Content uploaded by Kevin E. M. Church

Author content

All content in this area was uploaded by Kevin E. M. Church on Sep 19, 2023

Content may be subject to copyright.

Uniqueness of solutions and linearized stability for impulsive

diﬀerential equations with state-dependent delay

Kevin E. M. Church

*

August 16, 2022

Abstract

We prove that under fairly natural conditions on the state space and nonlinearities, it is typical for

an impulsive diﬀerential equation with state-dependent delay to exhibit non-uniqueness of solutions.

On a constructive note, we show that uniqueness of solutions can be recovered using a Winston-type

condition on the state-dependent delay. Irrespective of uniqueness of solutions, we prove a result

on linearized stability. As a speciﬁc application, we consider a scalar equation on the positive half-

line with continuous-time negative feedback, non-negative state-dependent delayed nonlinearity and

impulse eﬀect functional satisfying aﬃne bounds.

1 Introduction

Diﬀerential equations with state-dependent delay are notorious for their lack of smoothness properties and

the wealth of associated open problems pertaining to the semiﬂow and their invariant manifolds. See [19,

25] for background. As for the associated Cauchy problem, continuity of the initial condition is typically

not enough to ensure uniqueness of solutions, as was demonstrated by the classical counterexample of

Winston [42]. Under various deﬁnitions of solution, absolute continuity [15] or Lipschitz continuity [2, 33,

42] have been imposed to guarantee uniqueness of solutions.

State-dependent delay arises naturally in several areas of scientiﬁc interest, including cell biology

[17, 32], structured population models [13], infectious diseases [45], electromagnetism [12] and turning

processes [23]. In this setting, there is a fairly mature theory of solutions, with the most robust perhaps

being the solution manifold approach originally developed by Walther [39]. General results concerning the

Cauchy problem for impulsive delay diﬀerential equations have been known for some time [3, 4, 24], but

they do not grant uniqueness if state-dependent delays are present. State-dependent delay is important

in such control engineering problems as multi-agent consensus [14, 26], and as these protocols operate

discretely in time, there is a need to understand how such systems behave in the presence of impulses.

Constant discrete delays are mathematically convenient, but their use is not always justiﬁed by the

physical problem being studied. The same is true for delay diﬀerential equations with impulses, of which

there numerous applications in biology and control [16, 21, 30, 36, 43]. Understanding the solution sets

of impulsive systems with state-dependent delay therefore has practical implications.

In another direction, stability (in the sense of Lyapunov) is a fundamental topic in dynamical systems.

It is especially important in impulsive diﬀerential equations literature, due to the applications of such

systems in control theory; see the 2018 survey article [44] for background. Several articles have recently

considered state-dependent delayed impulses from the point of view of stability [27, 28, 37, 46] using Lya-

punov functions/functionals. To compare, stability analysis of diﬀerential equations with state-dependent

*

Universit´e de Montr´eal, Centre de Recherches Math´ematiques. Email: kevin.church@umontreal.ca

1

delay (without impulses) has been studied for several decades [11, 20, 18, 29, 31]. Stability of impulsive

functional diﬀerential equations have been considered variously using Lyapunov functional-type methods

[41, 40, 44, 47] and a linearized stability result has been proven [9], but these require the continuous-time

functional to be at least Lipschitz continuous with domain being a phase space of discontinuous functions.

As we remark in Section 1.1, this means state-dependent delays in the continuous-time dynamics can not

be handled using the extant literature. As such, it seems as though stability of impulsive systems with

state-dependent delay in the continuous-time dynamics has not been well-studied.

With this discussion in mind, in this paper we will study uniqueness of solutions and linearized stability

for the impulsive diﬀerential equation with state-dependent delay

x′(t) = f(t, x(t), x(t−τ1(x(t)))), t =tk(1)

∆x(t) = g(t, xt−), t =tk,(2)

where τ1:Rn→[0, r], and the functions fand functional gwill be described later (Section 1.5). The

jump is deﬁned by ∆x(t) = x(t)−lims→t−x(s). For x:I → Rnfor an interval I, the history xtfor

[t−r, t]⊂ I is deﬁned as usual: xt(θ) = x(t+θ) for θ∈[−r, 0]. The left-limit xt−is deﬁned as follows:

xt−(θ) = x(t+θ), θ < 0

x(t−), θ = 0,

where x(t−) = lims→t−x(s) is the usual left-limit. In this paper, the sequence of impulse times tkis

always assumed increasing, and unbounded as k→ ∞. It can be either ﬁnite or inﬁnite on the left (i.e.

it may be indexed by Zor N), but in the case it is bi-inﬁnite, we require limk→±∞ |tk|=∞.

1.1 The uniqueness problem

We argue that thus far, uniqueness of solutions has been an elusive topic for impulsive systems with

state-dependent delay. While there are certainly contributions in the literature, we claim that many do

not thoroughly address state-dependent delay. Before surveying the literature, we will illustrate the main

problem. Let Xdenote a space of right-continuous functions mapping into (a subset of) Rd, deﬁned on

an interval of the form [−r, 0], possibly with additional structure (e.g. only ﬁnitely-many discontinuities).

Consider for simplicity the impulsive diﬀerential equation with state-dependent delay

x′(t) = f(x(t−τ(xt))), t =tk

∆x(t) = g(x(t−

k)), t =tk,

where τ:X→Ris non-negative. The functional deﬁning the right-hand side can be identiﬁed with

F=f◦ev ◦(id ×(−τ)),

where f:Rd→Rd, and ev :X×R→Rdis the evaluation map deﬁned by ev(ϕ, s) = ϕ(s). The evaluation

map is generally not locally Lipschitz continuous [25], even when Xis given the structure of containing only

continuous functions. Local Lipschitz conditions can be recovered if Xcontains only Lipschitz continuous

(or higher smoothness) functions, but this excludes functions with discontinuities. Consequently, for the

purposes of assuring uniqueness of solutions for impulsive systems with state-dependent delay, assuming

a priori that F:X→Rdhas a local Lipschitz property, is inappropriate.

With the above discussion in mind, let us survey some classical and more recent contributions to

the Cauchy problem for impulsive functional diﬀerential equations, framing them within the scope of

state-dependent delay. There is the work of Ballinger and Liu [3, 4], which is stated in terms of general

impulsive functional diﬀerential equations. To obtain uniqueness of solutions, Lipschitz-like conditions

are assumed at the level of the functional, and this is incompatible with state-dependent delays. Ouahab

2

[34, 35] uses a result on contraction maps in Fr´echet spaces to prove a global existence and uniqueness

result for impulsive systems with multiple (ﬁxed) delays and a general functional nonlinearity. They

work in spaces of functions with at most countably-many discontinuities, but once again, a local Lipschitz

condition is needed on the functional term to ensure uniqueness. In a recent paper of Chen and Ma [8], the

authors aim to extend the solution manifold concept from evolution equation with state-dependent delay

to the case of systems with impulses. However, the manifold the authors construct consists of continuous

(in fact, C1) functions and as such, any impulse eﬀect will move the solution oﬀ the manifold. Similarly,

discontinuous initial conditions are not permitted.

We can gain some additional insight by surveying the literature on abstract impulsive functional

evolution equations. With respect to the state-dependent delay, Azevedo [1] proves local existence and

uniqueness of solutions in a setting where state-dependent delay is permissible — that is, in a phase space

of functions with Lipschitz conditions and some discontinuities — but the state-dependence is only in

the impulse term. In the continuous-time dynamics, only a time-varying delay is permitted, and it is not

clear at present how to extend this to allow state-dependent delay. There is also the work of Benchohra

and Ziane [6] and Benchohra and Henderson [5] on impulsive evolution inclusions with state-dependent

delay, but as this is a multivalued setting, only existence of solutions is considered. Neutral equations

are considered in [22] by Hern´andez, Rabello and Henr´ıquez, again with local Lipschitz conditions for

uniqueness.

1.2 A simple, typical example

We claim that even the simplest impulsive equations with state-dependent delay can have multiple solu-

tions when we allow for discontinuous initial conditions. Since the latter is strictly necessary in discussions

of continuation of solutions, any pathology in this class of system should be observable from an example

initial-value problem that features the following two ingredients:

a state-dependent delay in the continuous-time dynamics;

a discontinuous initial condition or a non-trivial impulse eﬀect.

