Manuscript submitted to
AIMS’ Journals
Volume X, Number 0X, XX 200X
Website: http://AIMsciences.org
pp. X–XX
EXPONENTIAL STABILITY OF A STATE-DEPENDENT DELAY
SYSTEM
István Győri
Department of Mathematics and Computing
University of Pannonia
H-8201 Veszprém, P.O.Box 158, Hungary
Ferenc Hartung
Department of Mathematics and Computing
University of Pannonia
H-8201 Veszprém, P.O.Box 158, Hungary
(Communicated by Aim Sciences)
Abstract. In this paper we study exponential stability of the trivial solution
Pm
of the state-dependent delay system ẋ(t) =
i=1 Ai (t)x(t − τi (t, xt )). We
show that under mild assumptions, the trivial solution of the state-dependent
system is exponentially stable, if and only if the trivial
P solution of the corresponding linear time-dependent delay system ẏ(t) = m
i=1 Ai (t)y(t − τi (t, 0))
is exponentially stable. We also compare the order of the exponential stability
of the nonlinear equation to that of its linearized equation. We show, that in
some cases, the two orders are equal. As an application of our main result, we
formulate a necessary and sufficient condition for the exponential stability of
the trivial solution of a threshold-type delay system.
1. Introduction. In this paper we study exponential stability of the trivial solution of delay systems with state-dependent delays (SD-DDEs) of the form
ẋ(t) =
m
X
i=1
Ai (t)x(t − τi (t, xt )).
(1)
We obtain sharp linearized stability results for SD-DDEs of the form (1) by comparing the stability of the trivial solution to that of an associated linear delay equation.
Cooke obtained sufficient stability conditions for the scalar and autonomous version
of (1) with m = 1 first comparing the stability the trivial solution to that of an
associated ODE in [3], and later in [4], to that of an associated linear differential
equation with time-dependent delay.
In [11] we studied scalar SD-DDEs of the form ẋ(t) = a(t)x(t − τ (t, x(t))), where
in addition to a sufficient condition, a necessary condition was obtained to guarantee
exponential stability of the trivial solution of the SD-DDE. To obtain the necessary
part, it was assumed that τ is differentiable with respect to both variables, and
2000 Mathematics Subject Classification. Primary: 34K20 Secondary: 34K25.
Key words and phrases. state-dependent delay, exponential stability, order of exponential stability, threshold delay.
This research was partially supported by Hungarian National Foundation for Scientific Research
Grant No. T046929.
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the partial derivatives are bounded and satisfy a certain smallness condition to
guarantee monotonicity of the function t 7→ t − τ (t, x(t)). In this paper we extend
these results to the more general equation (1), and using a different technique,
we significantly relax the condition used for the necessary part of the main result:
instead of the differentiability and the monotonicity conditions cited above a much
weaker condition (H5) (see Section 2) is assumed. This weaker condition is satisfied
for treshold-type delays and for delays of the form
ci |ψ(0)|,
|ψ(0)| < γi ,
τi (ψ) =
ci γi ,
|ψ(0)| ≥ γi ,
where ci ≥ 0 and γi > 0 are constants.
Stability of more general SD-DDEs was investigated, e.g., in [5], [23], [30]. Stability conditions for general nonlinear differential equations with state-dependent delay
using linearization techniques were obtained for different classes of SD-DDEs in [6],
[18]–[20]. For a recent review on basic theory of state-dependent delay equations
and related applications we refer to [21].
The organization of the paper is as follows. In Section 2 we give our assumptions
and formulate our main results (Theorem 2.4 below). We also investigate the relation between the order of stability of the trivial solution of the SD-DDE and that
of its linearized equation, and also give estimate of the domain of attraction of the
nonlinear equation. Several explicit exponential stability conditions are formulated
as corollaries of our main theorem applying known stability results for the comparison equation. In Section 3 we show how our main theorem can be applied for
threshold-type delay equations. Section 4 introduces some notations and lemmas
and contains the proofs of the main results.
2. Formulation of the Main Results. Throughout this paper a fixed norm on
Rn and its induced matrix norm on Rn×n is denoted by | · |. The Banach space of
continuous functions ψ : [−r, 0] → Rn equipped with the norm kψk = sup{|ψ(s)| :
s ∈ [−r, 0]} is denoted by C. The solution segment function xt : [−r, 0] → Rn is
defined by xt (s) = x(t + s).
Consider the delay system
ẋ(t) =
m
X
i=1
Ai (t)x(t − τi (t, xt )),
t ≥ t0 .
(1)
We assume that r > 0 is fixed, t0 ≥ 0, Ai : [0, ∞) → Rn×n and τi : [0, ∞)×C → [0, r]
(i = 1, . . . , m).
We compare the exponential stability of the trivial solution of (1) to that of the
associated linear system
ẏ(t) =
m
X
i=1
Ai (t)y(t − τi (t, 0)),
t ≥ t0 ,
(2)
where 0 is the constant 0 function in C.
We associate the initial condition
x(t) = ϕ(t − t0 ),
to Equations (1) and (2).
t ∈ [t0 − r, t0 ],
ϕ∈C
(3)
EXPONENTIAL STABILITY OF SD-FDES
3
Definition 2.1. We say that the trivial (zero) solution of the linear equation (2)
is exponentially stable, if there exist constants K1 ≥ 0 and α > 0 independent of t0
such that
|y(t)| ≤ K1 e−α(t−t0 ) kϕk,
t ≥ t0 ≥ 0.
(4)
If (4) holds, then we say that the order of exponential stability is α.
We note that this notion is also called in the literature as uniform exponential
stability. It would be more precise to say the order of exponential stability is at
least α, since (4) may hold with larger α, as well, but we use this terminology for
simplicity.
We define the fundamental solution of (2) as the n × n matrix solution of the
initial value problem
m
X
∂
Ai (t)V (t − τi (t, 0), s),
t ≥ s,
(5)
V (t, s) =
∂t
i=1
I,
t = s,
V (t, s) =
(6)
0
t < s.
Here I and 0 denote the identity and the zero matrices, respectively.
