞

∥u(· +

τ

)∥T

=

0. We defineU_γ0

:= {u

∈

Uγ

:

u

→

0}

as the set of all the admissible inputs that converge to 0.

ByProposition 2.2, it follows that if lim_τ→∞

∥u(· +

τ

)∥T

=

0 for

someT

>

0 then lim_τ→∞

∥u(· +

τ

)∥T

=

0 for allT

>

0. Therefore,

neither the setU_γ0 nor the convergence ofuto 0 depend on the numberTin their definition.

3. Zero-input stability inheritance

In Section 3.1, we state and prove our main result, namely Theorem 3.2. Some particular cases are addressed in Sections3.2 and3.3. Specifically, Section3.2contains results for finite-energy inputs and Section 3.3for essentially bounded ones. Along this section, we show thatTheorem 3.2subsumes and extends many existing results under weaker assumptions.

3.1. Main result

We require the following definition.

Definition 3.1. A maximally defined forward solution x

:

[t

0

,

tx)

→

X of(1)is said to be forward complete iftx

= ∞. A

forward complete solutionx

: [t

0

,

∞)

→

Xof(1)converges toC, denoted byx

→

C, if limt→∞

|x(t

)|_C

=

0.

We recall that the

ω

-limit set Ω(x) of a forward complete solutionx

: [t

0

,

∞)

→

Xof(1)is the set of all the points

ξ

∈ ¯

X (X

¯

being the closure ofX) for which there exists a sequence

{t

k

} ⊂

R≥0such thattk

↗ ∞

andx(tk)

→

ξ

.

The following is the main result of the paper.

Theorem 3.2. LetAssumptions1–3hold. Let x be a forward complete solution of (1)corresponding to some input u

∈

U_γ0[where

γ

is as in (C1) inAssumption3]. Then, the following are equivalent:

(i) x

→

C , (ii) Ω(x)

∩

G

̸= ∅,

(iii)

∅ ̸=

Ω(x)

⊂

C .

The proof of Theorem 3.2 requires Lemmas 3.3 and 3.4. Lemma 3.3bounds the difference between solutions of(1)and(2) having the same initial condition. The bound given by this lemma is useful only for small values oft

−t

0because it already assumes that solutions lie in the compact setΓ in the time interval of interest. The proof ofLemma 3.3is similar to part of the proof of Lemma 3 in [15] but is included here for the reader’s convenience.

Lemma 3.3. LetAssumptions1and3hold. LetΓ

⊂

Xbe a compact set and let L_Γ

:

_R≥0

→

R≥0 be as in(A2)of Assumption1with K

=

Γ. Then, for every

η >

0there exists a positive constant

κ

=

κ

(Γ

, η

)such that the following holds: if x is a solution of (1) corresponding to u

∈

U_γ, z is a solution of (2)such that x(t)and z(t) belong toΓ for all t

∈ [t

0

,

t0

+

T

], and x(t

0)

=

z(t0)then for all t

∈ [t

0

,

t0

+

T

],

|x(t)

−

z(t)| ≤

[

η

(t

−

t0)

+

κ

∫

t

t0

γ

(u(

τ

))d

τ

]

e

∫t t₀LΓ(s)ds

.

Proof. LetΓ

⊂

Xbe compact and

η >

0.

Claim:there exists

κ

=

κ

(Γ

, η

)

>

0 such that for allt

≥

0,

ξ

∈

Γ and

µ

∈

U,

|f

(t

, ξ, µ

)

−

f(t

, ξ,

0)| ≤

η

+

κγ

(

µ

)

.

(4) From (C2) there exists 0

< δ

=

δ

(Γ

, η

)

<

1 such

that for all t

≥

0, all

ξ

∈

Γ and all

µ

∈

U such

that

|

µ

|

< δ

, it happens that

|f

(t

, ξ, µ

)

−

f(t

, ξ,

0)|

<

η

. If

ξ

∈

Γ and

|

µ

|

≥

δ

, using (C1) it follows that

|f

(t

, ξ, µ

)

−

f(t

, ξ,

0)| ≤ |f(t

, ξ, µ

)| + |f(t

, ξ,

0)| ≤2M(1+

γ

(

µ

)) and hence

|f

(t

, ξ, µ

)

−

f(t

, ξ,

0)|

/γ

(

µ

)

≤

2M[1

/γ

_δ

+

1] =:

κ

. In consequence

|f

(t

, ξ, µ

)

−

f(t

, ξ,

0)| ≤

κγ

(

µ

)

∀

ξ

∈

Γ

,

|

µ

| ≥

δ.