With this in mind, consider the following “trivially” impulsive diﬀerential equation with state-dependent

delay and initial condition x0at time zero:

x′(t) = x(t−τ(x(t))), t =tk,(3)

∆x(t)=0, t =tk,

x0(θ) =

1, θ = 0

0,−1≤θ < 0

2,−2≤θ < −1,

(4)

with delay function τ(y) = (2y)2/(1 + y2). The delay has range in the interval [0,2], and τis C1. As the

impulse eﬀect is trivial, the Cauchy problem with Lipschitz continuous initial data is well-posed and has

a unique solution [2, 33]. The data x0is not continuous, but it is piecewise smooth. We claim

x(1)(t) =

1, t ∈[0,1]

0, t ∈[−1,0)

2, t ∈[−2,−1),

x(2)(t) =

1 + √2, t ∈(1/√2,1]

1+2t, t ∈[0,1/√2]

0, t ∈[−1,0)

2, t ∈[−2,−1),

(5)

are both solutions of the initial-value problem (3)–(4) deﬁned on the common domain [−2,1]. To verify

this, observe that for t∈(0,1/√2),

z(1)(t) := t−τ(x(1) (t)) = t−1, z(2) (t) := t−τ(x(2) (t)) = t−2(1 + 2t)2

1 + (1 + 2t)2<−1,(6)

3

-2 -1.5 -1 -0.5 0 0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Figure 1: Two distinct solutions (left) of the initial-value problem (3)–(4) and their associated time lags

(right). Discontinuities in the initial condition are indicated by solid dots (function value) and hollow dots

(left-limit). The time lag for the green solution (dashed-dot line) initially ﬂows according to the continuous

history segment on [−1,0), while the time lag for the purple solution (dotted line) ﬂows according to the

one on [−2,−1).

so x(1)(t−τ(x(1) (t))) = 0 and x(2)(t−τ(x(2) (t))) = 2 on this interval. Hence, for t∈[0,1/√2],

x0(0) + Zt

0

x(1)(s−τ(x(1) (s)))ds = 1 + Zt

0

0ds =1=x(1)(t)

x0(0) + Zt

0

x(2)(s−τ(x(2) (s)))ds = 1 + Zt

0

2ds = 1 + 2t=x(2) (t).

Also, z(1)(t) = t−1 and z(2) (t) = t−1−1/√2 for t∈[1/√2,1], so in particular, x(1)(t−τ(x(1) (t))) = 0

and x(1)(t−τ(x(1) (t))) = 0 on this interval. Taking into account,

x0(0) + Z1/√2

0

x(2)(s−τ(x(2) (s)))ds +Zt

1/√2

x(2)(s−τ(x(2) (s)))ds

=x(2)(1/√2) + Zt

1/√2

0ds =x(2)(t).

Therefore, x(2) is a solution (in an integrated sense) of the initial-value problem. Similarly, one can check

that x(1) is a solution. The initial condition is piecewise-constant, so the lack of uniqueness is entirely

due to the discontinuity.

A bit more analysis can give hints about why this non-uniqueness happened. First, τ(x0(0)) = 1, and

−1 is a point of discontinuity of x0. Second, we have the rather suggestive equalities

d

dtτ(x(1) (t))t=0+= 0 = lim

s→−1+x0(s),d

dtτ(x(2) (t))t=0+= 2 = lim

s→−1−x0(s).

That is, it seems as though the “diﬀerential equation for delay”, τ(x(t)), is itself ill-posed. The discon-

tinuity in x0results in two separate directions the time lag t−τ(x(t)) can ﬂow. See Figure 1.2 for a

visualization. We show in Section 3.3 that this phenomenon is fairly typical, and additional conditions

on the delay τand the functional fmust generally be imposed to prevent it from occurring.

4

1.3 Winston’s monotone lag condition

One of the earliest papers on uniqueness of solutions for diﬀerential equations with state-dependent delay

is due to Elliot Winston [42] in 1974. He considers the initial-value problem

x′(t) = F(x(t), x(t−g(x(t)))), x0=ϕ,

for ϕcontinuous, and proves the following: if Dis a domain in Rn,F:D×D→Rnis locally Lipschitz

continuous, g:D→R+has Lipschitz ﬁrst derivative and there exists η > 0 such that |y| ≤ ηimplies

∇g(x)F(x, y)<1

for all x∈D, then the above initial-value problem has a unique solution provided ||ϕ|| < η. His proof is

based on the observation that under this condition, the lag function t7→ t−g(x(t)) is strictly increasing

along any solution and that this lag function in some sense determines the solution for small time. Our

observation with the present paper is that Winston’s lag condition can also be exploited in the case of

discontinuous initial functions ϕ, and the result is once again uniqueness of solutions. As a consequence,

it can be adapted to equations with impulses.

1.4 Linearized stability

As mentioned in the third paragraph of Section 1, stability analysis of impulsive diﬀerential equations

with state-dependent delay in the continuous-time dynamics has yet to be studied in any depth. It can

be argued that the most direct approach to stability is to infer this information from linearization –

that is, through a principle of linearized stability. This has been accomplished for impulsive functional

diﬀerential equations for C1right-hand sides with Lipschitz derivatives [9], but of course this situation

does not accommodate state-dependent delay. We will remedy this by introducing a formal linearization

approach analogous to that of Cooke and Huang [11]. While that paper does indeed prove a linearized

stability, the conceptual linearization done in that paper was not fully resolved until the work of Walther

[38] rigorously derived the linear variational equation and interpreted it in the context of the solution

manifold. In the present paper we will consider only linearized stability, and make no eﬀort to formalize

the linearization process itself.

1.5 The phase space, auxiliary assertions and deﬁnitions

Let G+(I,Ω) be the space of right-continuous regulated functions (continuous from the right with ﬁnite

limits on the left at each point in the domain) deﬁned on an interval Iand mapping into Ω ⊂Rn. For I

compact, this space is complete with respect the the supremum norm (provided Ω is closed). We write

G+(Ω) ≡G+([−r, 0],Ω), and when we use this symbol without any modiﬁers, we will be referring to the

Banach space (G+(Ω,||·||∞), where Rnis itself interpreted as the normed vector space (Rn,|·|), with |·|

any suitable norm on Rn.

For a function f:X1× ··· × Xk→Yfor Banach spaces X1, . . . , Xkand Y, the partial Fr´echet

derivative with respect to the jth variable is denoted Djf.

Deﬁne the function space G+,Lip(Ω) = Sk≥0G+,Lip(k)(Ω), with

G+,Lip(k)(Ω) = ϕ∈G+(Ω) : ∀x∈[−r, 0) and y∈(−r, 0],∃ϵ1, ϵ2>0 such that ϕ|[x,x+ϵ1]

and ϕ|[y−ϵ2,y)are Lipschitz continuous with Lipschitz constant at most k.

Deﬁne the upper (vector) Dini derivative of a function ϕ:R→Rncomponent-wise as D+ϕ(t) =

(D+ϕ1(t), . . . , D+ϕn(t)) whenever it exists, where

D+ϕi(t) = lim sup

h→0+

ϕ(t+h)−ϕ(t)

h.

5

Note that if ϕis locally Lipschitz from the right at t— that is, ϕ|[t,t+ϵ)is Lipschitz continuous for some

ϵ > 0 — then D+ϕ(t) exists.

In the following sections, we will typically assume fand gfrom (1)–(2) are functions of the form

f:R×Ω×Ω→Rnand g:R×G+(Ω) →Rn, for some Ω ⊂Rnopen. Speciﬁc conditions of regularity

will be speciﬁed as needed.

Remark 1. As we will see in Theorem 5, the functional form of ghas no impact on local uniqueness of

solutions. As such, we have left it very general. For example, the state-dependent delayed impulse eﬀect

of the form

∆x(t) = g(t, x(t−), x(t−τ(x(t−)))), τ(x(t−)) = 0

g(x(t−), x(t−)), τ(x(t−)) = 0

can be included by imposing g(t, ϕ) = g(t, ϕ(0), ϕ(−τ(ϕ(0)))). The “piecewise” deﬁnition here is needed

to resolve the ambiguity in the composition xt−(−τ(x(t−)) when τ(x(t−)) = 0. The impulse eﬀect above

can be equivalently written in a more functional form as

∆x(t) = g(t, xt−(0), xt−(−τ(x(t−)))).

The form of gwill, however, be relevant in Section 4 for linearized stability.

1.6 Structure of the paper

Section 2 is concerned with existence and continuability of solutions; the results in this section are not new,

but are needed for further discussions. Uniqueness is considered in Section 3, with our converse result

appearing in Section 3.3, where we show that a Winston-type lag monotonicity condition is typically

necessary if one wishes to ensure uniqueness of solutions. We prove a linearized stability result in Section

4. We conclude with an application in Section 5 for a scalar equation with negative feedback, nonlinear

state-dependent delays, and aﬃne-bounded impulses on the positive half-line. Section 6 concludes with a

discussion.

2 Existence of solutions

Let Ω ⊂Rnbe open. Let f:R×Ω×Ω→Rnand g:R×G+(Ω) →RnFor some s∈Rand a > 0, a

function x: [s−r, s +a]→Uis a solution of (1)–(2) if x∈G+([s−r, s +a],Ω) and

x(t) = x(0) + Zt

s

f(µ, x(µ), x(µ−τ1(x(µ))))dµ +X

s<tk≤t

gtk, xt−

k, t ∈[s, s +a],(7)

with the integral interpreted in the Lebesgue sense. We can similarly allow solutions to be deﬁned on

right-open intervals [s−r, s +a). We say xsatisﬁes the initial condition (s, ϕ)∈R×G+(Ω) if xs=ϕ.

2.1 The local existence result

The proof of the following existence result can be considered an extension of the proof of Lemma 3.3 from

[4], streamlined to make use of the Schauder ﬁxed point theorem and the assumptions H.1–H.6. As the

proof is in some sense “typical”, we will merely provide a brief outline.

Lemma 1. Suppose τ1: Ω →R+is continuous and fis composite-integrable and locally bounded: for

any x, y ∈G+([s, s +a],Ω) and s∈R,a > 0, the function t7→ f(t, x(t), y (t)) is integrable, the image of

a bounded set by fis bounded. For each (s, ϕ)∈R×G+(Ω), there exists a solution x: [s−r, s +a]→Ω

of (1)–(2) satisfying the initial condition xs=ϕ, for some a > 0. Moreover, this solution is Lipschitz

continuous on [s, s +a).

