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Uniqueness of solutions and linearized stability for impulsive differential equations with state-dependent delay



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.
Uniqueness of solutions and linearized stability for impulsive
differential equations with state-dependent delay
Kevin E. M. Church
August 16, 2022
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:
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(tτ1(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)limstx(s). For x:I Rnfor an interval I, the history xtfor
[tr, t] I is defined as usual: xt(θ) = x(t+θ) for θ[r, 0]. The left-limit xtis defined as follows:
xt(θ) = x(t+θ), θ < 0
x(t), θ = 0,
where x(t) = limstx(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(tτ(xt))), t =tk
x(t) = g(x(t
k)), t =tk,
where τ:XRis non-negative. The functional defining the right-hand side can be identified with
F=fev (id ×(τ)),
where f:RdRd, and ev :X×RRdis 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:XRdhas 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 Fechet 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
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(tτ(x(t))), t =tk,(3)
x(t)=0, t =tk,
x0(θ) =
1, θ = 0
0,1θ < 0
2,2θ < 1,
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),
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) := tτ(x(1) (t)) = t1, z(2) (t) := tτ(x(2) (t)) = t2(1 + 2t)2
1 + (1 + 2t)2<1,(6)
-2 -1.5 -1 -0.5 0 0.5
0 0.2 0.4 0.6 0.8 1
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)(tτ(x(1) (t))) = 0 and x(2)(tτ(x(2) (t))) = 2 on this interval. Hence, for t[0,1/2],
x0(0) + Zt
x(1)(sτ(x(1) (s)))ds = 1 + Zt
0ds =1=x(1)(t)
x0(0) + Zt
x(2)(sτ(x(2) (s)))ds = 1 + Zt
2ds = 1 + 2t=x(2) (t).
Also, z(1)(t) = t1 and z(2) (t) = t11/2 for t[1/2,1], so in particular, x(1)(tτ(x(1) (t))) = 0
and x(1)(tτ(x(1) (t))) = 0 on this interval. Taking into account,
x0(0) + Z1/2
x(2)(sτ(x(2) (s)))ds +Zt
x(2)(sτ(x(2) (s)))ds
=x(2)(1/2) + Zt
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
dtτ(x(1) (t))t=0+= 0 = lim
dtτ(x(2) (t))t=0+= 2 = lim
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 tτ(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.
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(tg(x(t)))), x0=ϕ,
for ϕcontinuous, and proves the following: if Dis a domain in Rn,F:D×DRnis locally Lipschitz
continuous, g:DR+has Lipschitz first derivative and there exists η > 0 such that |y| ηimplies
g(x)F(x, y)<1
for all xD, 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→ tg(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 Rn. For I
compact, this space is complete with respect the the supremum norm (provided 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× ··· × XkYfor 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) = Sk0G+,Lip(k)(Ω), with
G+,Lip(k)(Ω) = ϕG+(Ω) : x[r, 0) and y(r, 0],ϵ1, ϵ2>0 such that ϕ|[x,x+ϵ1]
and ϕ|[yϵ2,y)are Lipschitz continuous with Lipschitz constant at most k.
Define the upper (vector) Dini derivative of a function ϕ:RRncomponent-wise as D+ϕ(t) =
(D+ϕ1(t), . . . , D+ϕn(t)) whenever it exists, where
D+ϕi(t) = lim sup
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××Rnand g:R×G+(Ω) Rn, for some 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(tτ(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
2 Existence of solutions
Let Rnbe open. Let f:R××Rnand g:R×G+(Ω) RnFor some sRand a > 0, a
function x: [sr, s +a]Uis a solution of (1)–(2) if xG+([sr, s +a],Ω) and
x(t) = x(0) + Zt
f(µ, x(µ), x(µτ1(x(µ)))) +X
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 [sr, 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: R+is continuous and fis composite-integrable and locally bounded: for
any x, y G+([s, s +a],Ω) and sR,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: [sr, s +a]
of (1)(2) satisfying the initial condition xs=ϕ, for some a > 0. Moreover, this solution is Lipschitz
continuous on [s, s +a).
Proof (Outline). Without loss of generality, let s= 0. Define the function space
X={ψC([0, a],Rn) : ||ψϕ(0)||δ, |ψ(t2)ψ(t1)| k|t2t1| t1, t2[0, a]}
parameterized by some constants a,δand k. Define a map j:C([0, a],Rn)G+([r, a],Rn) by
(t) = ψ(t), t (0, a]
ϕ(t), t [r, 0].