Combining the inequalities obtained, the claim is established. Letxbe a solution of(1)corresponding to someu

∈

U_γ and letzbe a solution of (2)such thatx(t) andz(t) lie inΓ for all t

∈ [t

0

,

t0

+

T

]

for somet0

≥

0 andT

>

0. Fixt

∈ [t

0

,

t0

+

T

].

Then, for allt0

≤

τ

≤

t, we have

|x(

τ

)

−

z(

τ

)| ≤

∫

τ

t0

|f

(s

,

x(s)

,

u(s))

−

f(s

,

z(s)

,

0)|ds

≤

∫

τ

t0

|f

(s

,

x(s)

,

u(s))

−

f(s

,

x(s)

,

0)|ds

+

∫

τ

t0

|f

(s

,

x(s)

,

0)

−

f(s

,

z(s)

,

0)|ds

≤

∫

τ

t0

[

η

+

κγ

(u(s))]ds

+

∫

τ

t0

L_Γ(s)

|x(s)

−

z(s)|ds

≤

η

(t

−

t0)

+

κ

∫

t

t0

γ

(u(s))ds

+

∫

τ

t0

L_Γ(s)

|x(s)

−

z(s)|ds

.

Using Gronwall’s inequality, it follows that

|x(t)

−

z(t)| ≤

[

η

(t

−

t0)

+

κ

∫

t t0

γ

(u(s))ds

]

e ∫t t0LΓ(s)ds

for allt

∈ [t

0

,

t0

+

T

].

■

Lemma 3.4shows that solutions of(1)that begin sufficiently close to the compact setC and correspond to inputs with suffi-ciently small power are forward complete, and thatCis uniformly Lyapunov stable under the dynamics of(1).

Lemma 3.4. LetAssumptions1–3hold. For every

ε >

0and every T

>

0there exists

δ

=

δ

(

ε,

T)

>

0so that the following holds: if x is a solution of (1)corresponding to an input u

∈

U_γ such that

∥u∥

T

≤

δ

and

|x(t

0)|C

≤

δ

for some t0

≥

0, then x is forward complete and

|x(t)|

_C

≤

ε

for all t

≥

t0.

Proof. Let

ε >

0 andT

>

0. We can assume, without loss of generality, that

ε

is small enough so thatΓ

:=

C_ε

⊂

G. From (B1), there exists

δ

∗

_∈

(0

, ε/

2) such that for allt0

≥

0 and all

ζ

∈

C_δ∗, the unique maximal solution of the zero-input system

satisfies

ϕ

(t

,

t0

, ζ

)

∈

Cε/2for allt

≥

t0. From (B3), there exists T∗

₌

T∗

(C_δ∗

, δ

∗

/

2)

>

0 such that for allt₀

≥

0 and all

ζ

∈

C_δ∗,

ϕ

(t

,

t0

, ζ

)

∈

Cδ∗_/2for allt

≥

t₀

+

T∗.

Let L_Γ

:

_R≥0

→

R≥0 be as in (A2) withK

=

Γ and let L∗

=

sup_t≥0

∫

t+T∗

t LΓ(s)ds. Consider

η

=

δ

∗

4T∗_eL∗. Let

κ

=

κ

(Γ

, η

)

be given byLemma 3.3. Pickc

>

0 such that

∥u∥

T∗

≤

c∥u∥_Tholds

for everyu

∈

U_γ. Letxbe a solution of(1)corresponding to some u

∈

U_γsuch that

κ

c∥u∥TeL

∗

< δ

∗

_/

4 and such that

|x(t

0)|C

≤

δ

∗

for somet0

≥

0. Define

τ

∗

=

sup{

τ

≥

t0

:

x(s)

∈

Γ

∀s

∈ [t

0

, τ

]}.