If the trivial solution of (2) is exponentially stable with order α, then it is known
(see, e.g., [15]), that there exists K2 ≥ 1 such that
|V (t, s)| ≤ K2 e−α(t−s) ,
t ≥ 0,
s ∈ R.
(7)
Definition 2.2. We say the trivial solution of (1) is exponentially stable, if there
exist K3 ≥ 0, β > 0 and σ > 0 independent of t0 such that
|x(t)| ≤ K3 e−β(t−t0 ) kϕk,
t ≥ t0 ≥ 0,
kϕk ≤ σ.
Definition 2.3. We say the trivial solution of (1) is exponentially stable in the
large with order β, if for every σ > 0 there exists K4 = K4 (σ) ≥ 0 such that
|x(t)| ≤ K4 e−β(t−t0 ) kϕk,
t ≥ t0 ≥ 0,
kϕk ≤ σ.
We assume throughout the paper
(H1) Ai : [0, ∞) → Rn×n is continuous, and |Ai (t)| ≤ bi , t ∈ [0, ∞) for i = 1, . . . , m;
(H2) ϕ ∈ C;
(H3) the delay functions τi : [0, ∞) × C → [0, r] are continuous for i = 1, . . . , m;
(H4) there exist a constant 0 < γ ≤ ∞ and continuous functions ωi : [0, γ) → [0, ∞),
such that
|τi (t, ψ) − τi (t, 0)| ≤ ωi (kψk),
t ≥ 0,
kψk < γ,
i = 1, . . . , m,
where ωi (0) = 0 (i = 1, . . . , m).
(H5) the sets {s ∈ [0, r] : s − τi (s + t0 , 0) = 0} have Lebesgue measure 0 for
i = 1, . . . , m and t0 ≥ 0.
We remark that if the value of the function s 7→ s − τi (s + t0 , 0) is equal to 0 at
most for countably many s, then (H5) holds.
We note that conditions (H1)–(H3) guarantee the existence, but not the uniqueness of the solution (see, e.g., [7], [14], [19]).
Our main result is formulated in the following theorem.
Theorem 2.4. Suppose (H1)–(H5). Then the trivial solution of (1) is exponentially
stable, if and only if the trivial solution of (2) is exponentially stable.
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In the proof of the necessary part it was important the we “linearize” around
the trivial solution. It would be interesting to extend this result to more general
solutions, e.g., to periodic solutions (see [18] where a sufficient condition was formulated in this case). It is also an open problem to relax or omit condition (H5) in
Theorem 2.4.
We note that assumption (H5) is used only in the necessary part of the proof of
the previous theorem. Therefore, we can formulate the sufficient part of Theorem 2.4
as follows.
Theorem 2.5. Suppose (H1)–(H4). If the trivial solution of (2) is exponentially
stable, then the trivial solution of (1) is exponentially stable, as well.
The proof of Theorem 2.5 yields the next corollary.
Corollary 1. Suppose (H1)–(H4). If the trivial solution of (2) is exponentially stable with order α, then for every 0 < β < α the trivial solution of (1) is exponentially
stable with order β, as well.
Note that the proof of the necessary part of Theorem 2.4 shows that the exponential stability of the trivial solution of (1) with order α implies the exponential
stability for the trivial solution of (3) with the same order.
Corollary 2. Suppose (H1)–(H5). If the trivial solution of (1) is exponentially
stable with order α, then the trivial solution of (2) is exponentially stable with order
α, as well.
Under some more restriction on the functions ωi , we can prove that the order of
the exponential stability of the trivial solution of the linear equation (2) is preserved
for that of the SD-DDE (1). We also give an explicit estimate for the domain of
attraction of the trivial solution.
Theorem 2.6. Assume (H1)–(H4), moreover there exists c0 > 0 such that
Z c0
ωi (u)
du < ∞,
u
0
(8)
ωi is monotone increasing on [0, c0 ] (i = 1, . . . , m), and suppose the trivial solution
of (2) is exponentiallyPstable with order α. Let K1 and K2 be defined by (4) and
m
(7), respectively, b = i=1 bi , M2 = max{1, K1 + 2K2 be(α+b)r /α}. Let
(
)
Z c
m
K2 be2αr X
ωi (u) c bi
1−
σ0 = sup
du
,
(9)
M2
α
u
c∈U
0
i=1
where
U=
(
)
Z c
m
K2 be2αr X
ωi (u)
du < 1 .
bi
c > 0:
α
u
0
i=1
Then for every 0 < σ < σ0 there exists K ≥ 0 such that the solution of (1) satisfies
|x(t)| ≤ Ke−α(t−t0 ) kϕk,
t ≥ t0 ≥ 0,
kϕk ≤ σ,
i.e., the trivial solution of (1) is exponentially stable with order α, as well.
This theorem implies immediately the next corollary.
EXPONENTIAL STABILITY OF SD-FDES
5
Corollary 3. Assume (H1)–(H4), moreover
Z ∞
m
K2 be2αr X
ωi (u)
bi
du < ∞,
α
u
0
i=1
(10)
ωi is monotone increasing on [0, c0 ] (i = 1, . . . , m), and suppose the trivial solution of (2) is exponentially stable with order α. Then the trivial solution of (1) is
exponentially stable in the large with order α.
As an application of the previous corollary, consider
ẋ(t) = −bx(t − τ (x(t))),
−|u|
t ≥ 0,
where b > 0, τ (u) = re|u|e
. It is easy to check that ω(u) = reue
(H4), K1 = K2 = 1, α = b and
Z ∞
m
ωi (u)
K2 be2αr X
bi
du = ebre2br .
α
u
0
i=1
(11)
−u
satisfies
Therefore, if ebre2br < 1, then the trivial solution of (11) is exponentially stable in
the large with order b.
Next we apply known conditions in the scalar case to check the exponential
stability of the trivial solution of (2), and we obtain several corollaries of our main
results. Note that in the single delay case similar results are formulated in [11].
Next result follows from conditions of Krisztin [22].