For ψC([0, a],Rn), we will write ˜
ψ=. Using the conditions of the lemma, one can find constants δ,
kand asuch that P:X X ,
P ψ(t) = ϕ(0) + Zt
f(µ, ψ(µ), ψ(µτ1(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:kZ}=, and in this way, we conclude that x: [r, a]Rndefined
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(µτ1(x(µ))))| |s2s1|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 of (1)–(2) with xs=ϕ, is a function
yG+(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 tR,ϕG+(Ω),
we have the inclusion ϕ(0) + g(t, ϕ). Let x:I be a solution of (1)(2) with xs=ϕG+(Ω).
If sup I=b < ,xadmits a prolongation if and only if limtbx(t).
Again a typical result, we have a statement concerning maximal prolongations of any given solution.
A prolongation y:I2U0of x:I1U0is maximal if there is no prolongation z:I3U0with
I3 I2. The following can be proven using the standard argument (e.g. based on Zorn’s lemma), and is
Lemma 4. Suppose the conditions of Lemma 3 are satisfied. Any solution x:I of (1)(2) satisfying
xs=ϕfor some ϕG+(Ω) admits a maximal prolongation.
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 URand Kcompact, there exists L > 0such that |f(t, x1, y1)f(t, x2, y2)|
L(|x1x2|+|y1y2|)for x1, x2, y1, y2Kand tU.
2. τ1: [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 and tR.
Then, for each sRand ϕG+,Lip(Ω), there exists a > 0such that (1)(2) has a unique solution
x: [sr, s +a]satisfying 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:kZ}=. Suppose there
exists another solution y: [r, a] 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
f(µ, x(µ), x(µτ1(x(µ)))) f(µ, x(µ), x(µτ1(y(µ)))) (10)
f(µ, x(µ), x(µτ1(y(µ)))) f(µ, x(µ), y(µτ1(y(µ))))
f(µ, x(µ), y(µτ1(y(µ)))) f(µ, y(µ), y (µτ1(y(µ))))dµ,
while x(t) = y(t) for t[r, t]. For t(t, a), we have
D+(tτ1(y(t))) = 1 τ1(y(t))f(t, y(t), y(tτ1(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→ tτ1(y(t)) is non-decreasing using (Corollary 11.4.1, [7]) for i= 1, . . . , . The same is true
for t7→ tτ1(x(t)). Define u(t) = tτ(x(t)) and v(t) = tτ(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 ϕG+,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)|≤|tt|L ksup
s[t,t]|u(s)v(s)|+|x(u(s)) y(u(s))|+|x(s)y(s)|!
for t[t, δ]. Note that sups[t,t]|x(u(s)) y(u(s))| sups[0,t]|x(s)y(s)|for tδ. 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)|≤|tt|L(k+L+ 1) sup
s[0,t]|x(s)y(s)|, t [t, δ].
Let ϵ=1
2L1(k+L+ 1)1. Then
t[0,t+ϵ]|x(t)y(t)|= sup
t[t,t+ϵ]|x(t)y(t)| 1
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
and URthe existence of an integrable function L:URsuch that
f(s, x1(s), x2(s)) f(s, y1(s), y2(s))dsZU
for all s1=s2U, 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 . Suppose condition 1. of Theorem 5 is satisfied, τ1: [0, r]
is C1, and (9) holds for tRand x, y D. For all sRand (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 [sr, 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 tR,ϕG+(Ω), we have the
inclusion ϕ(0) + g(t, ϕ). A solution x: [sr, s +a]of (1)(2) admits a prolongation if and
only if limt(s+a)x(s). In this case, there exists a> a such that there is a unique prolongation
y: [sr, s +a]of x.
Subsequently, we can obtain some global uniqueness results. The proofs are straightforward and
Corollary 8. Under the assumptions of Lemma 7, exactly one of the following occurs:
the unique solution is defined on [sr, ), or
there is a unique solution x: [sr, s +a)satisfying xs=ϕwith a > 0finite, and it
admits no prolongation: that is, x(t)either becomes unbounded or approaches the boundary of as
Corollary 9. Let the conditions of Corollary 6 hold, and additionally, for tR,ϕG+(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 [sr, ), or
there is a unique solution x: [sr, s +a)Dwith a > 0finite, and it admits no prolongation:
specifically, x(t)becomes unbounded or approaches the boundary of Das t(s+a).
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×RR.
