Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Construction of Liapunov Functionals for Linear
Volterra Integrodifferential Equations and
Stability of Delay Systems
Bo Zhang
Department of Mathematics and Computer Science
Fayetteville State University
Fayetteville, NC 28301
Abstract Inspired by the construction of a new Liapunov functional for linear
Volterra integrodifferential equations, we prove some general stability theorems for
functional differential equations with infinite delay and weaken the usual requirement
for positive definiteness of Liapunov functionals involved in stability theory.
AMS (MOS) subject classification: 34K20, 45J05
Key words: stability of delay systems, construction of Liapunov functionals, linear
Volterra equations
1
Introduction.
Consider a system of functional differential equations
x′ (t) = F (t, xt ),
(1.1)
in which F (t, φ), F (t, 0) = 0, is a functional defined for t ≥ 0 and φ ∈ C, where C
is the set of bounded continuous functions φ : R− → Rn , R− = (−∞, 0], with the
supremum norm k · k = sup{|φ(s)| : s ∈ R− }, where | · | is the Euclidean norm on
This paper is in final form and no version of it will be submitted for publication elsewhere
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 1
Rn . We assume that for each t0 ≥ 0 and each φ ∈ CH , CH = {φ ∈ C : kφk < H}
for some H > 0, there is at least one solution x = x(t, t0 , φ) of (1.1) defined on an
interval [t0 , α) with xt0 = φ. We may also denote the solution by xt = xt (t0 , φ). Here
xt (s) = x(t + s) for s ≤ 0. Moreover, if the solution remains bounded, then α = +∞.
We denote by C(X, Y ) the set of continuous functions φ : X → Y .
The object of this paper is to give conditions on Liapunov functionals to ensure
the asymptotic stability of the zero solution of (1.1) and to provide a new method of
constructing Liapunov functionals for the linear Volterra integrodifferential equation
x′ = Ax(t) +
Z
t
0
C(t − s)x(s)ds
(1.2)
where A is a constant and C ∈ C(R+ , R), R+ = [0, +∞). In fact, it was equation
(1.2) and its nonlinear perturbations which inspired this investigation. There are
many known results and applications on stability and basic theory of systems (1.1)
and (1.2) in the literature. For reference and history, we refer to the books of Burton
[3,4], Driver [8], Gripenberg, Londen, and Staffans [9], Hale [11], Hino, Murakami and
Naito [12], Kolmanovskii [16], Krasovskii [17], Kuang [19], Lakshmikantham, Wen,
and Zhang [20], Yoshizawa [24].
Let V : R × C → R+ be continuous. The upper right-hand derivative along a
solution of (1.1) is defined by
′
V(1.1)
(t, φ) = lim sup{V (t + δ, xt+δ (t, φ)) − V (t, φ)}/δ
δ→0+
Definition 1.1. The zero solution of (1.1) is said to be stable if for each ε > 0
and t0 ≥ 0, there is a δ = δ(ε, t0) > 0 such that [φ ∈ C, kφk < δ, t ≥ t0 ] imply
|x(t, t0 , φ)| < ε. The zero solution of (1.1) is uniformly stable if it is stable and δ is
independent of t0 .
Definition 1.2. The zero solution of (1.1) is said to be asymptotically stable if
it is stable and if for each t0 ≥ 0 there exists δ0 > 0 such that kφk < δ0 implies that
x(t, t0 , φ) → 0 as t → +∞. The zero solution is uniformly asymptotically stable if it
is uniformly stable and if there is δ1 > 0 and for each ε > 0 there exists T > 0 such
that [t0 ≥ 0, φ ∈ C, kφk < δ1 , t ≥ T + t0 ] imply that |x(t, t0 , φ)| < ε.
Definition 1.3. W : R+ → R+ is called a wedge if W is continuous and strictly
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 2
increasing with W (0) = 0. Throughout of this paper W and Wj (j = 1, 2, · · ·) will
denote the wedges.
Definition 1.4. A continuous function g : R+ → R+ is convex downward if
g[(t + s)/2] ≤ [g(t) + g(s)]/2 for all t, s ∈ R+ .
Jensen’s inequality.
Let g be convex downward and let x, p : [a, b] → R+ be
Rb
continuous with a p(s)ds > 0. Then
Z
b
a
p(s)ds g
hZ
b
a
p(s)x(s)ds
.Z
b
a
Z
i
p(s)ds ≤
b
a
p(s)g(x(s))ds.
For reference on Jensen’s inequality and its applications in stability theory, we refer
to Becker, Burton, and Zhang [1] and Natanson [22].