Sincex(t0)

∈

Cδ∗

⊂

C_ε

=

Γ,

δ

∗

< ε

andxis continuous, it follows

thatx(s)

∈

Γ for allsin some interval

[t

0

,

t0′

]

witht

′

0

>

t0. So

τ

∗

_>

t0. We claim that

τ

∗

>

t0

+

T∗. Suppose on the contrary that

τ

∗

_≤

t0

+

T∗. From the definition of

τ

∗, the continuity ofxand the definition and compactness ofΓ it follows thatx(t)

∈

Γ for allt

∈ [t

0

, τ

∗

]

and that

|x(

τ

∗)|_C

=

ε

. Sincez(t)

:=

ϕ

(t

,

t0

,

x(t0))

∈

C_ε/2

⊂

Γ for allt

≥

t0, by applyingLemma 3.3it follows that for allt

∈ [t

0

, τ

∗

]

|x(t

)

−

z(t)| ≤

[

η

(t

−

t0)

+

κ

∫

t

t0

γ

(u(

τ

))d

τ

]

e

∫t t0LΓ(s)ds

≤

[

η

T∗

+

κ

∥u∥

T∗

]

eL ∗

≤

[

η

T∗

+

κ

c∥u∥T

]

eL∗

<

δ

∗

2

.

(5)

The latter and the fact that

|z(t

)|_C

≤

ε/

2 for allt

≥

t0yield, for all t

∈ [t

0

, τ

∗

],

|x(t

)|_C

≤ |z(t

)|_C

+ |x(t

)

−

z(t)| ≤

ε

2

+

δ

∗

2

< ε.

In particular, we have that

|x(

τ

∗

)|_C

< ε

, which contradicts the fact that

|x(

τ

∗

)|_C

=

ε

. Thus

τ

∗

_>

t0

+

T∗ as claimed. Besides, since

|z(t

0

+

T∗)|C

≤

δ

∗

_/

2 and(5)holds fort

=

t0

+

T∗, we also have that

|x(t

0

+

T∗)|C

≤

δ

∗

. Repeating the same reasoning in a recursive manner we obtain that for allj

∈

_N0,

|x(t

)|C

< ε

for all

t

∈ [t

0

+

jT∗

,

t0

+

(j

+

1)T∗

]

and

|x(t

0

+

(j

+

1)T∗)|C

≤

δ

∗

. Taking

δ

=

min{

δ

∗

_,

δ∗

4κceL∗

}

thus establishes the first assertion. Sincexis contained in the compact setC_ε

⊂

X for as long as it is defined, thenxis forward complete. ■

We are now ready to proveTheorem 3.2.

Proof of Theorem 3.2. (i)

⇒

(ii) Sincex

→

C andCis compact, then

∅ ̸=

Ω(x)

∩

C

⊂

Ω(x)

∩

G.

(ii)

⇒

(iii) Firstly we will prove thatΩ(x)

∩

C_δ

̸= ∅

for every

δ >

0. Let

δ >

0. We can assume without loss of generality that C_δ

⊂

G. Pick any

ξ

∈

Ω(x)

∩

Gand anyr

>

0 such thatBr(

ξ

)

⊂

G.

Due to (B2) there exists a compact setΓ

⊂

Xso that for allt0

≥

0 and all

ζ

∈

Br(

ξ

),

ϕ

(t

,

t0

, ζ

)

∈

Γ for allt

≥

t0. From (B3), there existsT

>

0 so that for everyt0

≥

0 and every

ζ

∈

Br(

ξ

),

ϕ

(t

,

t0

, ζ

)

∈

Cδ/2for allt

≥

t0

+

T. Pick any 0

< δ

1

< δ/

2 such thatΓ_δ1

⊂

Xis compact. LetL_Γδ

1

:

R≥0

→

R≥0be as in (A2) with K

=

Γ_δ1 and letL

=

sup_t≥0

∫

t+T

t LΓδ1(s)ds. Define

η

=

δ1 4TeL. Let

κ

=

κ

(Γ_δ1

, η

) be given byLemma 3.3.

Since

ξ

∈

Ω(x)

∩

G, there exists a sequence

{t

k

}

inR≥0so that x(tk)

→

ξ

. Then

ξ

k

=

x(tk)

∈

Br(

ξ

) forklarge enough, sayk

≥

k0. Since

ρ

k

:= ∥u(· +

tk)∥T

→

0,

κρ

keL

< δ

1

/

4 forklarge enough, sayk

≥

k1. Letk

≥

max{k0

,

k1

}

and

τ

k

=

sup{

τ

≥

tk

:

x(s)

∈

Γδ1

∀s

∈ [t

k

, τ

]}. Since

x(tk)

∈

Γ andxis continuous, it follows thatx(s)