Suppose (0, a] {tk:kZ}=. If x: [sr, s +a]Rfor a > 0satisfies xs=ϕand tτ(x(t)) 0
for t[s, s +a], then xis a solution of (1)(2) if and only
x(t) = ϕ(0) + Zt
f0(ϕ(µ), ϕ(µτ1(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
Theorem 11. Let f(t, x, y) = f0(x, y)for some locally Lipschitz continuous function f0:R×RR.
Suppose τ1:R[0, r]is C1. Let ϕG+,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 ϕG+,Lip (R), there exists b1>0 and b2>0 such
that ϕ|[v,v+b1]and ϕ|[vb2,v)are Lipschitz continuous. Denote ϕ(v) = limsv, 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 [vb2, v)
ϕ(vb2), 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:kZ} =. In particular, as these solutions are classical (i.e., differentiable on (0, a)),
dtx(1) (t) = f0(ϕ(0), ϕ(v)),lim
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) = tτ(x(i)(t)) for i= 1,2. These functions are continuous (in fact,
differentiable), and we have
x(t) = 1 τ(x(i)(t))f(x(i)(t), x(i)(tτ(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)[vb2, v] for
t[0, ϵ
2]. Then, for t[0, ϵ
x(i)(t) = ϕ(0) + Zt
f(x(i)(µ), x(i)(z(i)
x(µ))) =ϕ(0) + Zt
f(x(i)(µ), ϕi(z(i)
x(µ)))dµ. (12)
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→ tτ(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 t0, and ˜x(i)(t) = ϕ(t) for t < 0. By
definition of 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××Rnand τ2: [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 and the following baseline hypotheses on fand g.
H.1 For all tR,f(t, 0,0) = 0 f(t, ·,·) : ×Rnis C1, and Df (t, ·,·) is locally Lipschitz continuous
uniformly in tR.
H.2 For all t=tk,g(t, 0,0) = 0, g(t, ·,·) : ×Rnis C1, and Dg0(t, ·,·) is locally Lipschitz
continuous uniformly in t, and x+g(t, x, y) whenever x, y Ω.
H.3 There exists ξ > 0 such that tk+1 tkξfor all kZ(respectively, kN).
Definition 1. Suppose f(t, 0,0) = 0 and g(t, 0,0) = 0 for all tR. The solution x= 0 of the impulsive
differential equation with state-dependent delay (1)(2) is exponentially stable if there exist constants
K1, 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|| Keα(ts)||ϕ|| whenever
||ϕ|| η.
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(tτ1(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 K01and α0>0such that for (s, ϕ)R×G+(Rn)all
solutions y: [sr, )Rnsatisfy the exponential bound ||yt|| K0eα0(ts)||ϕ|| for ts 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)|| K0eα0(ts). 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 xG+,Lip(k)(Rn)and D={u[r, 0] : x(u)=x(u)}.If Dis finite, then
|x(t)x(s)| k|ts|+X
Proof. Let stand [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(dN) +
(x(dn+1)x(dn)) + x(d1)x(s)
x(t)x(dN) +
n+1) + x(dn+1 )x(dn)) + x(d1)x(s)
x(t)x(dN) +
n+1)x(dn)) + x(d
x(dn+1) + x(d1)
and the result follows because xis k-Lipschitz continuous on the intervals [s, d1), [d1, d2),...,[dN1, dN),
[dN, t].
Proposition 14. Let τ1: [0, r]be C1. For any ϵ > 0, there exists δ > 0such that for sR,a > 0
and xG+([sr, s +a],Ω) the function
F(t) = f(t, x(t), x(tτ1(x(t)))) D2f(s, 0,0)x(t)D3f(s, 0,0)x(tτ1(0))
defined for t[s, s +a]satisfies
|F(t)| ϵ||xt|| +||D3f(t, 0,0)|| · |x(tτ1(x(t))) x(tτ1(0))|(16)
provided supv[sr,s+a]|x(v)| δ.
Proof. We can write
F(t) = Z1
D2f(t, µx(t), µx(tτ1(x(t))))x(t) + D3f(t, µx(t), µx(tτ1(x(t))))x(tτ1(x(t)))
D2f(t, 0,0)x(t)D3f(t, 0,0)x(tτ1(0))
[D2f(t, µx(t), µx(tτ1(x(t)))) D2f(t, 0,0)]x(t)
[D3f(t, µx(t), µx(tτ1(x(t)))) D3f(t, 0,0)]x(tτ1(x(t)))
D3f(t, 0,0)[x(tτ1(x(t))) x(tτ1(0))]
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: [0, r]is C1. For all ϵ > 0, there
exist δ, N, M > 0such that if xG+([sr, s +a],Ω) has only finitely-many discontinuities, the bound
(16) can be refined further to
|F(t)| ϵ||xt|| +N
Mk||xt|| +X
provided xtG+,Lip(k)(Ω) and supv[sr,s+a]|x(v)| δ, where
Dt(x) = {vH(τ1(x(t)),τ1(0)) : xt(v)=xt(v)}.