Lemma 1.1. Let W1 be a wedge. For any L > 0, define W0 (r) = 0r W1 (s)ds/L
on [0, L]. Then W0 is a convex downward wedge such that W0 (r) ≤ W1 (r) for all
r ∈ [0, L].
R
It is well-known (see [7]) that conditions
W1 (|φ(0)|) ≤ V (t, φ), V (t, 0) = 0,
′
V(1.1)
(t, φ) ≤ 0
(1.3)
guarantee the stability of the zero solution of (1.1) and
W1 (|x(t)|) ≤ V (t, xt ) ≤ W2 (|x(t)|) + W3 (
′
V(1.1)
(t, xt ) ≤ −W5 (|x(t)|)
Z
0
t
Φ(t − s)W4 (|x(s)|)ds),
(1.4)
yield the uniform asymptotic stability, where Φ : R+ → R+ is continuous, bounded,
and Φ ∈ L1 (R+ ).
The goal here is to weaken the lower bound of V (t, φ) to
W1 (|x(t) +
Z
0
t
G(t − s)x(s)ds|) ≤ V (t, xt )
(1.5)
and drop the boundedness condition on Φ(t). This will be done in section 2. As an
application of the main theorem, we construct a new Liapunov functional for (1.2) in
section 3.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 3
2
A General Stability Theorem
Liapunov functional of the form
V (t, xt ) = |x(t)|2 +
Z
t
Φ(t − s)|x(s)|2 ds, Φ(s) ≥ 0
0
may work for (1.2), but the stability condition may not be the best. Many concrete
examples and applications (see [5]) suggest that V (t, φ) should take the form
V (t, xt ) = V1 (t, xt ) + V2 (t, xt )
where
Z
V1 (t, xt ) = (x(t) +
t
0
G(t − s)x(s)ds)2
or a more general integral operator. A complete discussion on the construction of
such functionals for (1.2) will be given in Section 3. The following lemmas are needed
for our main theorem.
Lemma 2.1 ([9], p.14). Let G, Z : R+ → Rn×n be continuous and satisfy
Z(t) = G(t) −
t
Z
G(t − s)Z(s)ds.
0
(2.1)
If y, x : R+ → Rn satisfy
t
y(t) = x(t) +
Z
t
x(t) = y(t) −
Z
then
G(t − s)x(s)ds,
0
0
Z(t − s)y(s)ds
for all t ∈ R+ .
Lemma 2.2. For all the functions given in Lemma 2.1, if 0+∞ |G(u)|du < 1, then
|Z(u)|du < +∞ and if y(t) → 0 as t → +∞, then x(t) → 0 as t → +∞.
R
R +∞
0
Proof. It follows from (2.1) that
|Z(t)| ≤ |G(t)| +
Z
0
t
|G(t − s)||Z(s)|ds.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 4
Integrate the above inequality from 0 to ∞ and interchange variables of the second
term to obtain
Z
+∞
0
|Z(t)|dt ≤
Z
+∞
0
|G(t)|dt +
Z
+∞
0
Z
|Z(s)|ds
0
+∞
|G(t)|dt.
This yields 0+∞ |Z(t)|dt < +∞ since 0+∞ |G(t)|dt < 1. The statement y(t) → 0 as
t → +∞ implies x(t) → 0 as t → +∞ is a known result ([3], p.48). The proof is
complete.
R
R
The following result has a general lower bound on V (t, φ) and removes the boundedness restriction on Φ(t) in ([7], p.144).
Theorem 2.1. Suppose that there exists a continuousR functional V : R+ ×
C → R+ , V (t, 0) = 0, functions G, Φ ∈ C(R+ , R) with 0+∞ |G(u)|du < 1 and
Φ ∈ L1 (R+ , R+ ), and wedges Wj such that the following conditions hold for t ≥ 0
and kxt k < H,
(i) W1 (|x(t) +
Rt
0
G(t − s)x(s)ds|) ≤ V (t, xt ),
(ii) V (t, xt ) ≤ W2 (|x(t)|) + W3 (
′
(iii) V(1.1)
(t, xt ) ≤ −W5 (|x(t)|).
Rt
0
Φ(t − s)W4 (|x(s)|)ds)
Then the zero solution of (1.1) is uniformly asymptotically stable.
Proof. Let J =
0 < δ < B and
R +∞
0
Φ(u)du. For each 0 < B < H, we choose δ > 0 with
W2 (δ) + W3 [JW4 (δ)] < W1 [B(1 −
Z
+∞
0
|G(u)|du)].