∈

Γ_δ1 for allsin some interval

[t

k

,

t

′ k

], with

t

′

k

>

tk. So

τ

k

>

tk. We claim that

τ

k

>

tk

+

T. Suppose on the contrary that

τ

k

≤

tk

+

T. From the definition of

τ

k, the continuity ofxand the

that

|x(

τ

k)|_Γ

=

δ

1. Sincezk(t)

:=

ϕ

(t

,

tk

, ξ

k)

∈

Γ

⊂

Γδ1 for all t

≥

tk, by applyingLemma 3.3it follows that for allt

∈ [t

k

, τ

k

]

|x(t)

−

zk(t)| ≤

[

η

(t

−

tk)

+

κ

∫

t

tk

γ

(u(

τ

))d

τ

]

e

∫t tkLΓδ1(s)ds

≤

[

η

T

+

κρ

_k]eL

<

δ

1

4

+

δ

1

4

=

δ

1

2

.

(6)

In consequence, the latter and the fact that

|z

k(t)|Γ

=

0 for all t

≥

tkyield

|x(

τ

k)|Γ

≤ |z

k(

τ

k)|Γ

+ |x(

τ

k)

−

zk(

τ

k)|

<

δ

1 2

,

which contradicts

|x(

τ

k)|_Γ

=

δ

1. Thus

τ

k

>

tk

+

T.

From the facts that(6)holds fort′

k

=

tk

+

Tandzk(tk

+

T)

∈

Cδ/2 it follows that

|x(t

_k′)|_C

≤ |z

k(t ′

k)|C

+ |x(t

′ k)

−

zk(t

′ k)|

<

δ

2

+

δ

1 2

≤

δ.

We then have that for allk

≥

max{k0

,

k1

},

x(tk′)

∈

Cδ. From the

compactness ofC_δ and the fact thatt′

k

↗ ∞, the existence of an

ω

-limit point ofxlying inC_δfollows.

Next we will prove thatΩ(x)

⊂

C. Let

ε >

0 and pick any T′

>

0. Let

δ

=

δ

(

ε,

T′

)

>

0 as inLemma 3.4. Let

ζ

∈

Ω(x)

∩

C_δ/2 and let

{t

k

}

be a sequence such that

ζ

k

=

x(tk)

→

ζ

. Then

ζ

k

∈

Cδ

forklarge enough, sayk

≥

k2. Sinceu

→

0, then there existsk3 such that

∥u(· +

tk)∥T′

≤

δ

for allk

≥

k₃. Letk₄

=

max{k₂

,

k₃

}

and

let

v

:

_R≥0

→

Ube the input defined by

v

(t)

=

0 ift

∈ [0

,

tk4) and

v

(t)

=

u(t) ift

∈ [t

k4

,

∞). Then the restriction of

xto

[t

k4

,

∞)

is a solution of(1)corresponding to the input

v

,

|x(t

k4)|C

≤

δ

and

∥

v

∥

_T′

≤

δ

. Then, by applyingLemma 3.4it follows that

|x(t

)|

C

≤

ε

for allt

≥

tk4. ThereforeΩ(x)

⊂

Cε for all

ε >

0 andΩ(x)

⊂

C follows.

(iii)

⇒

(i). Sincex is continuous and Ω(x) is nonempty and compact, well-known properties of

ω

-limits (see, e.g. Proposition 2.1 in [11]) imply thatΩ(x) is approached byxand thereforex

→

C. ■

We next consider the case of globally defined systems for which C is globally uniformly asymptotically stable for the zero-input system(2). The following corollary extends Corollary 4.4 of [11] to time-varying systems and under weaker assumptions (see Sec-tion3.2).

Corollary 3.5. LetAssumptions1–3hold withG

=

X

=

_Rn_{. If x is a}

forward complete solution of(1)corresponding to some u

∈

U_γ0, then as t

→ ∞

either

|x(t)|

_C

→

0or

|x(t

)|_C

→ ∞.

Proof. Suppose that lim inft→∞

|x(t)|

_C

<

∞. Then

∅ ̸=

Ω(x)

=

Ω(x)

∩

G. ByTheorem 3.2, then

|x(t)|

_C

→

0. If lim inft→∞

|x(t)|

_C

=

∞, then lim

t→∞

|x(t)|

_C

= ∞.

■

3.2. Bounded-energy inputs

In this subsection, we consider the case in which the inputsu have finite energy, that is

∫

∞

0

γ

(u(s))ds

<

∞

for some function

γ

as in condition (C1).