6

Proof (Outline). Without loss of generality, let s= 0. Deﬁne the function space

X={ψ∈C([0, a],Rn) : ||ψ−ϕ(0)||∞≤δ, |ψ(t2)−ψ(t1)| ≤ k|t2−t1| ∀t1, t2∈[0, a]}

parameterized by some constants a,δand k. Deﬁne a map j:C([0, a],Rn)→G+([−r, a],Rn) by

jψ(t) = ψ(t), t ∈(0, a]

ϕ(t), t ∈[−r, 0].

For ψ∈C([0, a],Rn), we will write ˜

ψ=jψ. Using the conditions of the lemma, one can ﬁnd constants δ,

kand asuch that P:X → X ,

P ψ(t) = ϕ(0) + Zt

0

f(µ, ψ(µ), ψ(µ−τ1(ψ(µ))))dµ, (8)

is well-deﬁned and continuous. As Xis compact, Phas a unique ﬁxed point. In particular, one can take

asmall enough so that (0, a]∩{tk:k∈Z}=∅, and in this way, we conclude that x: [−r, a]→Rndeﬁned

by

x(t) = z(t), t ∈[0, a]

ϕ(t), t ∈[−r, 0)

satisﬁes x0=ϕ, the integral equation (7), and is Lipschitz continuous on [0, a] with constant k. Also,

δ > 0 can be chosen small enough so that elements of Xhave range in U0.

Corollary 2. Suppose the conditions of Lemma 1 are satisﬁed. The restriction of xto any interval of

the form [tk, tk+v]⊂[s, s +a]with tk+v < tk+1, is Lipschitz continuous.

Proof (Outline). Since fmaps bounded sets to bounded sets, one can always extract a Lipschitz constant

from the integral formulation of the solution. Let s1, s2∈[tk, tk+v]. Then

|x(s1)−x(s2)| ≤ Zs2

s1|f(µ, x(µ), x(µ−τ1(x(µ))))dµ| ≤ |s2−s1|K

for some constant Kthat depends on xand the enclosing interval [tk, tk+v].

2.2 Prolongation of solutions and maximal interval of existence

For intervals I1and I2, a prolongation of a solution x:I1→Ω of (1)–(2) with xs=ϕ, is a function

y∈G+(I2,Ω) that satisﬁes (7), such that I1⊂ I2and y|I1=x. Again, the proof of the following lemma

is “typical”, and we omit the proof.

Lemma 3. Suppose the conditions of Lemma 1 are satisﬁed and, additionally, for all t∈R,ϕ∈G+(Ω),

we have the inclusion ϕ(0) + g(t, ϕ)∈Ω. Let x:I → Ωbe a solution of (1)–(2) with xs=ϕ∈G+(Ω).

If sup I=b < ∞,xadmits a prolongation if and only if limt→b−x(t)∈Ω.

Again a typical result, we have a statement concerning maximal prolongations of any given solution.

A prolongation y:I2→U0of x:I1→U0is maximal if there is no prolongation z:I3→U0with

I3⊃ I2. The following can be proven using the standard argument (e.g. based on Zorn’s lemma), and is

omitted.

Lemma 4. Suppose the conditions of Lemma 3 are satisﬁed. Any solution x:I → Ωof (1)–(2) satisfying

xs=ϕfor some ϕ∈G+(Ω) admits a maximal prolongation.

7

3 Uniqueness of solutions

In this section we will prove local and global uniqueness of solutions of (1)–(2) under a Winston-type

monotone lag condition, plus some expected regularity conditions on fand τ1.

3.1 Local uniqueness of solutions

Our ﬁrst result concerns local uniqueness of solutions.

Theorem 5. Suppose the following conditions are satisﬁed.

1. For all U⊂Rand K⊂Ωcompact, there exists L > 0such that |f(t, x1, y1)−f(t, x2, y2)| ≤

L(|x1−x2|+|y1−y2|)for x1, x2, y1, y2∈Kand t∈U.

2. τ1: Ω →[0, r]is continuously diﬀerentiable and the monotone lag condition is satisﬁed:

1− ∇τ1(x)f(t, x, y)≥0 (9)

for all x, y ∈Ωand t∈R.

Then, for each s∈Rand ϕ∈G+,Lip(Ω), there exists a > 0such that (1)–(2) has a unique solution

x: [s−r, s +a]→Ωsatisfying the initial condition xs=ϕ.

Proof. As usual, let s= 0 without loss of generality. By Lemma 1, there exists a solution x: [−r, a]→Ω.

We may without loss of generality choose asmall enough so that (0, a]∩{tk:k∈Z}=∅. Suppose there

exists another solution y: [−r, a]→Ω and that x=y. Deﬁne t∗= inf {t∈[0, a] : x(t)=y(t)}. Then

t∗∈[0, a), and using (7), we have that for t∈[t∗, a],

x(t)−y(t) = Zt

t∗

f(µ, x(µ), x(µ−τ1(x(µ)))) −f(µ, x(µ), x(µ−τ1(y(µ))))dµ (10)

+Zt

t∗

f(µ, x(µ), x(µ−τ1(y(µ)))) −f(µ, x(µ), y(µ−τ1(y(µ))))dµ

+Zt

t∗

f(µ, x(µ), y(µ−τ1(y(µ)))) −f(µ, y(µ), y (µ−τ1(y(µ))))dµ,

while x(t) = y(t) for t∈[−r, t∗]. For t∈(t∗, a), we have

D+(t−τ1(y(t))) = 1 − ∇τ1(y(t))f(t, y(t), y(t−τ1(y(t))))

and by the monotone lag condition, this is non-negative. Since t7→ τ1(y(t)) is continuous on [0, a), we

conclude t7→ t−τ1(y(t)) is non-decreasing using (Corollary 11.4.1, [7]) for i= 1, . . . , ℓ. The same is true

for t7→ t−τ1(x(t)). Deﬁne u(t) = t−τ(x(t)) and v(t) = t−τ(y(t)). Then each of uand vare continuous

and non-decreasing on [t∗, a]. Let ϵ > 0 be small enough so that x|[u(t∗),u(t∗)+ϵ]is Lipschitz continuous

with some constant k > 0. Note that this can always be accomplished by using either the assumption

that ϕ∈G+,Lip(Ω) (if u(t∗)<0) or Corollary 2 (if u(t∗)≥0). Deﬁne

δ= sup{s∈[t∗, a] : max{u(s), v(s)} ≤ u(t∗) + ϵ}.

Since u(t∗) = v(t∗) and each of uand vis continuous and non-decreasing, we have δ > t∗. Applying this

to (10) and using condition 1. of the theorem, there is a constant L > 0 such that

|x(t)−y(t)|≤|t−t∗|L ksup

s∈[t∗,t]|u(s)−v(s)|+|x(u(s)) −y(u(s))|+|x(s)−y(s)|!

8

for t∈[t∗, δ]. Note that sups∈[t∗,t]|x(u(s)) −y(u(s))| ≤ sups∈[0,t]|x(s)−y(s)|for t≤δ. Since τis C1,

there exists another constant L′>0 such that |u(s)−v(s)|=|τ(y(s)) −τ(x(s))| ≤ L′|x(s)−y(s)|for

s∈[t∗, δ]. From here, we conclude that

|x(t)−y(t)|≤|t−t∗|L(k+L′+ 1) sup

s∈[0,t]|x(s)−y(s)|, t ∈[t∗, δ].

Let ϵ′=1

2L−1(k+L′+ 1)−1. Then

sup

t∈[0,t∗+ϵ′]|x(t)−y(t)|= sup

t∈[t∗,t∗+ϵ′]|x(t)−y(t)| ≤ 1

2sup

s∈[0,t∗+ϵ′]|x(s)−y(s)|,

which contradicts the deﬁnition of t∗. Therefore, x=y.

Remark 2. Condition 1 of Theorem 5 could be weakened from local Lipschitz continuity to a local

Lipschitz-like integrability condition. For example, it is enough to require for each compact K⊂Ω

and U⊂Rthe existence of an integrable function L:U→Rsuch that

ZU

f(s, x1(s), x2(s)) −f(s, y1(s), y2(s))ds≤ZU

L(s)||x(s)−y(s)||ds

for all s1=s2∈U, for a suitable norm || · ||, where x= (x1, x2)and y= (y1, y2)are G+(U, K ×K).

The following corollary can be useful in applications. Its proof is a straightforward adjustment to the

previous, and is omitted.

Corollary 6. Let Dbe a closed subset of Ω. Suppose condition 1. of Theorem 5 is satisﬁed, τ1: Ω →[0, r]

is C1, and (9) holds for t∈Rand x, y ∈D. For all s∈Rand (s, ϕ)∈R×G+,Lip(D), there exists a > 0

such that there is at most one solution of (1)–(2) deﬁned on the interval [s−r, s +a]and having range

in D.

3.2 Prolongation and global uniqueness

Similarly to Corollary 3, one can prove the following prolongation result.

