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Uniqueness of solutions and linearized stability for impulsive

diﬀerential equations with state-dependent delay

Kevin E. M. Church∗

April 7, 2021

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, in order to guarantee uniqueness of

solutions it is necessary to impose at the very least absolute continuity [15] or, barring that, Lipschitz

[2, 33, 42] continuity of the initial data.

State-dependent delay arises naturally in problems in various areas, 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 that have

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

using Lyapunov functions/functionals. To compare, stability analysis of diﬀerential equations with state-

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

∗McGill University, Department of Mathematics and Statistics. Email: kevin.church@mcgill.ca

1

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

x0(t) = f(t, x(t), x(t−τ1(x(t)))), t 6=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(t). 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

x0(t) = f(x(t−τ(xt))), t 6=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

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

2

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:

x0(t) = x(t−τ(x(t))), t 6=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

x0(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 the observation that this lag function uniquely 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 and some auxiliary assertions

Let RCR(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

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

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

with |·|any suitable norm on Rn.

Deﬁne the function space RCRLip (Ω) = Sk≥0RCRLip(k)(Ω), with

RCRLip(k)(Ω) = φ∈ RCR(Ω) : ∀x∈[−r, 0) and y∈(−r, 0],∃1, 2>0 such that φ|[x,x+1]

and φ|[y−2,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+)is Lipschitz continuous for some

> 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×RCR(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 gno 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−)) 6= 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× RCR(Ω) →RnFor some s∈Rand a > 0, a

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

we have the inclusion φ(0) + g(t, φ)∈Ω. Let x:I → Ωbe a solution of (1)–(2) with xs=φ∈ RCR(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∈ RCR(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∈ RCRLip(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 x6=y. Deﬁne t∗= inf {t∈[0, a] : x(t)6=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, . . . , `. 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 > 0 be small enough so that x|[u(t∗),u(t∗)+]is Lipschitz continuous

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

that φ∈ RCRLip (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∗) + }.

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 L0>0 such that |u(s)−v(s)|=|τ(y(s)) −τ(x(s))| ≤ L0|x(s)−y(s)|for

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

|x(t)−y(t)|≤|t−t∗|L(k+L0+ 1) sup

s∈[0,t]|x(s)−y(s)|, t ∈[t∗, δ].

Let 0=1

2L−1(k+L0+ 1)−1. Then

sup

t∈[0,t∗+0]|x(t)−y(t)|= sup

t∈[t∗,t∗+0]|x(t)−y(t)| ≤ 1

2sup

s∈[0,t∗+0]|x(s)−y(s)|,

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

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× RCRLip(D), there exists

a > 0such that there is at most one solution of (1)–(2) 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∈ RCR(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 a0> a such that there is a unique prolongation

y: [s−r, s +a0]→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∈ RCR(D)and

τ∈[0, r], we have the inclusion φ(0) + g(t, φ)∈D. For any (s, φ)∈R×RCRLip(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 as 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 RCRLip 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∈ RCRLip(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∈ RCRLip(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} 6=∅. 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)6=x(2)(t) for t > 0 small enough,

so the solutions are distinct. Denote z(i)

x(t) = t−τ(x(i)(t)). 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, a0], for a0∈(0, a].

Let 0

1and 0

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

x(t)∈[v, v +b1] for t∈[0, 0

1] and z(2)

x(t)∈[v−b2, v] for

t∈[0, 0

2]. Then, for t∈[0, 0

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{0

1, 0

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

deﬁnition of 0

1,0

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× RCR(Rn), any solution x:I → Rnof (1)–

(2) satisfying the initial condition xs=φadmits the exponential bound ||xt|| ≤ Keβ(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

y0(t) = D2f(t, 0,0)y(t) + D3f(t, 0,0)y(t−τ1(0)), t 6=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×RCR(Rn)all

solutions y: [s−r, ∞)→Rnsatisfy the exponential estimate ||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.

4.1 Preparatory results

Proposition 13. Let x∈ RCRLip(k)(Rn)and D={u∈[−r, 0] : x(u)6=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 > 0, there exists δ > 0such that for s∈R,a > 0

and x∈ RCR([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))

deﬁned for t∈[s, s +a]satisﬁes

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

provided supv∈[s−r,s+a]|x(v)| ≤ δ.