Note that any inputu

∈

U such that

∫

₀∞

γ

(u(s))ds

<

∞

belongs toU_γ0, and that thereforeTheorem 3.2andCorollary 3.5 remain valid if we replace the conditionu

∈

U_γ0by the stronger one

∫

∞

0

γ

(u(s))ds

<

∞. A consequence of this simple observation

is that theLp_{-input converging-state results for time-invariant}

sys-tems in [11], namely Theorem 4.2 and Corollary 4.4, can be easily deduced fromTheorem 3.2andCorollary 3.5, respectively. Those results straightforwardly follow by observing that: (a) the continu-ity offand the zero-input local Lipschitz condition assumed in [11] imply thatf satisfiesAssumption 1withU

=

_Rm_{and condition}

(C2) ofAssumption 3; (b) from the zero-input asymptotic stability of the compact setCassumed in [11] it follows thatAssumption 2 holds withCandG

=

A, whereAis domain of attraction ofC(see Remark 1); (c) the continuity off, the growth condition assumed in [11, eq. (1)] and the fact that

|

µ

| ≤

1

+ |

µ

|

pfor all

µ

∈

_Rnand allp

≥

1 imply that condition (C1) holds with

γ

(

µ

)

= |

µ

|

p, for everyp

≥

1; and (d) the inputsuconsidered in [11] belong toLpm

for somep

≥

1.

We also remark that the growth condition assumed in [11, eq. (1)] in conjunction with the continuity off constitute condi-tions that are more restrictive than (C1) inAssumption 3. Also, such growth condition is a kind of Lipschitz continuity requirement on f that we do not assume (recallRemark 3).

The main proof technique in [11] requires a converse Lyapunov argument that is not valid for time-varying systems. Consequently, the proof in the current paper is completely novel even for this particular case.

Theorem 3.6 provides a result in the caseG

=

X

=

_Rn_{, related}

to the BEICS property. Let

γ

be as in condition (C1). We say that(1) has the

γ

-BEICS property with respect to a compact subsetC

⊂

_Rn

if every solutionxof(1)corresponding to an inputu

∈

Usuch that

∫

∞

0

γ

(u(s))ds

<

∞

satisfiesx

→

C.

Theorem 3.6.LetAssumptions1–3hold withG

=

X

=

_Rn. Suppose there exists a continuously differentiable function V

:

_R≥0

×

Rn

→

R such that

1. there exists

φ

∈

K∞so that for all t

≥

0and

ξ

∈

Rn

φ

(|

ξ

|

_C)

≤

V(t

, ξ

); (7)

2. there exists R

≥

0such that for all t

≥

0,

µ

∈

U and

ξ

∈

_Rn

the following implication holds

V(t

, ξ

)

≥

R

⇒

∂

V

∂

t

+

∂

V

∂ξ

f(t

, ξ, µ

)

≤

γ

(

µ

)

,

(8)

with

γ

as in (C1) inAssumption3.

Then(1)has the

γ

-BEICS property with respect to C .

Proof. Letu

∈

Ube such that

∫

₀∞

γ

(u(s))ds

<

∞. Then,

u

∈

U_γ0. Letxbe a solution of(1)corresponding tou, maximally defined on

[t

0

,

tx). The existence of a functionVsatisfying(8)implies that

V(t

,

x(t))

≤

R

+

V(t0

,

x(t0))

+

∫

∞

t0

γ

(u(s))ds

∀t

∈ [t

0

,

tx)

.

Then(7)and the compactness ofC imply thatxis bounded on

[t

0

,

tx) and thattx

= ∞. Application of

Corollary 3.5shows that

|x|

_C

→

0. ■

The BEICS part of the main result in [9] (Theorem 3.1) is a particular case of Theorem 3.6, corresponding toC

= {0}

and U

=

_Rm_{. This can be seen as follows. Theorem 3.1 of [9] assumes}

thatf in(1) is time-invariant and locally Lipschitz, and that(1) is zero-input globally asymptotically stable and dissipative with supply function

σ

∈

K. As a consequence, Assumptions 1and 2, and (C2) of Assumption 3 are clearly satisfied. Theorem 3.1 of [9] also requires a condition named (A), which implies that (C1) ofAssumption 3is satisfied with

γ

(·)

=

σ

(|·|). Finally, the dissipativity assumption with supply function

σ

, implies (but it is not equivalent to) the existence of the functionV as required in Theorem 3.6. Therefore, application ofTheorem 3.6recovers the