Lemma 7. Suppose the conditions of Theorem 5 and additionally, for t∈R,ϕ∈G+(Ω), we have the

inclusion ϕ(0) + g(t, ϕ)∈Ω. A solution x: [s−r, s +a]→Ωof (1)–(2) admits a prolongation if and

only if limt→(s+a)−x(s)∈Ω. In this case, there exists a′> a such that there is a unique prolongation

y: [s−r, s +a′]→Ωof x.

Subsequently, we can obtain some global uniqueness results. The proofs are straightforward and

omitted.

Corollary 8. Under the assumptions of Lemma 7, exactly one of the following occurs:

the unique solution is deﬁned on [s−r, ∞), or

there is a unique solution x: [s−r, s +a)→Ωsatisfying xs=ϕwith a > 0ﬁnite, and it

admits no prolongation: that is, x(t)either becomes unbounded or approaches the boundary of Ωas

t→(s+a)−.

Corollary 9. Let the conditions of Corollary 6 hold, and additionally, for t∈R,ϕ∈G+(D)and

τ∈[0, r], we have the inclusion ϕ(0) + g(t, ϕ)∈D. For any (s, ϕ)∈R×G+,Lip(D), there exists at most

one solution of (1)–(2) satisfying xs=ϕhaving range in D, and if such a solution exists, exactly one of

the following occurs:

the unique solution is deﬁned on [s−r, ∞), or

there is a unique solution x: [s−r, s +a)→Dwith a > 0ﬁnite, and it admits no prolongation:

speciﬁcally, x(t)becomes unbounded or approaches the boundary of Das t→(s+a)−.

9

3.3 A converse theorem

In this section, we demonstrate that a constraint akin to the monotone lag condition is necessary if one

wishes to guarantee uniqueness of solutions, at least for the case of discrete state-dependent delays. First,

a preparatory lemma. Its proof is straightforward.

Lemma 10. Let f(t, x, y) = f0(x, y)for some locally Lipschitz continuous function f0:R×R→R.

Suppose (0, a]∩ {tk:k∈Z}=∅. If x: [s−r, s +a]→Rfor a > 0satisﬁes xs=ϕand t−τ(x(t)) ≤0

for t∈[s, s +a], then xis a solution of (1)–(2) if and only

x(t) = ϕ(0) + Zt

s

f0(ϕ(µ), ϕ(µ−τ1(x(µ))))dµ, t ∈[s, s +a].

With this lemma at hand, we will show that if the monotone lag condition is violated, it is generally

possible to construct an initial condition in G+,Lip such that the associated Cauchy problem is ill-posed.

This is the content of the following theorem, which is in some sense “constructive”. See Figure 3.3 for a

visualization.

Theorem 11. Let f(t, x, y) = f0(x, y)for some locally Lipschitz continuous function f0:R×R→R.

Suppose τ1:R→[0, r]is C1. Let ϕ∈G+,Lip(R)and denote v=−τ(ϕ(0)). If v∈(−r, 0) and

1− ∇τ(ϕ(0))f0(ϕ(0), ϕ(v−)) <0<1− ∇τ(ϕ(0))f(ϕ(0), ϕ(v)),(11)

then there are at least two distinct solutions of (1)–(2) satisfying the initial condition (s, ϕ). The same

holds if the above inequalities are reversed.

Proof. Without loss of generality, let s= 0. Since ϕ∈G+,Lip (R), there exists b1>0 and b2>0 such

that ϕ|[v,v+b1]and ϕ|[v−b2,v)are Lipschitz continuous. Denote ϕ(v−) = lims→v−, and deﬁne

ϕ1(θ) =

ϕ(0) + θ

v+b1(ϕ(v+b1)−ϕ(0)), θ ∈[v+b1,0]

ϕ(θ), θ ∈[v, v +b1)

ϕ(v), θ ∈[−r, v)

ϕ2(θ) =

ϕ(0) + θ

v(ϕ(v−)−ϕ(0)), θ ∈[v, 0]

ϕ(θ), θ ∈[v−b2, v)

ϕ(v−b2), θ ∈[−r, v −b2).

Each of these functions is Lipschitz continuous, from which it follows [2, 33] that (1)–(2) has a unique

solution x(i): [−r, a]→Rsatisfying x(i)

0=ϕi, for i= 1,2, deﬁned on a (mutual) interval [−r, a] such that

(0, a]∩ {tk:k∈Z} =∅. In particular, as these solutions are classical (i.e., diﬀerentiable on (0, a)),

lim

t→0+

d

dtx(1) (t) = f0(ϕ(0), ϕ(v)),lim

t→0+

d

dtx(2) (t) = f0(ϕ(0), ϕ(v−)).

By (11), these derivatives are not equal and it follows that x(1)(t)=x(2) (t) for t > 0 small enough, so the

solutions are distinct. Denote z(i)

x(t) = t−τ(x(i)(t)) for i= 1,2. These functions are continuous (in fact,

diﬀerentiable), and we have

D+z(i)

x(t) = 1 − ∇τ(x(i)(t))f(x(i)(t), x(i)(t−τ(x(i)(t)))).

By continuity and (11), z(1)

xis increasing while z(2)

xis decreasing on some interval [0, a′], for a′∈(0, a].

Let ϵ′

1and ϵ′

2>0 be small enough so that z(1)

x(t)∈[v, v +b1] for t∈[0, ϵ′

1] and z(2)

x(t)∈[v−b2, v] for

t∈[0, ϵ′

2]. Then, for t∈[0, ϵ′

i],

x(i)(t) = ϕ(0) + Zt

0

f(x(i)(µ), x(i)(z(i)

x(µ)))dµ =ϕ(0) + Zt

0

f(x(i)(µ), ϕi(z(i)

x(µ)))dµ. (12)

10

Figure 2: Visual description of the proof of Theorem 11. A discontinuity at v∈[−r, 0] is used to construct

two Lipschitz continuous “surrogate” initial conditions ϕ1(purple, bottom curve) and ϕ2(green, top curve)

based on the true, discontinuous initial condition ϕ(black). The ﬂow of the lagged variable t7→ t−τ(x(t))

is initially (for t > 0 small) increasing for the surrogate initial condition ϕ1, while for ϕ2it is decreasing.

This is indicated by arrows in the ﬁgure. Since these portions of ϕ1and ϕ2track the true initial condition

for t > 0 small, the solutions of the modiﬁed initial-value problems (for initial conditions ϕ1and ϕ2) still

satisfy the original initial-value problem.

Now deﬁne ˜x(i): [−r, min{ϵ′

1, ϵ′

2}]→Rby ˜x(i)(t) = x(i)(t) for t≥0, and ˜x(i)(t) = ϕ(t) for t < 0. By

deﬁnition of ϵ′

1,ϵ′

2and the respective ϕ1and ϕ2, these “modiﬁed” solutions still satisfy (12). By Lemma

10, each of the ˜x(i)are solutions of (1)–(2), satisfy ˜x(i)

0=ϕ, and are not equal for t > 0 small.

4 Linearized stability

In this section we will establish a linearized stability result that holds regardless of whether we have

uniqueness of solutions. The functional form of gwill be relevant in this instance, and we will assume

throughout that

g(t, ϕ, τ ) = ˜g(t, ϕ(0), ϕ(−τ2(ϕ(0))))

for a suitable ˜g:R×Ω×Ω→Rnand τ2: Ω →[0.r]. In what follows we will abuse notation and drop

the tilde, writing formally g(t, ϕ, τ ) = g(t, ϕ(0), ϕ(−τ)). This should not cause confusion.

In this section, we will assume 0 ∈Ω and the following baseline hypotheses on fand g.

H.1 For all t∈R,f(t, 0,0) = 0 f(t, ·,·) : Ω×Ω→Rnis C1, and Df (t, ·,·) is locally Lipschitz continuous

uniformly in t∈R.

H.2 For all t=tk,g(t, 0,0) = 0, g(t, ·,·) : Ω ×Ω→Rnis C1, and Dg0(t, ·,·) is locally Lipschitz

continuous uniformly in t, and x+g(t, x, y)∈Ω whenever x, y ∈Ω.

H.3 There exists ξ > 0 such that tk+1 −tk≥ξfor all k∈Z(respectively, k∈N).

Deﬁnition 1. Suppose f(t, 0,0) = 0 and g(t, 0,0) = 0 for all t∈R. The solution x= 0 of the impulsive

diﬀerential equation with state-dependent delay (1)–(2) is exponentially stable if there exist constants

K≥1, α > 0and η > 0such that for all (s, ϕ)∈R×G+(Rn), any solution x:I → Rnof (1)–(2)

satisfying the initial condition xs=ϕadmits the exponential bound ||xt|| ≤ Ke−α(t−s)||ϕ|| whenever

||ϕ|| ≤ η.

11

Note that assuming x= 0 is an equilibrium solution (in H.1 and H.2) and deﬁning stability relative

to this solution is not a great restriction. If ˜xis a bounded solution, one can perform a time-dependent

aﬃne change of coordinates to translate it to zero.

The result we will prove is the following.

Theorem 12. Suppose each of τ1and τ2are continuously diﬀerentiable and conditions H.1–H.3 are

satisﬁed. If the linear impulsive delay diﬀerential equation

y′(t) = D2f(t, 0,0)y(t) + D3f(t, 0,0)y(t−τ1(0)), t =tk(13)

∆y(t) = D2g(t, 0,0)y(t−) + D3g(t, 0,0)yt−(−τ2(0)), t =tk(14)

is exponentially stable — that is, there exists K0≥1and α0>0such that for (s, ϕ)∈R×G+(Rn)all

solutions y: [s−r, ∞)→Rnsatisfy the exponential bound ||yt|| ≤ K0e−α0(t−s)||ϕ|| for t≥s— then the

solution x= 0 of (1)–(2) is also exponentially stable.