Corollary 4. Suppose (H1)–(H4) hold with n = 1, Ai (t) ≤ 0 for i = 1, . . . , m.
Then the trivial solution of (1) is exponentially stable, if
m
X
bi sup τi (t, 0) < 1.
i=1
t≥0
Moreover, if Ai (t) = −bi (t ≥ 0), then
m
X
3
bi sup τi (t, 0) <
2
t≥0
i=1
yields the exponential stability of the trivial solution of (1). If, in addition, τi (t, 0) =
ri are constant, then
m
X
π
(12)
bi ri <
2
i=1
implies the exponential stability of the trivial solution of (1). If m = 1, then (12) is
also necessary for the exponential stability of the trivial solution (1).
Next result is based on Theorem 4.1 of [12], which generalizes a condition of
Yoneyama [30] for the multiple delay case.
Corollary 5. Suppose (H1)–(H4) hold with n = 1, Ai (t) ≤ 0 for i = 1, . . . , m.
Moreover, suppose
Z t
m
X
lim sup
−Ak (s) ds < 1,
i,k=1
t→∞
t−τi (t,0)
and there exists α > 0 such that
Z t
1
−Ak (s) ds ≥ α,
t − t0 t0
t > t0 ≥ 0,
k = 1, . . . , m.
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Then the trivial solution of (1) is exponentially stable.
Corollary 6. Suppose (H1)–(H4) hold with n = 1, and τi (t, 0) = 0 for all t ≥ 0
and i = 1, . . . , m. Then the trivial solution of (1) is exponentially stable if and only
if there exists α > 0 such that
m Z t
X
Ai (s) ds ≤ −α(t − t0 ),
t > t0 ≥ 0.
i=1
t0
Note this result for m = 1 was obtained in [11] assuming additional smoothness
on the delay.
For the special case when the linearized system (2) is two-dimensional (n = 2),
autonomous and has a single constant delay, we can use a condition of Hara and
Sugie [16] to check uniform asymptotic stability, i.e., exponential stability of the
trivial solution.
Corollary 7. Suppose
Pm (H1)–(H4) hold with n = 2, τi (t, 0) = τ for all t ≥ 0 and
i = 1, . . . , m and
i=1 Ai (t) = A for t ≥ 0. Then the trivial solution of (1) is
exponentially stable if and only if
√
√
2τ det A
π
+
2 det A sin τ det A < − tr A <
2τ
π
and
π 2
.
0 < det A <
2τ
Similar sufficient or sufficient and necessary conditions can be formulated for
SD-DDEs by combining Theorem 2.4 or 2.5 and conditions from, e.g., [13], [17],
[25], [26], [28].
3. Applications for threshold-type delay equations. In this section we consider the delay system
m
X
Ai (t)x(t − σ̃i (t)),
t ≥ t0 ,
(1)
ẋ(t) = A0 (t)x(t) +
i=1
where the delay functions σ̃i are defined by the threshold relations
Z t
fi (t, u − t, x(t), x(u)) du = 1
(2)
t−σ̃i (t)
for i = 1, . . . , m, where fi are given nonnegative scalar functions for i = 1, . . . , m.
Similar equations, so-called threshold-type delay equations, were frequently used in
biological models ([2], [21], [29]), and were investigated, e.g., in [8], [9], [10], [12],
[24], [27].
We show that under the following assumptions the delay functions σ̃i are welldefined and can be rewritten in the form of τi in (1), i.e., σ̃i depends on xt , as well.
We assume F0 > 0, t0 ≥ 0 are given, and let r = 1/F0 ; moreover,
(A1) Ai : [0, ∞) → Rn are continuous, and |Ai (t)| ≤ bi , t ∈ [0, ∞), for i =
0, 1, . . . , m;
(A2) ϕ ∈ C;
EXPONENTIAL STABILITY OF SD-FDES
7
(A3) fi : [0, ∞) × [−r, 0] × Rn × Rn → (0, ∞) continuous functions for i = 1, . . . , m,
and
fi (t, s, u, v) ≥ F0 ,
t ≥ 0,
s ∈ [−r, 0],
u, v ∈ Rn ;
(A4) there exist a constant γ > 0 and a function ω̃ : [0, γ) → [0, ∞), such that
ω̃(0) = 0 and
Z 0
fi (t, s, ψ(0), ψ(s))−fi (t, s, 0, 0) ds ≤ ω̃(kψk), t ≥ 0, kψk < γ, i = 1, . . . , m;
−r
Z
n
(A5) the sets t ∈ [0, r] :
0
−t
o
fi (t + t0 , s, 0, 0) ds = 1 have Lebesgue-measure 0 for
i = 1, . . . , m and t0 ≥ 0.
Introducing the new variable s = u − t we can rewrite (2) as
Z t
Z 0
fi (t, u − t, x(t), x(u)) du =
fi (t, s, x(t), x(t + s)) ds = 1.
t−σ̃i (t)
−σ̃i (t)
Note that such unique σ̃i (t) ∈ [0, r] exists, since by (A3)
Z 0
fi (t, s, x(t), x(t + s)) ds ≥ F0 r = 1,
i = 1, . . . , m.
−r
Now we can reformulate the problem: We rewrite (1)-(2) in the form
ẋ(t) = A0 (t)x(t) +
m
X
i=1
Ai (t)x(t − σi (t, xt )),
t ≥ t0 ,
(3)
where the delay functions σi : [0, ∞)×C → [0, r] are defined by the treshold relation
Z 0
fi (t, s, ψ(0), ψ(s)) ds = 1.
(4)
−σi (t,ψ)
The solution of (3) corresponds to an initial condition of the form (3).
In the case when fi (t, s, u, v) does not depend on s and v, i.e., has the form
fi (t, s, u, v) = gi (t, u), relation (4) reduces to
σi (t, ψ) =
1
.
gi (t, ψ(0))
Therefore such formulation of threshold delays includes a large class of ”usual”
state-dependent delays.
As in Section 2, we associate the linear equation
ẋ(t) = A0 (t)x(t) +
m
X
i=1
Ai (t)x(t − σi (t, 0)),
t ≥ t0
(5)
to (3).