3.3. Essentially bounded inputs

By considering the case of essentially bounded inputs, we are able to relax the assumptions ofTheorem 3.2even further. We recall that an inputu

∈

U is locally essentially bounded if for each T

>

0 there exists a compact set BT

⊂

U such that

|{t

∈ [0

,

T

] :

u(t)

̸∈

BT

}| =

0. Also,u

∈

Uis essentially bounded

if it is locally essentially bounded and the setBT can be selected

independently ofT

>

0. We also consider the following type of ‘meagreness’ condition onu

∈

U(cf. [18,24]):

(M) for everyT

>

0 and

λ >

0, lim

t→∞

|{s

∈ [t

,

t

+

T

] : |u(s)| ≥

λ

}| =

0

.

Lemma 3.7 characterizes essentially bounded inputs which belong toU_γ0in terms of Property (M).

Lemma 3.7.Let

γ

:

U

→

_R≥0be continuous and such that

γ

(0)

=

0 andinf|µ|≥r

γ

(

µ

)

>

0for all r

>

0. Let u

∈

Ube essentially bounded.

Then, the following are equivalent:

(a) u

∈

U_γ0.

(b) u satisfies condition (M).

Proof. (a)

⇒

(b). Suppose thatudoes not satisfy condition (M). Then there existT

>

0,

λ >

0,

ε

0

>

0 and a sequencetk

↗ ∞

such that

|{s

∈ [t

k

,

tk

+

T

] : |u(s)| ≥

λ

}| ≥

ε

0for allk. Let

γ

λ

=

inf|µ|≥λ

γ

(

µ

)

>

0. Then

∫

tk+T

tk

γ

(u(s))ds

≥

γ

_λ

ε

0

∀k

which contradicts the fact thatu

∈

U_γ0.

(b)

⇒

(a). Sinceuis essentially bounded and

γ

is continuous it follows that

γ

◦

uis essentially bounded, hence

γ

(u(t))

≤

Mfor almost allt

≥

0 for someM

>

0. LetT

>

0 and

ε >

0. From the continuity of

γ

and the fact that

γ

(0)

=

0, there exists

δ >

0 such that

γ

(

µ

)

<

₂ε_Tfor all

|

µ

|

< δ

. Fort

≥

0, letJ(t)

= {s

∈ [t

,

t

+

T

] :

|u(s)| ≥

δ

}

andI(t)

= {s

∈ [t

,

t

+

T

] : |u(s)|

< δ

}. Since

usatisfies condition (M) there existst0

>

0 such that

|J(t

)|

<

2εMfor allt

≥

t0. Then, for everyt

≥

t0we have that

∫

t+T

t

γ

(u(s))ds

=

∫

J(t)

γ

(u(s))ds

+

∫

I(t)

γ

(u(s))ds

≤

M

|J(t)| +

ε

2T

|I(t

)| ≤

ε

2

+

ε

2

=

ε.

■

The following is the main result of this subsection.

Theorem 3.8. LetAssumptions1,2and4hold. Let x be a forward complete solution of (1)corresponding to some essentially bounded input u

∈

Uwhich satisfies condition (M). Then the statements(i), (ii)and(iii)of Theorem3.2are equivalent.

Proof. Sinceuis essentially bounded, there exists a compact set B

⊂

Usuch thatu(t)

∈

Bfor almost allt

≥

0. Then, there exists uB

∈

Usuch thatuB(t)

∈

Bfor allt

≥

0 anduB(t)

=

u(t) for almost

allt

≥

0. It is clear thatxis a solution of(1)corresponding touif and only if it also is a solution corresponding touB. By replacingu

byuBandUbyB, and restrictingftoR≥0

×

X

×

Bwe can suppose without loss of generality thatUis a compact metric space and that fsatisfiesAssumptions 1,2and4. By applyingLemma 2.1it follows thatfalso satisfiesAssumption 3. The theorem then follows from Lemma 3.7andTheorem 3.2. ■

The converging-input converging-state result in Theorem 1 of [6] is a simple consequence ofTheorem 3.8and the following result.

Lemma 3.9. Let u

∈

U be locally essentially bounded and such thatlimt→∞

|u(t

)| = 0. Then u is essentially bounded and satisfies

condition (M).