The proof appears in Section 4.2. A brief remark: exponential stability of the linear system in Theorem

12 is equivalent to the associated evolution family U(t, s) satisfying ||U(t, s)|| ≤ K0e−α0(t−s). See later

Proposition 20. We do not prove the natural converse of this theorem, which is that if the formal linearized

system is strongly unstable (the unstable ﬁbre bundle is non-empty; see [10]), then the trivial solution is

unstable in (1)–(2). The proof of this result is rather more technical and is postponed to future research.

Before we continue, we need to deﬁne a few extra pieces of notation. If Sis a set, we denote its

cardinality by #S. If x, y ∈R, we deﬁne the convex hull H(x, y) = [min{x, y},max{x, y}].

4.1 Preparatory results

Proposition 13. Let x∈G+,Lip(k)(Rn)and D={u∈[−r, 0] : x(u)=x(u−)}.If Dis ﬁnite, then

|x(t)−x(s)| ≤ k|t−s|+X

u∈D|x(u)−x(u−)|.(15)

Proof. Let s≤tand [s, t]∩D={d1, . . . , dN}with d1< d2<··· < dN. Denote ∆x(t) = x(t)−x(t−).

Without loss of generality, assume s < d1< dN< t. Then

|x(t)−x(s)|=

x(t)−x(dN) +

N−1

X

n=1

(x(dn+1)−x(dn)) + x(d1)−x(s)

=

x(t)−x(dN) +

N−1

X

n=1

(x(d−

n+1) + ∆x(dn+1 )−x(dn)) + x(d1)−x(s)

≤

x(t)−x(dN) +

N−1

X

n=1

(x(d−

n+1)−x(dn)) + x(d−

1)−x(s)

+

N−1

X

n=1

∆x(dn+1) + ∆x(d1)

≤ |x(t)−x(dN)|+

N−1

X

n=1

|x(d−

n+1)−x(dn)|+|x(d−

1)−x(s)|+

N

X

m=1

|∆x(dm)|,

and the result follows because xis k-Lipschitz continuous on the intervals [s, d1), [d1, d2),...,[dN−1, dN),

[dN, t].

Proposition 14. Let τ1: Ω →[0, r]be C1. For any ϵ > 0, there exists δ > 0such that for s∈R,a > 0

and x∈G+([s−r, s +a],Ω) the function

F(t) = f(t, x(t), x(t−τ1(x(t)))) −D2f(s, 0,0)x(t)−D3f(s, 0,0)x(t−τ1(0))

12

deﬁned for t∈[s, s +a]satisﬁes

|F(t)| ≤ ϵ||xt|| +||D3f(t, 0,0)|| · |x(t−τ1(x(t))) −x(t−τ1(0))|(16)

provided supv∈[s−r,s+a]|x(v)| ≤ δ.

Proof. We can write

F(t) = Z1

0

D2f(t, µx(t), µx(t−τ1(x(t))))x(t) + D3f(t, µx(t), µx(t−τ1(x(t))))x(t−τ1(x(t)))dµ

−D2f(t, 0,0)x(t)−D3f(t, 0,0)x(t−τ1(0))

=Z1

0

[D2f(t, µx(t), µx(t−τ1(x(t)))) −D2f(t, 0,0)]x(t)dµ

+Z1

0

[D3f(t, µx(t), µx(t−τ1(x(t)))) −D3f(t, 0,0)]x(t−τ1(x(t)))dµ

+Z1

0

D3f(t, 0,0)[x(t−τ1(x(t))) −x(t−τ1(0))]dµ

Using the uniform local Lipschitz continuity of Df (and hence the partial derivatives D2fand D3f), we

get the claimed result.

Lemma 15. Assume the hypotheses of Proposition 14 and that τ1: Ω →[0, r]is C1. For all ϵ > 0, there

exist δ, N, M > 0such that if x∈G+([s−r, s +a],Ω) has only ﬁnitely-many discontinuities, the bound

(16) can be reﬁned further to

|F(t)| ≤ ϵ||xt|| +N

Mk||xt|| +X

u∈Dt(x)|xt(u)−xt(u−)|

,(17)

provided xt∈G+,Lip(k)(Ω) and supv∈[s−r,s+a]|x(v)| ≤ δ, where

Dt(x) = {v∈H(−τ1(x(t)),−τ1(0)) : xt(v)=xt(v−)}.

Also, δ, N, M can be chosen such that they remain bounded as ϵ→0.

Proof. Let N= supv∈R||D3f(v, 0,0)||. Without loss of generality, assume τ1(x(t)) ≥τ1(0). We can

bound |x(t−τ1(x(t))) −x(t−τ1(0))|using Proposition 13, taking into account that the discontinuities of

xt(·) are precisely the elements of Dt(x). With sup|y|≤δ|∇τ(y)| ≤ M, we get (17) by applying Proposition

14.

Remark 3. The length of the interval H(−τ1(x(t)),−τ1(0)) is bounded above by M|x(t)| ≤ M||xt||.

Consequently, if x: [s−r, s+a]→Ωis a solution and t≥s+r, then the discontinuities of xt: [−r, 0] →Rn

are due only to the impulses. By assumption H.3, for such t≥s+r, the interval H(−τ1(x(t)),−τ1(0))

contains at most (the integer ﬂoor of) ξ−1M||xt|| discontinuities. Therefore #(Dt(x)) ≤ξ−1M||xt||.

This will be incredibly important later.

In an analogous fashion, one can obtain a bound for g.

Lemma 16. Let τ2: Ω →[0, r]be C1. For all ϵ > 0, there exist ν, N,M>0such that if x∈

G+([s−r, s +a],Ω) has only ﬁnitely-many discontinuities, the function

G(k) = g(tk, xt−

k(0), xt−

k(−τ(x(t−

k)))) −D2g(tk,0,0)xt−

k(0) −D3g(tk,0,0)xt−

k(−τ2(0))

13

deﬁned for k∈Z(or k∈N, for one-sided indexed impulses) satisﬁes

|G(k)| ≤ ϵ||xt−

k|| +N

Mk||xt−

k|| +X

u∈D−

tk(x)|xtk−(u)−xtk−(u−)|

,(18)

provided xt−

k∈G+,Lip(k)(Ω) and supv∈[s−r,s+a]|x(v)| ≤ ν, where

D−

t(x) = {v∈H(−τ2(x(t)), τ2(0)) : xt(v)=xt(v−)}.

Also, ν, N,Mcan be chosen such that they remain bounded as ϵ→0.

Remark 4. Analogous way to Remark 3, for x: [s−r, s +a]→Ωa solution, we have #(D−

t(x)) ≤

ξ−1M||xt−|| whenever t≥s+r.

Proposition 17. There exists J > 0and ρ > 0such that |f(t, x, y )| ≤ Jmax{|x|,|y|} and |g(t, x, y)| ≤

Jmax{|x|,|y|} for |x|,|y| ≤ ρ.

Lemma 18. If ||ϕ||exp aJ(1 + ξ−1)≤ρfor some a > 0, for the constants Jand ρfrom Proposition

17, then any solution xof (1)–(2) deﬁned on [s−r, s +a]and satisfying xs=ϕis uniformly bounded,

with ||xt|| ≤ ||ϕ||e(t−s)J(1+ξ−1)for t∈[s, s +a].

Proof. Deﬁne the “non-uniform” left-limit x−

t(θ) = lims→(t+θ)−x(s). We can easily get the bound

||xt|| ≤ ||ϕ|| +Zt

s

J||xµ||dµ +X

s<tk≤t

J||x−

tk||

for t∈[s, s +a] from the integral equation (7), for any solution deﬁned on the interval [s−r, s +a] that

remains bounded by ρ. Deﬁne X(t) = ||xt||. Then t7→ X(t) is right-continuous with limits on the left

(Lemma 3.1.1, [10]), and a straightforward veriﬁcation shows that limt→t−

kX(t) = ||x−

tk||. Therefore,

X(t)≤ ||ϕ|| +Zt

s

J|X(µ)|dµ +X

s<tk≤t

JX(t−

k).

By the impulsive Gronwall inequality (Lemma 3.2.1, [10]),

X(t)| ≤ ||ϕ||e(t−s)J(1 + J)#{k:tk∈(s,t]}≤ ||ϕ||e(t−s)J(1+ξ−1).

Suppose that there exists t∈(s, s +a] such that |x(t)|> ρ. Then t∗= inf{t > s :|x(t)| ≥ ρ}exists, and

t∗> s. By the above argument, |x(t)| ≤ ρfor t∈[s−r, t∗), so x(t)≤ ||ϕ||e(t−s)J(1+ξ−1)for t∈[s, t∗).

If t∗/∈ {tk:k∈Z}and t∗< s +a, then xis continuous at t∗, and the condition on ||ϕ|| implies that

|x(t∗)|< ρ. This contradicts the deﬁnition of t∗. Conversely, if t∗=tkfor some k, then

|x(tk)| ≤ (1 + J)X(t−

k)≤(1 + J)||ϕ||e(tk−s)J(1 + J)#{j:tj∈(s,tk)}≤ ||ϕ||e(tk−s)J(1+ξ−1)< ρ,

which by continuity of xon the right, contradicts the deﬁnition of t∗.