Theorem 2.4 has the folowing corollary.
Theorem 3.1. Assume (A1)–(A5). Then the trivial solution of (3) is exponentially
stable, if and only if the trivial solution of (5) is exponentially stable.
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Proof. It is enough to show that assumptions (A1)–(A5) imply assumptions (H1)–
(H5) of Theorem 2.4.
Clearly, (A1) and (A2) are identical to (H1) and (H2), respectively.
We show that σi is a continuous function. Fix an arbitrary t̄ ∈ [t0 , ∞) and ψ̄ ∈ C.
Then
Z 0
Z 0
1=
fi (t̄, s, ψ̄(0), ψ̄(s)) ds =
fi (t, s, ψ(0), ψ(s)) ds,
−σi (t̄,ψ̄)
and so
Z −σi (t,ψ)
−σi (t,ψ)
fi (t̄, s, ψ̄(0), ψ̄(s)) ds =
−σi (t̄,ψ̄)
Z
0
[fi (t, s, ψ(0), ψ(s))−fi (t̄, s, ψ̄(0), ψ̄(s))] ds.
−σi (t,ψ)
Consequently,
Z −σi (t,ψ)
Z
fi (t̄, s, ψ̄(0), ψ̄(s)) ds ≤
−σi (t̄,ψ̄)
0
−r
→ 0,
fi (t, s, ψ(0), ψ(s)) − fi (t̄, s, ψ̄(0), ψ̄(s)) ds
as |t − t̄| + kψ − ψ̄k → 0. On the other hand, using (A3)
Z −σi (t,ψ)
F0 |σi (t, ψ) − σi (t̄, ψ̄)| ≤ fi (t̄, s, ψ̄(0), ψ̄(s)) ds.
−σi (t̄,ψ̄)
Therefore
|σi (t, ψ) − σi (t̄, ψ̄)| → 0
as |t − t̄| + kψ − ψ̄k → 0,
and consequently, (H3) holds with τi = σi .
We get by the computation above with t̄ = t and ψ̄ = 0 that
Z 0
1
|σi (t, ψ) − σi (t, 0)| ≤
fi (t, s, ψ(0), ψ(s)) − fi (t, s, 0, 0) ds.
F0 −r
Therefore (A4) immediately implies (H4).
To show (H5) first note that t − σi (t + t0 , 0) = 0 implies
Z 0
Z 0
fi (t + t0 , s, 0, 0) ds =
fi (t + t0 , s, 0, 0) ds = 1.
−σi (t+t0 ,0)
−t
Therefore (A5) implies (H5).
Corollary 8. Assume (A1)–(A4). Then the exponential stability of the trivial
solution of (5) implies that for the trivial solution of (3), as well.
We note that Theorems 2.4 and 3.1 can be trivially combined for equations of
the form
ẋ(t) = A0 (t)x(t) +
m
X
i=1
Ai (t)x(t − τi (t, xt )) +
k
X
i=1
Bi (t)x(t − σi (t, xt )),
t ≥ t0 ,
where the ”explicit” delay functions τi satisfy (H3)-(H5), and the treshold delays
σi defined by (4) satisfy (A3)-(A5).
EXPONENTIAL STABILITY OF SD-FDES
9
4. Proof of the Main Results. The next lemma can be proved using Gronwall’s
inequality (see [11] for details in case of a similar equation).
Lemma 4.1. Assume (H1)–(H3). Then any solution of (1)–(3) satisfies
|x(t)| ≤ eb(t−t0 ) kϕk,
t ≥ t0 ≥ 0,
where b is defined by
b=
m
X
bi .
(1)
i=1
We can rewrite (1) as
ẋ(t) =
m
X
i=1
Ai (t)x(t − τi (t, 0)) + f (t),
t ≥ t0 ,
i = 1, . . . , n,
where
f (t) =
m
X
i=1
Ai (t) x(t − τi (t, xt )) − x(t − τi (t, 0)) .
(2)
Using the variation of constants formula we get
x(t) = y(t) +
Z
t
V (t, s)f (s) ds,
t0
t ≥ t0 ,
(3)
where y is the solution of (2) corresponding to the initial condition (3).
We will need an estimate of f .
Lemma 4.2. Assume (H1)–(H4), t0 ≥ 0, and suppose x is a solution of (1)-(3)
satisfying |x(t)| < γ for t ≥ t0 − r, where γ > 0 is defined in (H4). Then

m

 b max |x(u)| X b ω (kx k),
i i
t
|f (t)| ≤
u∈[t−2r,t]
i=1


2bebr kϕk,
t0 + r < t,
t ∈ [t0 , t0 + r],
where b is defined by (1).
Proof. We introduce the notations
ηi (t) = min{t−τi (t, 0), t−τi (t, xt )}
and ξi (t) = max{t−τi (t, 0), t−τi (t, xt )}. (4)
Then
|x(t − τi (t, 0)) − x(t − τi (t, xt ))| = |x(ξi (t)) − x(ηi (t))|.
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First suppose t > t0 + r. Then t0 ≤ ηi (t) ≤ ξi (t), and so
|f (t)|
Z
m
X
Ai (t)
= i=1
≤
=
≤
≤
=
m
X
i=1
m
X
i=1
m
X
i=1
m
X
i=1
m
X
i=1
≤ b
bi
ηi (t)
Z
ξi (t) ηi (t)
bi
Z
bi
bi
m
X
k=1
m
X
k=1
m
X
k=1
bk
ẋ(u) du
ẋ(u) du
m
ξi (t) X
ηi (t)
bi
ξi (t)
Ak (u)x(u − τk (u, xu )) du
k=1
ξi (t)
Z
ηi (t)
|x(u − τk (u, xu ))| du
bk (ξi (t) − ηi (t))
max
u∈[t−2r,t]
bk |τi (t, xt ) − τi (t, 0)|
max
u∈[t−2r,t]
|x(u)|
m
X
i=1
(5)
|x(u)|
max
u∈[t−2r,t]
|x(u)|
bi ωi (kxt k).