Proof. The fact thatusatisfies condition (M) is straightforward. Sinceuis locally essentially bounded and satisfies limt→∞

|u(t

)| =

0, application of Remark C.1.3 in [22] to the sequence

{u

i

} ⊂

U

withui

: [0

,

1] →U,ui(t)

:=

u(t

+

i) fori

=

0

,

1

, . . .

shows the

existence of a compact setB

⊂

Usuch thatu(t)

∈

Bfor almost all t

≥

0. ■

Theorem 1 of [6] assumes thatf in(1)is time-invariant, con-tinuous and locally Lipschitz inxuniformly inu, whenubelongs to a compact subset ofU. It is also assumed that

¯

x

∈

X is an asymptotically stable equilibrium point of the zero-input system with region of attractionO. Then, ifxis a forward complete solution of(1)corresponding to a locally essentially bounded inputusuch that limt→∞

|u(t

)| =0 andxsatisfies a certain recurrence condition

(see [6] for details), the aforementioned Theorem 1 asserts that x

→ ¯

x. It is clear that the assumptions made onf in [6] imply that f satisfiesAssumptions 1and4. In addition,Assumption 2holds withC

= {¯

x}andG

=

O. The facts thatuis locally essentially bounded and limt→∞

|u(t)| =

0 imply thatuis essentially bounded

and satisfies condition (M) in virtue ofLemma 3.9. The assumptions ofTheorem 3.8are thus fulfilled. In consequence, beingΩ(x)∩G

̸=

∅

due to the recurrence condition assumed in [6], application of Theorem 3.8shows thatx

→ ¯

x, as Theorem 1 of [6] asserts.

We note that our assumptions on the functionf, particularized to the time-invariant case, are slightly weaker than those assumed in [6], since we do not require the uniform-in-the-input local Lipschitz condition assumed in that paper.

We close the subsection with a convergence result for bounded-input bounded-state (BIBS) systems, easily derived from Theo-rem 3.8. We say that system(1)is BIBS if for every maximal solution x

: [t

0

,

tx)

→

X of(1)corresponding to an essentially bounded

inputu

∈

Uthere exists a compact setK

⊂

Xsuch thatx(t)

∈

K for allt

∈ [t

0

,

tx). Then, note thatxis forward complete.

Corollary 3.10.LetAssumptions1and4hold. Suppose that Assump-tion2holds withG

=

X and that(1)is BIBS. Let x be a maximal solution of(1)corresponding to some essentially bounded input u

∈

Uwhich satisfies condition (M). Then x

→

C .

4. Some technical results

Lemma 4.1. Let (S

,

d)be a separable and locally compact metric space and let s0

∈

S. Then, there exists a proper continuous function

ω

:

S

→

_R≥0such that

ω

(s0)

=

0and

ω

(s)

≥

d(s

,

s0)for all s

∈

S.

Proof. IfSis compact just take

ω

(s)

=

d(s

,

s0). Suppose thatSis locally compact and separable but not compact. Then there exists a sequence

{K

n

}

n≥1 of compact subsets ofS such thats0

∈

K1, Kn

⊂

int(Kn+1) for everyn

≥

1 and

∪

n≥1int(Kn)

=

S. Here

int(Kl) denotes the interior ofKlfor everyl

≥

1. SinceS is not

compact,S

\

Kn

̸= ∅

for alln

≥

1. From Urysohn’s lemma (see

e.g. Chapter II of [25]), for eachn

≥

1 there exists a continuous functiongn

:

S

→ [0

,

1]such thatgn(s)

=

0 for all

µ

∈

Kn

andgn(s)

=

1 for allsoutside int(Kn+1). Letg

:

U

→

R≥0be defined viag(s)

=

∑

∞

n=1gn(s). It is an exercise to show thatg

is well defined, continuous, proper andg(s0)

=

0. The lemma follows by taking

ω

:

S

→

_R≥0, with

ω

(s)

=

max{d(s

,

s0)

,

g(s)}. In fact,

ω

is continuous,

ω

(s0)

=

0 and

ω

(s)

≥

d(s

,

s0). That

ω

is proper follows from the fact that for everyr

∈

_R,

ω

−1_((−∞

_,

_r])

₌

g−1_((−∞

_,

_r])

_{∩ {s}

_∈

_S

_:

_d(s

_,

_s

Proof of Lemma 2.1. Pick any

ξ

0

∈

X. BothUandXare separable locally compact metric spaces, the first one by hypothesis and the second one becauseX is an open subset of a separable and locally compact metric space. Then, invoking Lemma 4.1, there exist proper continuous functions