To make the exponential bound of Lemma 18 useful, we need to ensure that all solutions from the

relevant initial condition are continuable to time s+a. This can be done by suitably restricting to a

smaller closed neighbourhood of zero for initial conditions ϕ. The proof is straightforward and therefore

omitted.

14

Corollary 19. If Ωis open, 0∈Ω, there exists ρ′∈(0, ρ]such that if ||ϕ||exp aJ(1 + ξ−1)≤ρ′for

some a > 0, every solution xof (1)–(2) satisfying xs=ϕis continuable to [x−s, x +a]and satisﬁes

||xt|| ≤ ||ϕ||e(t−s)J(1+ξ−1)for t∈[s, s +a].

Finally, we will require a result concerning the variation-of-constants representation of solutions for

impulsive delay diﬀerential equations in the phase space of right-continuous regulated functions. The

following is a consequence of the theory in [10].

Proposition 20. If f: [s, s +a]×G+(Ω) →Rnhas the property that t7→ f(t, xt)is regulated for any

x∈G+([s−r, s +a],Ω), then any solution z: [s−r, s +a]→Rnof the semilinear impulsive functional

diﬀerential equation

z′(t) = Azt+f(t, zt), t =tk

∆z(t) = Bzt−+g(k, zt−), t =tk,

satisﬁes the variation-of-constants formula

zt=U(t, s)zs+Zt

s

U(t, µ)χ0f(µ, zµ)dµ +X

s<tk≤t

U(t, tk)χ0g(k, zt−

k), t ≥s,

with the integral interpreted in the Gelfand-Pettis sense, χ0h(θ) = hfor θ= 0 and χ0h(θ) = 0 for θ < 0

and h∈Rn, and U(t, s) : G+(Rn)→G+(Rn)the evolution family associated to the linear system

u′(t) = Aut, t =tk

∆u(t) = But−, t =tk.

4.2 Proof of Theorem 12

Recall the constants δ, N, M introduced in Lemma 15, the constants ν, N,Mof Lemma 16, and ρ′from

Corollary 19. Choose ζ∈(0, ρ′] and ϵ > 0 small enough so that

ϵ+ max{N M, NM}J ζ(1 + ξ−1)< K −1

0(1 + ξ−1)−1α0.(19)

Let ||ϕ|| ≤ η, where η > 0 is chosen so that

ηerJ (1+ξ−1)K0<min{ζ , δ, ν}.(20)

Without loss of generality, we may take s= 0. First, suppose there exists some time t > 0 such that a

solution xsatisﬁes |x(t)|>min{ζ, δ, ν }. Deﬁne t∗= inf{t > 0 : |x(t)| ≥ min{ζ, δ, ν}}.Then t∗∈(0, t].

We ﬁrst prove that t∗> r. By Corollary 19, ||xt|| ≤ ||ϕ||etJ (1+ξ−1)for t∈[0, t∗), and by deﬁnition of η,

since K0≥1, it follows that ||xt|| <min{ζ, δ, ν }for t∈[0, r). By a similar argument to one appearing

near the end of the proof of Lemma 18, we can conclude that in fact, r < t∗.

By (7), for t∈[0, t∗) and c > 0 small,

|x(s1)−x(s2)| ≤ Zs2

s1|f(µ, x(µ), x(µ−τ1(x(µ))))|ds ≤ |s2−s1|Jζ

for any s1, s2∈[t, t +c]. Similarly, for t∈(0, t∗] and c > 0 small enough, |x(s1)−x(s2)|≤|s2−s1|Jζ for

s1, s2∈[t−c, t). Therefore, xt∈G+,Lip(Jζ)(Ω) for t∈[r, t∗), and by (7) has ﬁnitely-many discontinuities.

By Proposition 20, we can write

xt=U(t, r)xr+Zt

r

U(t, µ)χ0F(µ)dµ +X

r<tk≤t

U(t, tk)G(k),

15

for the functions Fand Gof Lemma 15 and Lemma 16. For t∈[r, t∗), we have ||xt|| <min{ζ, δ, ν},

which means by those previous lemmas we can majorize the above as follows:

||xt|| ≤ K0e−α0(t−r)||xr|| +Zt

r

K0e−α0(t−µ)

ϵ||xµ|| +N

MJ ζ||xµ|| +X

u∈Dµ(x)|xµ(u)−xµ(u−)|

dµ

+X

r<tk≤t

K0e−α0(t−tk)

ϵ||xt−

k|| +N

MJζ||xt−

k|| +X

u∈D−

tk

|xt−

k(u)−xt−

k(u−)

where the Lipschitz constant khas been replaced by ζJ due to the above discussion. Note that for

µ+u=tk< t∗for some k∈Z, we have

|xµ(u)−xµ(u−)| ≤ |g(tk, x(t−

k), xt−

k(−τ2(x(t−

k)))| ≤ J||xt−

k|| ≤ Jζ.

Now we make use the above and the observations of Remark 3 and Remark 4 to obtain the further bound

||xt|| ≤ K0e−α0(t−r)||xr|| +Zt

r

K0e−α0(t−µ)ϵ+NM J ζ(1 + ξ−1)||xµ||dµ

+X

t<tk≤t

K0e−α0(t−tk)ϵ+NMJ ζ(1 + ξ−1)||xt−

k||.

Applying the impulsive Gronwall inequality (Lemma 3.2.1, [10]) and our previous bound for ||xr||, it

follows that ||xt|| ≤ ||ϕ||erJ(1+ξ−1)K0eβ(t−r), where

β=−α0+K0(ϵ+NM J ζ(1 + ξ−1)) + ξ−1K0(ϵ+NMJ ζ (1 + ξ−1))

≤ −α0+ (1 + ξ−1)K0ϵ+ max{N M, NM}J ζ(1 + ξ−1

<0

by (19). Since ||ϕ||erJ (1+ξ−1)K0<min{ζ , δ, ν}by (20), it follows that ||xt|| <min{ζ, δ, ν }. By the

same lines as the proof of Lemma 18, we get a contradiction to the deﬁnition of t∗. Therefore, ||xt|| <

min{ζ, δ, ν }for all t≥0. The bound ||xt|| ≤ ||ϕ||erJ(1+ξ−1)K0eβ(t−r)can then be easily shown to be

satisﬁed for all t≥r, and combining this with the previous exponential bound on [0, r], we get the required

exponential stability.

5 Application: negative feedback and state-dependent nonlin-

earity with impulses

With the previous theorems and lemmas in place we will consider a speciﬁc application to a scalar

problem with negative feedback and bounded nonlinearity, proving a global existence and uniqueness and

the existence of a compact, attracting invariant set.

Proposition 21. Consider the following scalar nonlinear impulsive diﬀerential equation with state-

dependent delay and negative feedback:

x′(t) = −γx(t) + µF (x(t−h(x(t)))), t =tk(21)

∆x(t) = g(x(t−)), t =tk,(22)

with γ > 0and µ > 0. Let F:R→[0,1] be locally Lipschitz continuous. Assume there exists p > 0such

that p≤tk+1 −tkfor all k≥0, and g:G+([−r, 0],R)→Ris functional with the following properties.

16

gis non-negative: for ϕ≥0,ϕ(0−) + g(ϕ−)≥0, where ϕ−(θ) = ϕ(θ)for θ < 0and ϕ−(0) = ϕ(0−).

gmaps bounded sets to bounded sets, and g(x)≤α+βx for some α≥0and β > −1.

If h:R→[0, r]is continuously diﬀerentiable with 0≤h′(x)≤µ−1for x≥0, then for any ϕ∈G+,Lip(R)

with ϕ≥0, there exists a unique solution x: [−r, ∞)→R+of (21)–(22) satisfying x0=ϕ. If additionally

−γ+1

plog(1 + β)<0,(23)

the following assertions hold.

1. There exists a compact interval Ω0⊂[0,∞)that is attracting for nonnegative initial conditions:

if ϕ≥0, then limt→∞ dH(x(t),Ω0) = 0 for xthe solution from 1 and dHthe Hausdorﬀ distance

dH(x, B) = inf {|x−b|:b∈B}.

2. The semiﬂow on X=G+,Lip(Ω0)is well-deﬁned, in the sense that to any ϕ∈Xthere is a unique

solution x: [−r, ∞)→Ω0with x0=ϕand xt∈Xfor t≥0.

Finally, if F,gand hare continuously diﬀerentiable and g(0) = F(0) = 0, the solution x= 0 is

exponentially stable provided the same is true of the linear system

y′(t) = −γy(t) + µF ′(0)y(t−h(0)), t =tk

∆y(t) = g′(0)y(t−), t =tk,

and this holds regardless of whether (23) is satisﬁed.

Proof. Let Ω = R,D= [0,∞). It is straightforward to check the conditions of Corollary 9. Indeed, the

monotone lag condition is satisﬁed: 1 −h′(x)(−γx +µF (y)) ≥0 for all x, y ∈D. By Corollary 9, for each

s∈Rand ϕ∈G+,Lip(D), there exists at most one non-negative solution xof (21)–(22) satisfying xs=ϕ.