Now consider the case when t ∈ [t0 , t0 + r]. We use Lemma 4.1 to estimate
|f (t)| ≤
≤
≤
m
X
bi |x(ξi (t))| + |x(ηi (t))|
i=1
m
X
bi eb max{ξi (t)−t0 ,0} + eb max{ηi (t)−t0 ,0} kϕk
i=1
2bebr kϕk.
Proof of the sufficient part of Theorem 2.4 (proof of Theorem 2.5). First we assume
that the trivial solution of (2) is exponentially stable with order α. Let K1 , m, K2
and b be defined by (4), (7) and (1), respectively.
We first show that the trivial solution of (1) is stable. By assumption (H4), there
exists 0 < ε0 < γ such that
m
1
K2 b X
bi max ωi (u) < .
0≤u≤ε
α i=1
3
0
Let 0 < ε < ε0 be arbitrary, and δ > 0 is such that
εα
δ = min ε,
.
3(K1 α + 2K2 bebr )
Fix an initial function satisfying kϕk < δ, and let x be any solution of (1) corresponding to this initial function. Then |ϕ(0)| < δ ≤ ε, therefore there exists T > t0
such that |x(t)| < ε for t ∈ [t0 , T ). Suppose |x(T )| = ε. First consider the case
when T ≤ t0 + r. Then it follows from the variation of constants formula (3) and
EXPONENTIAL STABILITY OF SD-FDES
11
Lemma 4.2 that
|x(T )| ≤
|y(T )| +
Z
T
t0
|V (T, s)||f (s)| ds
Z
K1 e−α(T −t0 ) kϕk +
≤
T
t0
K2 e−α(T −s) 2bebr kϕk ds.
Therefore,
ε ≤ K1 e
−α(T −t0 )
δ + K2 e
−αT
br
2be δ
Z
T
t0
2K2 bebr ε
eαs ds ≤ K1 +
δ≤ ,
α
3
which is a contradiction. Now suppose T > t0 + r. Then Lemma 4.2 yields
Z T
Z t0 +r
|V (T, s)||f (s)| ds
|V (T, s)||f (s)| ds +
|x(T )| ≤ |y(T )| +
t0
≤
K1 e−α(T −t0 ) kϕk +
+
Z
T
Z
t0
K2 e−α(T −s) b
t0 +r
t0 +r
t0 +r
K2 e−α(T −s) 2bebr kϕk ds
max
u∈[s−2r,s]
|x(u)|
m
X
i=1
bi ωi (kxs k) ds,
and so
ε
≤
K1 e
−α(T −t0 )
+K2 e−αT bε
δ + K2 e
m
X
i=1
≤
≤
−αT
br
2be δ
bi max ωi (u)
0≤u≤ε
Z
Z
t0 +r
eαs ds
t0
T
eαs ds
t0 +r
m
2K2 bebr K2 bε X
K1 +
δ+
bi max ωi (u)
α
α i=1 0≤u≤ε0
ε ε
+ ,
3 3
which is a contradiction, again. Therefore |x(t)| < ε is satisfied for all t > t0 , i.e.,
the trivial solution of (1) is stable, moreover, it is uniformly stable, since δ does not
depend on t0 .
Let 0 < β < α be arbitrary. Next we show that the trivial solution of (1) is
exponentially stable with order β. Let 0 < ε < γ be such that the constant
m
K2 be2βr X
M1 :=
bi max ωi (u)
α − β i=1 0≤u≤ε
satisfies M1 < 1, and 0 < σ ≤ ε be such that |x(t)| < ε for t ≥ t0 and for
kϕk < σ. Fix any initial function satisfying kϕk < σ, and let x be any solution of
(1) corresponding to the initial function ϕ.
First consider the case when t ∈ [t0 , t0 + r]. Then multiplying the variation of
constants formula by eβ(t−t0 ) we get
Z t
β(t−t0 )
−(α−β)(t−t0 )
β(t−t0 )
e
|x(t)| ≤ K1 e
kϕk + K2 e
e−α(t−s) |f (s)| ds.
t0
12
ISTVÁN GYŐRI
AND
FERENC HARTUNG
Introduce the function z(t) := eβ(t−t0 ) |x(t)|, t ≥ t0 − r. With this notation and
using Lemma 4.2 we have
Z t0 +r
eαs ds,
t ∈ [t0 , t0 +r].
z(t) ≤ K1 e−(α−β)(t−t0 ) kϕk+2K2 bebr eβ(t−t0 )−αt kϕk
t0
Define M2 = max{1, K1 + 2K2 be
(α+b)r
/α}. Then it follows
z(t) ≤ M2 kϕk,
t ∈ [t0 , t0 + r].
Next suppose t > t0 + r. Then
z(t) ≤ K1 e−(α−β)(t−t0 ) kϕk
Z t0 +r
Z
β(t−t0 )
−α(t−s)
+K2 e
e
|f (s)| ds +
e
−α(t−s)
t0 +r
t0
t
|f (s)| ds .
The first two terms can be estimated as before, in the last term we apply inequality
(5):
Z t
Z ξi (s)
m
m
X
X
β(t−t0 )−αt
αs
bi
z(t) ≤ M2 kϕk + K2 e
bk
e
|x(u − τk (u, xu ))| du ds
t0 +r
i=1
ηi (s)
k=1
−(α−β)t
≤ M2 kϕk + K2 e
Z
Z t
m
m
X
X
bi
bk
×
eαs
t0 +r
i=1
ξi (s)
ηi (s)
k=1
βr −(α−β)t
≤ M2 kϕk + K2 be e
e−β(u−τk (u,xu )) z(u − τk (u, xu )) du ds
max
u∈[t0 −r,t]
Z
z(u)
t
e
αs
t0 +r
×
Z
t
eαs
t0 +r
≤ M2 kϕk
m
X
i=1
max
u∈[t0 −r,t]
bi
i=1
Using that ηi (s) ≥ s − r, we get
z(t) ≤ M2 kϕk + K2 beβr e−(α−β)t
m
X
Z
ξi (s)
e−βu du ds.