12

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.

If x, y ∈R, denote by H(x, y) = [min{x, y},max{x, y}].

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

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

(16) can be reﬁned further to

|F(t)| ≤ ||xt|| +N

Mk||xt|| +X

u∈Dt(x)|xt(u)−xt(u−)|

,(17)

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

Dt(x) = {v∈H(−τ1(x(t)), τ1(0)) : xt(v)6=xt(v−)}.

Also, δ, N, M can be chosen such that they remain bounded as →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 2. 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 > 0, there exist ν, N,M>0such that if x∈

RCR([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))

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

|G(k)| ≤ ||xt−

k|| +N

Mk||xt−

k|| +X

u∈D−

tk(x)|xtk−(u)−xtk−(u−)|

,(18)

13

provided xt−

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

D−

t(x) = {v∈H(−τ2(x(t)), τ2(0)) : xt(v)6=xt(v−)}.

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

Remark 3. Analogous way to Remark 2, 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 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.

Corollary 19. If Ωis open, 0∈Ω, there exists ρ0∈(0, ρ]such that if ||φ||exp aJ (1 + ξ−1)≤ρ0for

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].

14

Proposition 20. Any solution z: [s−r, s +a]→Rnof a semilinear impulsive functional diﬀerential

equation

z0(t) = Azt+f(zt), t 6=tk

∆z(t) = Bzt−+g(zt−), t =tk,

satisﬁes the variation-of-constants formula

zt=U(t, s)zs+Zt

s

U(t, µ)χ0f(zµ)dµ +X

s<tk≤t

U(t, tk)χ0g(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) : RCR(Rn)→ RCR(Rn)the evolution family associated to the linear system

u0(t) = Aut, t 6=tk

∆u(t) = But−, t =tk.

4.2 Proof of Theorem 12

Choose ζ∈(0, ρ0] and > 0 small enough so that

+ 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∈ RCRLip(J ζ)(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),

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−α(t−r)||xr|| +Zt

r

K0e−α0(t−µ)

||xµ|| +N

MJζ||xµ|| +X

u∈Dµ(x)|xµ(u)−xµ(u−)|

dµ

+X

r<tk≤t

K0e−α(t−tk)

||xt−

k|| +N

MJζ||xt−

k|| +X

u∈D−

tk

|xt−

k(u)−xt−

k(u−)

15

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

x+µ=tkfor 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 2 and Remark 3 to obtain the further bound

||xt|| ≤ K0e−α0(t−r)||xr|| +Zt

r

K0e−α0(t−µ)+NM J ζ (1 + ξ−1)||xµ||dµ

+X

t<tk≤t

K0e−α0(t−tk)+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(+NM J ζ (1 + ξ−1)) + ξ−1K0(+NMJζ (1 + ξ−1))

≤ −α0+ (1 + ξ−1)K0+ max{N M, N,M}Jζ(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:

x0(t) = −γx(t) + µF (x(t−h(x(t)))), t 6=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:RCR([−r, 0],R)→Ris functional with the following properties.

•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≤h0(x)≤µ−1for x≥0, then for any φ∈ RCRLip(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.

16

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=RCRLip(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

y0(t) = −γy(t) + µF 0(0)y(t−h(0)), t 6=tk

∆y(t) = g0(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 −h0(x)(−γx +µF (y)) ≥0 for all x, y ∈D. By Corollary 9, for

each s∈Rand φ∈ RCRLip(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∈ RCR(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 +] and > 0 small enough. xis diﬀerentiable from the right and continuous in this interval,

so

x0(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∈ RCRLip 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

u0(t) = −γu(t), t 6=tk

∆u(t) = βu(t−), t 6=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).

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)].

17

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∈ RCRLip (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

x0(t) = −γx(t) + F(x(t−h(x(t)))), t 6=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 h0(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.

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 F0(0) = 1, the formal

linearization is

y0(t) = (1 −γ)y(t), t 6=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.

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.

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

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−

19

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) by exhibiting a solution on the unstable manifold, but this machinery is not available in the

case of state-dependent delays.

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