By Lemma 1, one can show that at least one solution exists for any ϕ∈G+(D). Let b= inf{t≥

s:x(t)<0}. Since gis non-negative, bcan not be an impulse time. If b < ∞, then x(b) = 0 and by

appealing to the integral equation, we have

x(t) = Zt

b−γx(s) + µF (x(s−h(x(s))))ds

for t∈[b, b +ϵ] and ϵ > 0 small enough. xis diﬀerentiable from the right and continuous in this interval,

so

x′(b) = −γ(b) + µF (x(b−h(x(b)))) = µF (x(b−h(0))) ≥0

by deﬁnition of F. This contradicts the deﬁnition of b. Therefore, any solution with non-negative initial

condition ϕ∈G+,Lip remains in D. By Corollary 9, such a solution must either be globally deﬁned, or

becomes unbounded in ﬁnite time.

We will now show that solutions remain bounded for all time. Let U(t, s) be the fundamental solution

of the linear equation

u′(t) = −γu(t), t =tk

∆u(t) = βu(t−), t =tk.

Let −ψ=−γ+1

plog(1 + β)<0. It follows that for any t≥s≥0,

U(t, s) = e−γ(t−s)Y

s<tk≤t

(1 + β).

17

Then U(t, s)≥0. If β≤0, then U(t, s)≤e−γ(t−s)≤e−ψ(t−s). If β > 0, then we have

Y

s<tk≤t

(1 + β) = (1 + β)#{tk:s<tk≤t}<(1 + β)t−s

p= exp t−s

plog(1 + β),

from which we can obtain the bound U(t, s)≤e−ψ(t−s).We conclude that 0 ≤U(t, s)≤e−ψ(t−s). Any

solution xof (21)–(22) with x0=ϕ≥0 satisﬁes the variation-of-constants formula

x(t) = U(t, 0)ϕ(0) + µZt

0

U(t, s)F(x(s−τ(x(s))))ds +X

s<tk≤t

U(t, tk)[g(xt−

k)−βx(t−

k)].

By consequence of the conditions of the corollary – speciﬁcally, f≥0 on R+and the properties of g–

one can verify that x≥0 as long as the solution exists. On the other hand, we also have

x(t)≤U(t, 0)ϕ(0) + µZt

0

U(t, s)ds +αX

s<tk≤t

U(t, tk)

≤e−ψtϕ(0) + µZt

0

e−ψ(t−s)ds +αX

s<tk≤t

e−ψ(t−tk)

≤e−ψtϕ(0) + µ

ψ(1 −e−ψt) + α

pZt+p

0

e−ψ(t−s)ds

=e−ψtϕ(0) + µ

ψ(1 −e−ψt) + αepψ

pψ 1−e−ψ(t−p).

Regardless the sign of ψ, it follows that solutions remain exponentially bounded for all time, which

concludes the proof of the global existence and uniqueness assertion.

Suppose now that ψ > 0. That is, (23) is satisﬁed. We directly get from the previous bound for x(t)

that solutions exist and are bounded for all time, and that lim supt→∞ x(t)≤sup(Ω0), with

Ω0=0,µ

ψ+αepψ

pψ .

This proves assertion 1. Similarly, for ϕ∈G+,Lip(Ω0), the same analysis demonstrates that xt(θ)∈Ω0.

By our previous veriﬁcation of uniqueness of solutions, it follows that the semiﬂow on Xis well-deﬁned,

proving assertion 2. The assertion concerning stability follows directly from Theorem 12.

As a speciﬁc instance of the system from (21)–(22), consider

x′(t) = −γx(t) + F(x(t−h(x(t)))), t =k(24)

∆x(t)=(βx −(β+ 1)x2)θ(1 −x), t =k, (25)

for β≥0, h(x) = x2/(1 + x2), Fpiecewise-deﬁned by

F(x) = x

1+x, x ≥0

0x < 0,

and θthe Heaviside step function. In (25), each instance of xon the right-hand side should be interpreted

as x(t−). In the language of Proposition 21, we have µ= 1. The function Fis nonnegative on [0,∞),

Lipschitz continuous, and |F| ≤ 1. As for the jump map g(x) = (βx −(β+ 1)x2)θ(1 −x), one can verify

that 0 ≤x+g(x) for x≥0 and g(x)≤βx. We have h′(x)≤3√3/8<1, so the Cauchy problem is

well-posed and all solutions are globally deﬁned and unique for t≥0 by Proposition 21.

18

0 20 40

0

1

2

0 20 40

0.5

1

1.5

2

0 20 40

0

1

2

0 5 10

0

1

2

0 10 20

0

1

2

0 10 20

0

1

2

Figure 3: Simulations of (24)–(25) for β= 1,4,6 (left to right) from the constant initial condition ϕ= 2.

Top row: γ= 1. Bottom row: γ= 2. Time ton the horizontal axis, with x(t) on the vertical.

For stability, observe that we can smoothly extend Fto (−1,∞] by instead deﬁning it by F(x) =

x/(1 + x). Then, as h(0) = 0 the stability condition is very easy to derive. Since F′(0) = 1, the formal

linearization is

y′(t) = (1 −γ)y(t), t =k

∆y(t) = βy(t−), t =k.

We will have exponential stability of the solution x= 0 provided (1 + β)e1−γ<1. The condition (23)

for the attracting invariant set will be satisﬁed if and only if β < eγ−1. In this case, the interval

Ω0=0, ψ−1with ψ=γ−log(1 + β), will be attracting.

Figure 3 provides simulations from the constant initial condition ϕ= 2 for β= 1,4,6 and γ= 1,2.

In the case γ= 1, for β= 1 the region Ω0is attracting and appears to contain a periodic solution. With

β= 4, the condition (23) is violated and the solution seems to converge to a periodic solution. At β= 6

the dynamics may be chaotic. This makes some sense, since the impulse eﬀect is essentially a constrained

logistic update. In all cases, the trivial solution appears to be unstable, which is consistent with (but is not

proven by) the stability condition. To compare, when γ= 2 and β= 1, we have (1 + β)e1−γ≈0.735 <1,

and the trivial solution is exponentially stable, as expected. This indeed appears to be the case from the

ﬁgure. Since eγ−1≈6.389 >6 for γ= 2, the simulations in the γ= 2 case all feature attractivity of the

region Ω0, and for β= 4,6 it appears to contain a periodic solution.

6 Discussion

We have presented in Section 3.3 an argument that, absent any Winston-type constraints on the state-

dependent delay, the Cauchy problem for impulsive delay diﬀerential equations is fundamentally ill-posed.

With such lag monotonicity conditions present, however, uniqueness of solutions can be saved. We have

19

focused on the case of a single discrete delay in the continuous-time dynamics, but of course this could be

readily extended to multiple discrete state-dependent delays, or to other classes of functional dependence.

Our proof of linearized stability crucially uses the assumption that the time between successive impulses

is bounded below by a constant ξ > 0. The reason this is needed is because we wanted to ensure that the

number of discontinuities in any interval of the form [t−r, t] remains ﬁnite and, in particular, bounded

by some global constant.

A natural direction of further research could be to extend our linearized stability result to the case

where the maximum delay (in this paper, r) is not known a priori. To accomplish this, it would be

necessary to ensure that τ(x(t)) remains uniformly bounded along solutions for suﬃciently small initial

conditions, so that the previous argument concerning the number of impulses in intervals such as [t−

τ(x(t)), t] can be controlled. We do not foresee this being incredibly diﬃculty, and expect it to be more of

a technical exercise. However, the variation-of-constants formula of Proposition 20 has not been extended

to the case of unbounded delay, and this was used to initiate the Gronwall inequality argument that

ultimately provides stability. As such, it might be necessary to adjust the argument somewhat and

use a Euclidean space version of the variation-of-constants formula, rather than the one in the inﬁnite-

dimensional phase space that was used here.

Another question is whether the natural converse of Theorem 12 holds. That is, does the instability of

the formal linearization (13)–(14) imply the instability of the trivial solution in (1)–(2)? Such a result was

proven for impulsive functional diﬀerential equations [9] with strong instabilities (i.e. non-trivial unstable

ﬁbre bundle) in the case of diﬀerentiable functional (i.e. no state-dependent delay) by exhibiting a solution

on the unstable manifold, but this machinery is not available in the case of state-dependent delays.

Acknowledgments

Thank you to the reviewer for their helpful comments, which led to some improvements to the paper.

References

[1] Katia A.G. Azevedo. Existence and Uniqueness of Solution for Abstract Diﬀerential Equations with

State-Dependent Time Impulses. Mediterranean Journal of Mathematics, 16(2):1–10, 2019.

[2] Istv´an Bal´azs, Philipp Getto, and Gergely R¨ost. A continuous semiﬂow on a space of Lipschitz

functions for a diﬀerential equation with state-dependent delay from cell biology. mar 2019.

[3] G Ballinger and X Liu. Existence and uniqueness results for impulsive delay diﬀerential equations.

Dynamics of Continuous, Discrete and Impulsive Systems, 5:579–591, 1999.

[4] George Ballinger and Xinzhi Liu. Existence, uniqueness and boundedness results for impulsive delay

diﬀerential equations. Applicable Analysis, 74(1):71–93, 2000.