ηi (s)
z(u)
bi e−β(s−r) (ξi (s) − ηi (s)) ds
+K2 be2βr e−(α−β)t
max
u∈[t0 −r,t]
z(u)
Z
t
e(α−β)s
t0 +r
m
X
i=1
bi ωi (kxs k) ds.
Using the definition of M1 and that |x(t)| < ε for all t ≥ 0, we obtain
Z t
m
X
2βr −(α−β)t
bi max ωi (u)
z(t) ≤ M2 kϕk + K2 be e
max z(u)
u∈[t0 −r,t]
≤ M2 kϕk + M1
max
u∈[t0 −r,t]
i=1
0≤u≤ε
z(u).
(6)
e(α−β)s ds
t0 +r
(7)
The right-hand-side of (7) is monotone in t, M2 ≥ 1 and z(t) ≤ |ϕ(t − t0 )| ≤ kϕk
for t ∈ [t0 − r, t0 ], therefore (7) yields
max
u∈[t0 −r,t]
Hence z(t) ≤
M2
1−M1 kϕk,
|x(t)| ≤
z(u) ≤ M2 kϕk + M1
max
u∈[t0 −r,t]
z(u).
and consequently,
M2 −β(t−t0 )
e
kϕk,
1 − M1
t ≥ t0 ,
kϕk < σ,
(8)
EXPONENTIAL STABILITY OF SD-FDES
13
which completes the proof.
For the rest of the paper ϕ
e : [−r, ∞) → Rn denotes the extension of ϕ : [−r, 0] →
R to the right by the zero vector:
ϕ(s),
s ∈ [−r, 0],
ϕ(s)
e =
0,
t > 0.
n
The proof of the necessary part of Theorem 2.4 will be based on the next lemma,
where we prove the existence of a certain auxiliary function under condition (H5).
Lemma 4.3. Suppose (H3)–(H5). Then for any ε > 0 and any t0 ∈ R there exists
a continuously differentiable scalar function hε,t0 : [−r, 0] → [0, 1] such that
Z r
e
i = 1, . . . , m,
hε,t0 (s − τi (s + t0 , 0)) ds ≤ ε,
0
and khε,t0 k = hε,t0 (0) = 1.
Proof. The sets
Ei = {s ∈ [0, r] : s − τi (s + t0 , 0) = 0},
Fi = {s ∈ [0, r] : s − τi (s + t0 , 0) < 0}
and
n
1o
k = 1, 2, . . .
Gi,k = s ∈ [0, r] : s − τi (s + t0 , 0) < − ,
k
are Lebesgue measurable for i = 1, . . . , m and k = 1, 2, . . ., moreover
Gi,1 ⊂ Gi,2 ⊂ Gi,3 ⊂ · · ·
Therefore
lim µ(Gi,k ) = µ(Fi ),
k→∞
and
∞
[
Gi,k = Fi .
k=1
i = 1, . . . , m,
where µ denotes the Lebesgue-measure. Fix an arbitrary 0 < ε < 2r, and let
k0 > 1/r be such that
ε
i = 1, . . . , m.
0 ≤ µ(Fi ) − µ(Gi,k0 ) < ,
2
Let hε,t0 : [−r, 0] → R be a strictly monotone increasing continuously differentiable scalar function satisfying
ε
1
=
and hε,t0 (0) = 1.
hε,t0 −
hε,t0 (−r) = 0,
k0
2r
Since the set Ei has measure 0, we get
Z r
e
hε,t0 (s − τi (s + t0 , 0)) ds
0
Z
Z
hε,t0 (s − τi (s + t0 , 0)) ds
hε,t0 (s − τi (s + t0 , 0)) ds +
=
Fi
Ei
Z
Z
=
hε,t0 (s − τi (s + t0 , 0)) ds +
hε,t0 (s − τi (s + t0 , 0)) ds.
Gi,k0
Since
Fi \Gi,k0
1
ε
hε,t0 (s − τi (s + t0 , 0)) < hε,t0 −
=
k0
2r
for s ∈ Gi,k0 ,
14
ISTVÁN GYŐRI
AND
FERENC HARTUNG
and 0 ≤ hε,t0 (t) ≤ 1 for t ∈ [−r, 0], we get
Z r
ε
ε
ε
hε,t0 (s − τi (s + t0 , 0)) ds ≤ µ(Gi,k0 ) + µ(Fi \ Gi,k0 ) ≤ r + = ε.
2r
2r
2
0
The next lemma gives an estimate of the derivative of an exponentially decaying
solution of (1).
Lemma 4.4. Suppose (H1)–(H3), |x(t)| ≤ Ke−α(t−t0 ) kϕk for t ≥ t0 , assuming
kϕk < σ where K ≥ 1, and b be defined by (1). Then
|ẋ(t)| ≤ bKkϕk,
t ≥ t0 .
Proof. It follows from the assumptions and from (1)
|ẋ(t)| ≤
for t ≥ t0 .
m
X
i=1
bi |x(t − τi (t, xt ))| ≤
m
X
i=1
bi Ke−α max{t−τi (t,xt )−t0 ,0} kϕk ≤ bKkϕk,
Proof of the necessary part of Theorem 2.4. It is known (see, e.g., [15]) that the
trivial solution of (2) is exponentially stable with order α if and only if the fundamental solution of (2) (i.e., the solution of the initial value problem (5)-(6)) satisfies
an estimate of the form (7) for some positive constants K2 . Now suppose the trivial
solution of (1) is exponentially stable with order α, i.e., there exist K and σ > 0
independent of t0 such that any solution of (1) satisfies |x(t)| ≤ Ke−α(t−t0 ) kϕk for
t ≥ t0 , assuming kϕk < σ. We show, in two steps, that the fundamental solution of
(2) satisfies (7) with K2 = K, therefore the trivial solution of (2) is exponentially
stable with order α.