(2.2)
Let x(t) = x(t, t0 , φ) be a solution of (1.1) with kφk < δ and V (t) = V (t, xt ). Then
for t ≥ t0 , we have
W1 (|x(t) +
Z
0
t
G(t − s)x(s)ds|) ≤ V (t) ≤ V (t0 )
≤ W2 (δ) + W3 (JW4 (δ)).
We claim that |x(t)| < B for all t ≥ t0 . Note that |x(u)| < δ < B for all 0 ≤ u ≤ t0 .
If there exists the first tˆ > t0 such that |x(tˆ)| = B and |x(s)| < B for t0 ≤ s < tˆ, then
W1 [B(1 −
Z
0
+∞
|G(u)|du)] ≤ W1 (|x(tˆ) +
Z
0
tˆ
G(tˆ − s)x(s)ds|)
≤ W2 (δ) + W3 [JW4 (δ)],
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 5
which contradicts (2.2). Thus,
the zero solution of (1.1) is uniformly stable. By (iii),
R
we have V (t) − V (τ ) ≤ − τt W5 (|x(s)|)ds for t ≥ τ ≥ t0 . Since
V (τ ) ≤ W2 (B) + W3 (JW4 (B)) =: B ∗
for any τ ≥ t0 , we have
Z
t
τ
W5 (|x(s)|)ds ≤ V (τ ) ≤ B ∗ .
This implies for each σ > 0 there exists L > 0 such that W5 (|x(τ ∗ )|) < σ for some
τ ∗ ∈ [τ, τ + L]. Now let σ > 0 be fixed and find h > 0 so that
(B + W4 (B))
Z
+∞
h
Φ(u)du < σ.
(2.3)
By the definition of L, we can choose a sequence {tn } satisfying the following conditions:
tn−1 + h ≤ tn ≤ tn−1 + h + L
and
|x(tn )| < σ
for n = 1, 2, · · ·. Define K = sup{Φ(u) : 0 ≤ u ≤ h}. By (i), we have
V (tj ) ≤ W2 (|x(tj )|)
+W3
Z
tj −h
0
Φ(tj − s)W4 (|x(s)|)ds +
Z
Z
< W2 (σ) + W3 W4 (B)
≤ W2 (σ) + W3 σ + K
Since
V (tn ) − V (t0 ) ≤ −
we have
tn
Z
t0
+∞
X Z tj
j=1 tj −h
+∞
h
tj
tj −h
Z
Φ(s)ds + K
tj
tj −h
tj
Φ(tj − s)W4 (|x(s)|)ds
Z
tj −h
W4 (|x(s)|)ds
W4 (|x(s)|)ds .
W5 (|x(s)|)ds ≤ −
n Z
X
tj
j=1 tj −h
W5 (|x(s)|)ds,
W5 (|x(s)|)ds ≤ V (t0 ) ≤ B ∗ .
This implies there exists a positive integer N such that
Z
tN
tN −h
W5 (|x(s)|)ds < σ.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 6
By Lemma 1.1 , there exists a convex downward wedge W6 with W6 (r) ≤ W5 [W4−1 (r)]
for 0 ≤ r ≤ W4 (B). This yields
tN
Z
tN −h
W5 (|x(s)|)ds =
Z
tN
tN −h
W5 (W4−1)[W4 (|x(s)|)]ds
≥
Z
tN
tN −h
W6 [W4 (|x(s)|)ds.
Apply Jensen’s inequality to obtain
Z
tN
tN −h
W6 [W4 (|x(s)|)]ds ≥ hW6
Thus,
Z
tN
h1 Z
h
tN
tN −h
W4 (|x(s)|)ds].
W4 (|x(s)|)ds < hW6−1 (σ/h)
tN −h
and
V (t) ≤ V (tN ) ≤ W2 (σ) + W3 σ + KhW6−1 (σ/h)
for all t ≥ tN . Therefore,
W1 (|x(t) +
Z
0
t
G(t − s)x(s)ds|) ≤ W2 (σ) + W3 σ + KhW6−1 (σ/h)
and
|x(t) +
Z
t
0
G(t − s)x(s)ds| ≤ W1−1 W2 (σ) + W3 [σ + KhW6−1 (σ/h)] =: σ ∗ .
Let y(t) = x(t) +
Rt
0
G(t − s)x(s)ds. By Lemmas 2.1 and 2.2, we have
Z
|x(t)| = |y(t) −
≤ σ∗ +
∗
Z
t
tN
0
≤ σ + 2B
Z(t − s)y(s)ds|
0
|Z(t − s)||y(s)|ds +
Z
t
tN
|Z(u)|du + σ
∗
Z
0
Z
t
tN
+∞
|Z(t − s)||y(s)|ds
|Z(u)|du
for t ≥ tN . Note that tN ≤ t0 + N(h + L). We also choose N large enough so that
2B
Z
+∞
N (h+L)
|Z(u)|du < σ ∗ .