ω

1

:

U

→

R≥0and

ω

2

:

X

→

R≥0such that

ω

1(0)

=

0 and

ω

1(

µ

)

≥ |

µ

|

for all

µ

∈

U, and

ω

2(

ξ

0)

=

0 and

ω

2(

ξ

)

≥ |

ξ

−

ξ

0

|

for all

ξ

∈

X. Note that for eachr

>

0, the setsS1(r)

:= {

µ

∈

U

:

ω

1(

µ

)

≤

r}and S2(r)

:= {

ξ

∈

X

:

ω

2(

ξ

)

≤

r}are compact, because the functions

ω

i,i

=

1

,

2 are proper. Define, for allr

>

0,

ˆ

γ

(r)

:=

sup{|f(t

, ξ, µ

)| : t

≥

0

, µ

∈

S1(r)

, ξ

∈

S2(r)}

.

ˆ

γ

(r) is nonnegative, nondecreasing and finite for allr

>

0 due to the compactness ofS1(r) andS2(r) and the boundedness condition assumed. Let

γ

ˆ

(0)

:=

limr→0+

γ

ˆ

(r); this limit exists because

γ

ˆ

is

nondecreasing. Then the function

γ

˜

: [0

,

∞)

→

_R≥0, defined via

˜

γ

(r)

= ˆ

γ

(r)− ˆ

γ

(0) is nondecreasing, continuous at 0 and

γ

˜

(0)

=

0. Therefore, there exists a continuous and strictly increasing func-tion

γ

∗

_:

R≥0

→

R≥0such that

γ

∗(0)

=

0 and

γ

˜

(r)

≤

γ

∗(r) for all r

>

0. Then, for every

ξ

∈

Xand

µ

∈

Uand everyr

>

0 ands

>

0 such that

ξ

∈

S2(r) and

µ

∈

S1(s) we have

|f

(t

, ξ, µ

)| ≤ ˆ

γ

(max{r

,

s})

≤ ˆ

γ

(r)

+ ˆ

γ

(s)

≤

2

γ

ˆ

(0)

+

γ

∗(r)

+

γ

∗(s)

.

So, for all

ξ

∈

Xand

µ

∈

U

|f

(t

, ξ, µ

)| ≤inf

{

2

γ

ˆ

(0)

+

γ

∗(r)

+

γ

∗(s)

:

r

>

0

,

s

>

0

, ω

2(

ξ

)

≤

r

, ω

1(

µ

)

≤

s

}

=

2

γ

ˆ

(0)

+

γ

∗(

ω

1(

µ

))

+

γ

∗(

ω

2(

ξ

))

.

LetK

⊂

Xbe a compact set andc

=

max_ξ∈K

γ

∗(

ω

2(

ξ

))+2

γ

ˆ

(0)

+1.

Then we have that

|f

(t

, ξ, µ

)| ≤c(1

+

γ

∗(

ω

1(

µ

)))

∀

ξ

∈

K

,

∀

µ

∈

U

.

If we take

γ

=

γ

∗

_◦

_ω

1, it follows that

γ

(0)

=

0, inf|µ|≥r

γ

(

µ

)

≥

inf|µ|≥r

γ

∗

◦

ω

1(

µ

)

≥

inf|µ|≥r

γ

∗(|

µ

|)

≥

γ

∗(r)

>

0 for allr

>

0,

where we have used the facts that

ω

1(

µ

)

≥ |

µ

|

and that

γ

∗

is strictly increasing. Hence, condition (C1) is satisfied. ■

5. Conclusions

We have provided stability results for time-varying systems with inputs. More precisely, we have given conditions under which if the maximum average power of the input converges to zero, then the state trajectories will inherit stability properties from the corresponding zero-input system. For this property to hold, input power must be measured according to a function that bounds the growth of the functionf defining the system dynamics as the input value grows. Our results generalize to time-varying systems other existing results that are valid for time-invariant systems. Even when particularized to time-invariant systems, the assump-tions required are weaker than existing ones. This relaxation of the required assumptions is made possible by avoiding the use of converse Lyapunov arguments (which are not valid for time-varying systems), and by not requiring Lipschitz continuity other

than for the zero-input system. In addition, our results do not require uniqueness of solutions.

Acknowledgements

Work partially supported by ANPCyT Grants PICT 2014-2599 and PICT 2013-0852, Argentina.

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