Step 1. First we show that for any fixed t0 ≥ 0 and 0 < ε < 1
3
|V (t, t0 )| ≤ Ke−α(t−t0 ) + ε max |V (t, s)|,
4 t0 ≤s≤t
t ≥ t0
(9)
nε o
,γ .
4b
(10)
holds. Let δ0 > 0 be such that
m
bKeαr X
ε
bi max ωi (u) <
α i=1 0≤u≤δ0
4
and
δ0 <
Fix a continuously differentiable initial function hδ0 ,t0 : [−r, 0] → R defined by
Lemma 4.3, and let M = kḣδ0 ,t0 k. Let δ1 > 0 be such that
2(M + bK)r
m
X
i=1
Finally, let δ > 0 be such that
bi max ωi (u) <
0≤u≤δ1
δ1
δ < min σ,
.
K
ε
,
4
δ1 < δ0 .
(11)
(12)
Let a ∈ Rn with |a| = δ be fixed, and let ϕ(t) = hδ0 ,t0 (t)a. Then ϕ(0) = a
and kϕk = |a| = δ. Let x and y be a solution of (1) and (2), respectively, both
corresponding to this initial function, ϕ.
EXPONENTIAL STABILITY OF SD-FDES
Then the variation-of-constants formula yields
Z t
V (t, s)f (s) ds,
x(t) = y(t) +
15
t ≥ t0 ,
t0
where f is defined by (2). Theorem 1.2 from Section 6.1 of [15] (see also [1]) yields
the following relation
Z t0 +r
m
X
y(t) = V (t, t0 )ϕ(0) +
Ai (s)ϕ(s
e − τi (s, 0) − t0 ) ds,
t ≥ t0 . (13)
V (t, s)
t0
i=1
Then
V (t, t0 )ϕ(0)
= x(t) −
−
Z
Z
r
V (t, s + t0 )
0
t
V (t, s)f (s) ds,
t0
Ke−α(t−t0 ) kϕk +
+
Z
Ai (s + t0 )ϕ(s
e − τi (s + t0 , 0)) ds
i=1
and so for t > t0 + r it follows
|V (t, t0 )a| ≤
m
X
t0 +r
Z
r
0
|V (t, s + t0 )|
|V (t, s)||f (s)| ds +
t0
Z
t
t0 +r
m
X
i=1
bi |ϕ(s
e − τi (s + t0 , 0))| ds
|V (t, s)||f (s)| ds.
(14)
We denote the last three integrals by I1 , I2 and I3 , respectively, and we estimate
them separately. Using Lemma 4.3 and (10) we get
Z r
m
X
e
hδ0 ,t0 (s − τi (s + t0 , 0))|a| ds
bi
I1 ≤
max |V (t, s)|
t0 ≤s≤t0 +r
≤ bδ0 |a|
0
i=1
max
|V (t, s)|
t0 ≤s≤t0 +r
ε
|a| max |V (t, s)|.
(15)
4 t0 ≤s≤t0 +r
Relations (10), (11) and (12), |x(t)| ≤ Kkϕk = Kδ < δ1 < δ0 for t ≥ t0 , kϕk = |a|
and Lemma 4.2 yield
Z t
m
X
bi ωi (kxs k) ds
I3 ≤
max |V (t, s)|b
max |x(u)|
≤
t0 +r≤s≤t
≤
=
max
t0 +r≤s≤t
max
t0 +r≤s≤t
t0 +r s−2r≤u≤s
|V (t, s)|b
Z
i=1
t
Ke
−α(s−2r−t0 )
t0 +r
|V (t, s)|bKe
α(2r+t0 )
|a|
m
m
X
i=1
kϕk
m
X
i=1
bi max ωi (u) ds
0≤u≤δ0
bi max ωi (u)
0≤u≤δ0
Z
t
e−αs ds
t0 +r
bKeαr X
≤
max |V (t, s)||a|
bi max ωi (u)
t0 +r≤s≤t
α i=1 0≤u≤δ0
ε
≤
|a| max |V (t, s)|.
4 t0 +r≤s≤t
To estimate I2 first consider
Z t0 +r
m
X
bi
|x(ξi (s)) − x(ηi (s))| ds,
I2 ≤ max |V (t, s)|
t0 ≤s≤t0 +r
i=1
t0
(16)
16
ISTVÁN GYŐRI
AND
FERENC HARTUNG
where we used (2), and ξi and ηi are defined by (4). We divide the interval [t0 , t0 +r]
into three disjoint sets:
Ai = {s ∈ [t0 , t0 + r] : ξi (s) < 0},
Bi = {s ∈ [t0 , t0 + r] : 0 < ηi (s)},
and
Ci = {s ∈ [t0 , t0 + r] : ηi (s) ≤ 0 ≤ ξi (s)}.
We estimate the integral separately on the sets Ai , Bi and Ci . Using the Mean
Value Theorem, (H4) and the definitions of M and δ1 we get
|hδ0 ,t0 (ξi (s)) − hδ0 ,t0 (ηi (s))|
for s ∈ [t0 , t0 + r], therefore
Z
|x(ξi (s)) − x(ηi (s))| ds
=
Z
Ai
Ai
≤
=
≤
M (ξi (s)) − ηi (s))
M |τi (s, xs ) − τi (s, 0)|
M max ωi (u)
0≤u≤δ1
|hδ0 ,t0 (ξi (s)) − hδ0 ,t0 (ηi (s))||a| ds
≤
M |a| max ωi (u)µ(Ai )
≤
M |a|r max ωi (u).
0≤u≤δ1
0≤u≤δ1
Lemma 4.4, the Mean Value Theorem, (H4) and kϕk = |a| imply
Z
Z
(ξi (s) − ηi (s)) ds
|x(ξi (s)) − x(ηi (s))| ds ≤ bKkϕk
Bi
Bi
≤
bK|a| max ωi (u)µ(Bi )
≤
bK|a|r max ωi (u).