[5] M Benchohra, J Henderson, and S K Ntouyas. Existence results for impulsive multivalued semilinear

neutral functional diﬀerential inclusions in Banach spaces. Journal of mathematical analysis and

applications, 263(2):763–780, 2001.

[6] Mouﬀak Benchohraa and Mohamed Ziane. Impulsive evolution inclusions with state-dependent delay

and multivalued jumps. Electronic Journal of Qualitative Theory of Diﬀerential Equations, (42):1–21,

2013.

[7] Andrew Bruckner. Diﬀerentiation of Real Functions. American Mathematical Society, Centre de

Recherches Mathematiques, 2 edition, 1994.

20

[8] Pengyu Chen and Weifeng Ma. The solution manifolds of impulsive diﬀerential equations. Applied

Mathematics Letters, page 107000, dec 2020.

[9] Kevin Church. Invariant manifold theory for impulsive functional diﬀerential equations with appli-

cations. PhD thesis, University of Waterloo, 2019.

[10] Kevin E M Church and Xinzhi Liu. Smooth centre manifolds for impulsive delay diﬀerential equations.

Journal of Diﬀerential Equations, 265(4):1696–1759, 2018.

[11] Kenneth L. Cooke and Wenzhang Huang. On the problem of linearization for state-dependent delay

diﬀerential equations. Proceedings of the American Mathematical Society, 124(5):1417–1426, 1996.

[12] Jayme De Luca, Nicola Guglielmi, Tony Humphries, and Antonio Politi. Electromagnetic two-

body problem: recurrent dynamics in the presence of state-dependent delay. Journal of Physics A:

Mathematical and Theoretical, 43(20):205103, may 2010.

[13] Odo Diekmann, Mats Gyllenberg, J. A. J. Metz, Shinji Nakaoka, and Andre M. de Roos. Daphnia

revisited: local stability and bifurcation theory for physiologically structured population models

explained by way of an example. Journal of Mathematical Biology, 61(2):277–318, aug 2010.

[14] S. Djaidja, Q.H. Wu, and L. Cheng. Stochastic consensus of single-integrator multi-agent systems

under relative state-dependent measurement noises and time delays. International Journal of Robust

and Nonlinear Control, 27(5):860–872, mar 2017.

[15] R. D. Driver. A neutral system with state-dependent delay. Journal of Diﬀerential Equations,

54(1):73–86, 1984.

[16] Teresa Faria and Jos´e J. Oliveira. Global asymptotic stability for a periodic delay hematopoiesis

model with impulses. Applied Mathematical Modelling, 79(November 2019):843–864, 2020.

[17] Philipp Getto and Marcus Waurick. A diﬀerential equation with state-dependent delay from cell

population biology. Journal of Diﬀerential Equations, 260(7):6176–6200, 2016.

[18] Ferenc Hartung. Linearized stability in periodic functional diﬀerential equations with state-dependent

delays. Journal of Computational and Applied Mathematics, 174(2):201–211, 2005.

[19] Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther, and Jianhong Wu. Functional Diﬀerential Equa-

tions with State-Dependent Delays: Theory and Applications. In Handbook of diﬀerential equations:

ordinary diﬀerential equations, volume 3, pages 435–545. 2006.

[20] Ferenc Hartung and Janos Turi. Linearized stability in functional diﬀerential equations with state-

dependent delays. Proceedings of the International Conference on Dynamical Systems and Diﬀerential

Equations, (SPEC. ISSUE):416–425, 2001.

[21] Mengxin He, Fengde Chen, and Zhong Li. Permanence and global attractivity of an impulsive delay

Logistic model. Applied Mathematics Letters, 62:92–100, 2016.

[22] Eduardo Hern´andez M., Marco Rabello, and Hern´an R. Henr´ıquez. Existence of solutions for impul-

sive partial neutral functional diﬀerential equations. Journal of Mathematical Analysis and Applica-

tions, 331(2):1135–1158, 2007.

[23] Tam´as Insperger, G´abor St´ep´an, and Janos Turi. State-dependent delay in regenerative turning

processes. Nonlinear Dynamics, 47(1-3):275–283, dec 2006.

21

[24] S V Krishna and A V Anokhin. Delay diﬀerential systems with discontinuous initial data and existence

and uniqueness theorems for systems with impulse and delay. Journal of Applied Mathematics and

Stochastic Analysis, (C):49–67, 1994.

[25] Tibor Krisztin and Hans Otto Walther. Smoothness issues in diﬀerential equations with state-

dependent delay. Rendiconti dell’Istituto di Matematica dell’Universita di Trieste, 49:95–112, 2017.

[26] Tao Li, Fuke Wu, and Ji-Feng Zhang. Multi-Agent Consensus With Relative-State-Dependent Mea-

surement Noises. IEEE Transactions on Automatic Control, 59(9):2463–2468, sep 2014.

[27] Xiaodi Li and Jianhong Wu. Stability of nonlinear diﬀerential systems with state-dependent delayed

impulses. Automatica, 64:63–69, 2016.

[28] Xiaodi Li and Jianhong Wu. Suﬃcient Stability Conditions of Nonlinear Diﬀerential Systems Un-

der Impulsive Control With State-Dependent Delay. IEEE Transactions on Automatic Control,

63(1):306–311, jan 2018.

[29] Xiaodi Li and Xueyan Yang. Lyapunov stability analysis for nonlinear systems with state-dependent

state delay. Automatica, 112:108674, feb 2020.

[30] Jianquan Lu, Zidong Wang, Jinde Cao, Daniel W C Ho, and Jurgen Kurths. Pinning impulsive

stabilization of nonlinear dynamical networks with time-varying delay. International Journal of

Bifurcation and Chaos, 22(07):1250176, 2012.

[31] F. M. G. Magpantay and A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar

state-dependent delay diﬀerential equations. Discrete & Continuous Dynamical Systems - S, 13(1):85–

104, 2020.

[32] Joseph M. Mahaﬀy, Jacques B´elair, and Michael C. Mackey. Hematopoietic Model with Moving

Boundary Condition and State Dependent Delay: Applications in Erythropoiesis. Journal of Theo-

retical Biology, 190(2):135–146, jan 1998.

[33] John Mallet-Paret, Roger D. Nussbaum, and Panagiotis Paraskevopoulos. Periodic solutions for

functional diﬀerential equations with multiple state-dependent time lags. Topological Methods in

Nonlinear Analysis, 3(1):101, 1994.

[34] Abdelghani Ouahab. Local and global existence and uniqueness results for impulsive functional diﬀer-

ential equations with multiple delay. Journal of Mathematical Analysis and Applications, 323(1):456–

472, nov 2006.

[35] Abdelghani Ouahab. Existence and uniqueness results for impulsive functional diﬀerential equations

with scalar multiple delay and inﬁnite delay. Nonlinear Analysis: Theory, Methods & Applications,

67(4):1027–1041, aug 2007.

[36] Yongzhen Pei, Shaoying Liu, Changguo Li, and Lansun Chen. The dynamics of an impulsive delay

SI model with variable coeﬃcients. Applied Mathematical Modelling, 33(6):2766–2776, 2009.

[37] Wei Ren and Junlin Xiong. Stability Analysis of Impulsive Switched Time-Delay Systems With

State-Dependent Impulses. IEEE Transactions on Automatic Control, 64(9):3928–3935, sep 2019.

[38] H.-O. Walther. Smoothness Properties of Semiﬂows for Diﬀerential Equations with State-Dependent

Delays. Journal of Mathematical Sciences, 124(4):5193–5207, 2004.

[39] Hans Otto Walther. The solution manifold and C1-smoothness for diﬀerential equations with state-

dependent delay. Journal of Diﬀerential Equations, 195(1):46–65, 2003.

22

[40] Huamin Wang, Shukai Duan, Chuandong Li, Lidan Wang, and Tingwen Huang. Globally exponential

stability of delayed impulsive functional diﬀerential systems with impulse time windows. Nonlinear

Dynamics, 84(3):1655–1665, 2016.

[41] Qing Wang and Xinzhi Liu. Impulsive stabilization of delay diﬀerential systems via the Lyapunov-

Razumikhin method. Applied Mathematics Letters, 20(8):839–845, 2007.

[42] Elliot Winston. Uniqueness of solutions of state dependent delay diﬀerential equations. Journal of

Mathematical Analysis and Applications, 47(3):620–625, 1974.

[43] Liu Yang and Shouming Zhong. Dynamics of an impulsive diﬀusive ecological model with distributed

delay and additive Allee eﬀect. Journal of Applied Mathematics and Computing, 48(1-2), 2015.

[44] Xueyan Yang, Xiaodi Li, Qiang Xi, and Peiyong Duan. Review of stability and stabilization for

impulsive delayed systems. Mathematical Biosciences & Engineering, 15(6):1495–1515, 2018.

[45] Yuan Yuan and Jacques B´elair. Threshold dynamics in an SEIRS model with latency and temporary

immunity. Journal of Mathematical Biology, 69(4):875–904, oct 2014.

[46] Xiaoyu Zhang and Chuandong Li. Finite-time stability of nonlinear systems with state-dependent

delayed impulses. Nonlinear Dynamics, 102(1):197–210, sep 2020.

[47] Yu Zhang and Jitao Sun. Stability of impulsive functional diﬀerential equations. Nonlinear Analysis,

Theory, Methods and Applications, 68(12):3665–3678, 2008.

23