Define Γ = 0+∞ |Z(u)|du. Then |x(t)| < (2 + Γ)σ ∗ for all t ≥ tN . For any ε > 0,
choose σ > 0 so that σ ∗ = ε/(2 + Γ). Letting T = N(h + L), we get |x(t)| < ε for all
t ≥ T + t0 . This proves the zero solution of (1.1) is uniformly asymptotically stable.
R
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 7
Remark 2.1. It is clear from the proof that Theorem 2.1 can be extended to
nonconvolution type with modifications on G and Φ. For more discussions on such
extension, we refer the reader to Lakshmikantham, Wen, and Zhang [20].
3
Construction of a Liapunov functional
Consider the linear scalar Volterra integrodifferential equation
′
x (t) = Ax(t) +
t
Z
C(t − s)x(s)ds
0
(3.1)
where A is a constant and C : R+ → R is continuous. We will construct a Liapunov
functional that is more general than those exist in the literature (see [6], [14]) and
provides a stability region that is closer to the one given by the characteristic equation
of (3.1). Our method here is inspired by the work of Knyazhishche and Shcheglov
[15] who investigated the linear scalar differential equations with finite delay.
Theorem
3.1. Suppose there exists aRconstant α ≤ 0 such that
R
R
Gα (t) = t+∞ C(u)eαu due−αt exists with 0+∞ |Gα (u)|du < 1 and 0+∞ u|Gα(u)|du <
+∞. If
A+
Z
+∞
0
αu
C(u)e du + |α|
Z
0
+∞
|Gα (u)|du < 0,
(3.2)
then the zero solution of (3.1) is uniformly asymptotically stable.
Proof. Let x(t) = x(t, t0 , φ) be a solution of (3.1) and define
V1 (t) =
1
x(t) +
2
Z
t
0
Gα (t − s)x(s)ds
2
(3.3)
for t ≥ t0 . Taking the derivative along the solution, we obtain
V1′ (t) = (x(t) +
+
Z
0
+∞
Z
t
Gα (t − s)x(s)ds) Ax(t) +
0
C(u)eαu du x(t) −
= (x(t) +
Z
Z
0
t
0
Z
0
C(t − s)x(s)ds − α
Gα (t − s)x(s)ds) a1 x(t) − α
= a1 x2 (t) + (a1 − α)x(t)
0
C(t − s)x(s)ds
t
Z
t
Z
0
Z
t
Gα (t − s)x(s)ds
0
t
Gα (t − s)x(s)ds
t
Gα (t − s)x(s)ds − α
(3.4)
Z
0
t
Gα (t − s)x(s)ds
2
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 8
where a1 = A +
Z
t
0
R +∞
0
C(u)eαu du. It follows from Cauchy’s inequality
Gα (t − s)x(s)ds
2
t
Z
≤
|Gα (t − s)|ds
0
t
Z
0
|Gα (t − s)|x2 (s)ds.
Thus, for any constant δ > 0 we have
V1′ (t) ≤ a1 x2 (t) + (a1 − α)x(t)
+|α|
Z
0
t
|Gα (t−s)|ds
Z
Z
t
0
t
0
Gα(t−s)x(s)ds
|Gα (t−s)|x2 (s)ds
(a1 − α)2 2
x (t)
4δ
Z
≤ a1 x2 (t) +
t
+(|α| + δ)
|Gα (t − s)|ds
0
Z
t
|Gα (t − s)|x2 (s)ds.
0
(3.5)
Next, define
V2 (t) = p
Z tZ
0
+∞
t−s
|Gα (u)|dux2(s)ds
(3.6)
where p is a positive constant and set
V (t) = V1 (t) + V2 (t).
Observe that
V2′ (t)
=p
Z
+∞
0
2
|Gα (u)|dux (t) − p
Z
0
t
|Gα (t − s)|x2 (s)ds.
Combining with (3.5), this yields
′
V (t) ≤
(a1 − α)2
+p
a1 +
4δ
+ [(|α| + δ)
Z
+∞
0
Z
0
+∞
|Gα (u)|du x2 (t)
|Gα (u)|du − p]
Z
0
t
(3.7)
|Gα (t − s)|x2 (s)ds.
We will choose the best δ and p so that
Z +∞
(a1 − α)2
a1 +
|Gα (u)|du < 0
+p
4δ
0
(3.8)
and
(|α| + δ)
Z
0
+∞
|Gα (u)|du − p ≤ 0.