0≤u≤δ1
0≤u≤δ1
Now the previous two cases yield
Z
|x(ξi (s)) − x(ηi (s))| ds
Ci
Z
Z
|hδ0 ,t0 (0) − hδ0 ,t0 (ηi (s))||a| ds
|x(ξi (s)) − x(0)| ds +
≤
Ci
Ci
Z
Z
−ηi (s) ds
ξi (s) ds + M |a|
≤ bK|a|
Ci
Ci
Z
Z
(ξi (s) − ηi (s)) ds
(ξi (s) − ηi (s)) ds + M |a|
≤ bK|a|
Ci
Ci
≤
|a|(bK + M )r max ωi (u).
0≤u≤δ1
Therefore the three cases and (11) give
I2
≤
max
|V (t, s)||a|2(M + bK)r
t0 ≤s≤t0 +r
≤
ε
|a| max |V (t, s)|.
4 t0 ≤s≤t0 +r
m
X
i=1
bi max ωi (u)
0≤u≤δ1
(17)
EXPONENTIAL STABILITY OF SD-FDES
17
Combining (14) with (15), (16) and (17) we get for t > t0 + r that
ε
ε
|V (t, t0 )a| ≤ Ke−α(t−t0 ) |a| + |a| max |V (t, s)| + |a| max |V (t, s)|
2 t0 ≤s≤t0 +r
4 t0 +r≤s≤t
3ε
≤ Ke−α(t−t0 ) |a| + |a| max |V (t, s)|.
(18)
t0 ≤s≤t
4
It is easy to see that (18) holds for t ∈ [t0 , t0 + r], as well. Since (18) holds for all
a ∈ Rn with |a| = δ, it implies (9).
Step 2: Now we show |V (t, u)| ≤ Ke−α(t−u) holds for t ≥ u ≥ 0.
Let 0 ≤ u ≤ s̄ ≤ t. Applying (9) with t0 = s̄ and using 0 < ε < 1 , we get
|V (t, s̄)| ≤ K +
3
3
max |V (t, s)| ≤ K + max |V (t, s)|.
4 s̄≤s≤t
4 u≤s≤t
Taking the maximum of the left-hand-side for u ≤ s̄ ≤ t we get
max |V (t, s)| ≤ 4K.
u≤s≤t
Substituting this back to (9) with t0 = u we get
|V (t, u)| ≤ Ke−α(t−u) + 3Kε.
Since ε was arbitrary small, we get that |V (t, u)| ≤ Ke−α(t−u) holds for t ≥ u ≥ 0,
which yields that the trivial solution of (2) is exponentially stable with order α.
Proof of Theorem 2.6. We use all the notations introduced in the proof of Theorem 2.5 (the sufficient part of Theorem 2.4).
First note that condition (8) yields that U is not empty. Let 0 < σ < σ0 be fixed.
Then there exists 0 < c such that
Z c
m
K2 be2αr X
ωi (u)
M3 :=
bi
du < 1
α
u
0
i=1
and
c
σ≤
M2
Z c
m
K2 be2αr X
ωi (u)
bi
1−
du
α
u
0
i=1
!
≤ c.
In the last estimate we used that M2 ≥ 1. Suppose kϕk < σ.
If we look at the proof of Theorem 2.5, we can easily see that for the derivation
of (6) we have not used that β < α, it holds with β = α, as well, i.e.,
Z
m
t X
2αr
z(t) ≤ M2 kϕk + K2 be
max z(u) bi ωi (kxs k) ds ,
t ≥ t0 ≥ 0,
t0 +r
u∈[t0 −r,t]
i=1
(19)
where z(t) = eα(t−t0 ) |x(t)|. Since
|xs (u)| = |x(s + u)| = e−α(s+u−t0 ) eα(s+u−t0 ) |x(s + u)| = e−α(s+u−t0 ) z(s + u)
for u ∈ [−r, 0], it follows
kxs k ≤ eα(t0 +r) e−αs kzs k.
Consequently, by the monotonicity of ωi
z(t) ≤ M2 kϕk + K2 be
2αr
Z
m
t X
α(t0 +r) −αs
e
kzs k ds .
max z(u) bi ωi e
t0 +r
u∈[t0 −r,t]
i=1
18
ISTVÁN GYŐRI
AND
FERENC HARTUNG
Since z(s) = eα(s−t0 ) |ϕ(s − t0 )| ≤ kϕk < σ ≤ c for s ∈ [t0 − r, t0 ], it follows
z(t) < c for t close enough to t0 . Suppose there exists T > t0 such that z(t) < c for
t ∈ [t0 − r, T ) and z(T ) = c. Then
Z T
m
X
bi
ωi eα(t0 +r) e−αs c ds.
c ≤ M2 kϕk + K2 be2αr c
i=1
t0 +r
Introducing N1 = eα(t0 +r) , the new variable u = N1 ce−αs and the definition of σ,
we get
Z N1 ce−α(t0 +r)
m
K2 be2αr c X
ωi (u)
c ≤ M2 kϕk +
bi
du
α
u
N1 ce−αT
i=1
Z c
m
ωi (u)
K2 be2αr c X
bi
du
≤ M2 kϕk +
α
u
0
i=1
Z c
m
K2 be2αr c X
ωi (u)
< M2 σ +
bi
du
α
u
0
i=1
≤ c,
where we used the definition of σ at the last estimate. Therefore z(t) < c for all
t ≥ t0 ≥ 0. Then it follows from (19) repeating the above argument that
Z
m
t X
2αr
−αs
z(t) ≤ M2 kϕk + K2 be
max z(u) bi ωi (N1 ce
) ds
t0 +r
u∈[t0 −r,t]
i=1
Z
m
c
X
ωi (u)
du,
t ≥ t0 ≥ 0,
bi
≤ M2 kϕk + K2 be2αr max z(u)
u
u∈[t0 −r,t]
0
i=1
and therefore
max
u∈[t0 −r,t]
z(u) ≤ M2 kϕk + M3
max
u∈[t0 −r,t]
z(u).
It implies
eα(t−t0 ) |x(t)| ≤
max
u∈[t0 −r,t]
z(u) ≤
M2
kϕk,
1 − M3
i.e., the statement of the theorem holds with K =
t ≥ t0 ≥ 0,
M2
1−M3 .
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