Also, δ, N, M can be chosen such that they remain bounded as ϵ0.
Proof. Let N= supvR||D3f(v, 0,0)||. Without loss of generality, assume τ1(x(t)) τ1(0). We can
bound |x(tτ1(x(t))) x(tτ1(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
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: [sr, s+a]is a solution and ts+r, then the discontinuities of xt: [r, 0] Rn
are due only to the impulses. By assumption H.3, for such ts+r, the interval H(τ1(x(t)),τ1(0))
contains at most (the integer floor of) ξ1M||xt|| discontinuities. Therefore #(Dt(x)) ξ1M||xt||.
This will be incredibly important later.
In an analogous fashion, one can obtain a bound for g.
Lemma 16. Let τ2: [0, r]be C1. For all ϵ > 0, there exist ν, N,M>0such that if x
G+([sr, s +a],Ω) has only finitely-many discontinuities, the function
G(k) = g(tk, xt
k(0), xt
k)))) D2g(tk,0,0)xt
k(0) D3g(tk,0,0)xt
defined for kZ(or kN, for one-sided indexed impulses) satisfies
|G(k)| ϵ||xt
k|| +N
k|| +X
provided xt
kG+,Lip(k)(Ω) and supv[sr,s+a]|x(v)| ν, where
t(x) = {vH(τ2(x(t)), τ2(0)) : xt(v)=xt(v)}.
Also, ν, N,Mcan be chosen such that they remain bounded as ϵ0.
Remark 4. Analogous way to Remark 3, for x: [sr, s +a]a solution, we have #(D
ξ1M||xt|| whenever ts+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 + ξ1)ρfor some a > 0, for the constants Jand ρfrom Proposition
17, then any solution xof (1)(2) defined on [sr, s +a]and satisfying xs=ϕis uniformly bounded,
with ||xt|| ||ϕ||e(ts)J(1+ξ1)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
J||xµ|| +X
for t[s, s +a] from the integral equation (7), for any solution defined on the interval [sr, 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 limtt
kX(t) = ||x
tk||. Therefore,
X(t) ||ϕ|| +Zt
J|X(µ)| +X
By the impulsive Gronwall inequality (Lemma 3.2.1, [10]),
X(t)| ||ϕ||e(ts)J(1 + J)#{k:tk(s,t]} ||ϕ||e(ts)J(1+ξ1).
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[sr, t), so x(t) ||ϕ||e(ts)J(1+ξ1)for t[s, t).
If t/ {tk:kZ}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(tks)J(1 + J)#{j:tj(s,tk)} ||ϕ||e(tks)J(1+ξ1)< 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
Corollary 19. If is open, 0, there exists ρ(0, ρ]such that if ||ϕ||exp aJ(1 + ξ1)ρfor
some a > 0, every solution xof (1)(2) satisfying xs=ϕis continuable to [xs, x +a]and satisfies
||xt|| ||ϕ||e(ts)J(1+ξ1)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
xG+([sr, s +a],Ω), then any solution z: [sr, 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
U(t, µ)χ0f(µ, zµ) +X
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 hRn, 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 ρfrom
Corollary 19. Choose ζ(0, ρ] and ϵ > 0 small enough so that
ϵ+ max{N M, NM}J ζ(1 + ξ1)< K 1
0(1 + ξ1)1α0.(19)
Let ||ϕ|| η, where η > 0 is chosen so that
ηerJ (1+ξ1)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+ξ1)for t[0, t), and by definition of η,
since K01, 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(µτ1(x(µ))))|ds |s2s1|Jζ
for any s1, s2[t, t +c]. Similarly, for t(0, t] and c > 0 small enough, |x(s1)x(s2)|≤|s2s1|Jζ for
s1, s2[tc, t). Therefore, xtG+,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
U(t, µ)χ0F(µ) +X
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|| K0eα0(tr)||xr|| +Zt
ϵ||xµ|| +N
MJ ζ||xµ|| +X
k|| +N
k|| +X
where the Lipschitz constant khas been replaced by ζJ due to the above discussion. Note that for
µ+u=tk< tfor some kZ, we have
|xµ(u)xµ(u)| |g(tk, x(t
k), xt
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|| K0eα0(tr)||xr|| +Zt
K0eα0(tµ)ϵ+NM J ζ(1 + ξ1)||xµ||
K0eα0(ttk)ϵ+NMJ ζ(1 + ξ1)||xt
Applying the impulsive Gronwall inequality (Lemma 3.2.1, [10]) and our previous bound for ||xr||, it
follows that ||xt|| ||ϕ||erJ(1+ξ1)K0eβ(tr), where
β=α0+K0(ϵ+NM J ζ(1 + ξ1)) + ξ1K0(ϵ+NMJ ζ (1 + ξ1))
α0+ (1 + ξ1)K0ϵ+ max{N M, NM}J ζ(1 + ξ1
by (19). Since ||ϕ||erJ (1+ξ1)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 t0. The bound ||xt|| ||ϕ||erJ(1+ξ1)K0eβ(tr)can then be easily shown to be
satisfied for all tr, 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(th(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 ptk+1 tkfor all k0, and g:G+([r, 0],R)Ris functional with the following properties.
gis non-negative: for ϕ0,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 α0and β > 1.
If h:R[0, r]is continuously differentiable with 0h(x)µ1for x0, then for any ϕG+,Lip(R)
with ϕ0, there exists a unique solution x: [r, )R+of (21)(22) satisfying x0=ϕ. If additionally
plog(1 + β)<0,(23)
the following assertions hold.
1. There exists a compact interval 0[0,)that is attracting for nonnegative initial conditions:
if ϕ0, then limt→∞ dH(x(t),0) = 0 for xthe solution from 1 and dHthe Hausdorff distance
dH(x, B) = inf {|xb|:bB}.
2. The semiflow on X=G+,Lip(Ω0)is well-defined, in the sense that to any ϕXthere is a unique
solution x: [r, )0with x0=ϕand xtXfor t0.
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(th(0)), t =tk
y(t) = g(0)y(t), t =tk,
and this holds regardless of whether (23) is satisfied.
Proof. Let = 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
sRand ϕG+,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 ϕG+(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
bγx(s) + µF (x(sh(x(s))))ds
for t[b, b +ϵ] and ϵ > 0 small enough. xis differentiable from the right and continuous in this interval,
x(b) = γ(b) + µF (x(bh(x(b)))) = µF (x(bh(0))) 0
by definition of F. This contradicts the definition of b. Therefore, any solution with non-negative initial
condition ϕG+,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 ts0,
U(t, s) = eγ(ts)Y
(1 + β).
Then U(t, s)0. If β0, then U(t, s)eγ(ts)eψ(ts). If β > 0, then we have
(1 + β) = (1 + β)#{tk:s<tkt}<(1 + β)ts
p= exp ts
plog(1 + β),
from which we can obtain the bound U(t, s)eψ(ts).We conclude that 0 U(t, s)eψ(ts). Any
solution xof (21)–(22) with x0=ϕ0 satisfies the variation-of-constants formula
x(t) = U(t, 0)ϕ(0) + µZt
U(t, s)F(x(sτ(x(s))))ds +X
U(t, tk)[g(xt
By consequence of the conditions of the corollary specifically, f0 on R+and the properties of g
one can verify that x0 as long as the solution exists. On the other hand, we also have
x(t)U(t, 0)ϕ(0) + µZt
U(t, s)ds +αX
U(t, tk)
eψtϕ(0) + µZt
eψ(ts)ds +αX
eψtϕ(0) + µ
ψ(1 eψt) + α
=eψtϕ(0) + µ
ψ(1 eψt) + αe
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
This proves assertion 1. Similarly, for ϕG+,Lip(Ω0), the same analysis demonstrates that xt(θ)0.
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(th(x(t)))), t =k(24)
x(t)=(βx (β+ 1)x2)θ(1 x), t =k, (25)
for β0, 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 x0 and g(x)βx. We have h(x)33/8<1, so the Cauchy problem is
well-posed and all solutions are globally defined and unique for t0 by Proposition 21.
0 20 40
0 20 40
0 20 40
0 5 10
0 10 20
0 10 20
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)e1γ<1. The condition (23)
for the attracting invariant set will be satisfied if and only if β < eγ1. In this case, the interval
0=0, ψ1with ψ=γlog(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 0is 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)e1γ0.735 <1,
and the trivial solution is exponentially stable, as expected. This indeed appears to be the case from the
figure. Since eγ16.389 >6 for γ= 2, the simulations in the γ= 2 case all feature attractivity of the
region 0, 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 [tr, 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.
Thank you to the reviewer for their helpful comments, which led to some improvements to the paper.
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