(3.9)
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 9
Notice that if
R +∞
0
(|α| + δ)
|Gα (u)|du 6= 0, then (3.8) and (3.9) hold if and only if
Z
+∞
0
|Gα (u)|du
2
≤p
Z
+∞
0
|Gα (u)|du < |a1 | −
(a1 − α)2
.
4δ
This implies that we choose δ > 0 so that
(|α| + δ)
Z
+∞
0
|Gα (u)|du
2
< |a1 | −
(a1 − α)2
.
4δ
(3.10)
Let Q = ( 0+∞ |Gα (u)|du)2. We may assume that Gα (u) 6≡ 0 since Gα (u) ≡ 0 for all
u ∈ R+ implies C(u) ≡ 0 for all u ∈ R+ . Thus we assume that Q > 0. It is clear that
(3.10) is equivalent to
R
4Qδ 2 + 4(|α|Q − |a1 |)δ + (a1 − α)2 < 0.
(3.11)
The quadratic function of δ on the left-hand side of (3.11) has its minimum at
|a1 | − |α|Q
.
2Q
δ∗ =
Since
R +∞
0
|Gα (u)|du < 1, we have 0 < Q < 1 and
|a1 | − |α|Q = −A −
Z
≥ − A+
+∞
0
Z
0
C(u)eαu du − |α|Q
+∞
C(u)eαu du + |α|
Z
0
+∞
|Gα (u)|du > 0
by (3.2). Thus δ∗ > 0 and
4Qδ∗2 + 4(|α|Q − |a1 |)δ∗ + (a1 − α)2
(|a1 | − |α|Q)2 2(|a1 | − |α|Q)2
−
+ (a1 − α)2
Q
Q
Q−1
=
[|a1 |2 − Q|α|2 ]
Q
q
q
Q−1
=
(|a1 | + Q|α|)(|a1| − Q|α|)
Q
Z +∞
Z +∞
q
1 − Q
αu
=
|a1 | + Q|α|)(A +
|Gα (u)|du < 0.
C(u)e du + |α|
Q
0
0
=
Let
γ1 =: (|α| + δ∗ )Q = (|α| +
|a1 | − |α|Q
|a1 | + |α|Q
)Q =
2Q
2
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 10
and
γ2
(a1 − α)2
|a1 |2 − α2 Q + (1 − Q)|a1 |2
=: |a1 | −
=
.
4δ∗
2(|a1 | − |α|Q)
p=
γ1 + γ2
(1 − Q)|a1 |2 + 2|a1 |2 − Q(1 + Q)|α|2
√
√
=
.
2 Q
4(|a1 | − |α|Q) Q
Define
We now obtain
a1 +
and
(3.12)
q
(a1 − α)2
(1 − Q)(|a1 |2 − α2 Q)
+p Q=−
4δ∗
4(|a1 | − |α|Q)
q
(|α| + δ∗ )Q − p Q = −
Define
(1 − Q)(|a1 |2 − α2 Q)
.
4(|a1 | − |α|Q)
(1 − Q)(|a1 |2 − α2 Q)
1
min{1, √ }.
4(|a1 | − |α|Q)
Q
Combining (3.3), (3.6), and (3.7), we have
β=
Z t
Z t Z +∞
2
1
|Gα (u)|dux2 (s)ds
x(t) +
Gα (t − s)x(s)ds + p
2
0
0 t−s
V (t) =
≤ x2 (t) +
Z t
=: x2 (t) +
Z
0
0
t
|Gα (t − s)| + p
+∞
Z
t−s
|Gα (u)|du x2 (s)ds
Φ(t − s)x2 (s)ds
(3.13)
and
′
2
V (t) ≤ −βx (t) − β
Z
t
0
|Gα (t − s)|x2 (s)ds
(3.14)
where p is given in (3.12) and
Φ(t) = |Gα (t)| + p
Z
+∞
t
|Gα (u)|du.
One can show by interchanging the order of integration that
Z
0
+∞
Z
+∞
s
|Gα (u)|duds =
Z
0
+∞
u|Gα (u)|du < +∞.
(3.15)
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 11
Thus, Φ ∈ L1 (R+ ). By Theorem 2.1, the zero solution of (3.1) is uniformly asymptotically stable. This completes the proof.
When α = 0, Theorem 3.1 takes the following form.
Corollary 3.1. Suppose that
A+
∞
Z
0
∞
Z
0
∞
+∞
Z
s
0
+∞
t
and
Z
Z
s
C(u)du < 0,
C(u)du dt < 1,
C(u)du ds < +∞.
(3.16)
Then the zero solution of (3.1) is uniformly asymptotically stable.
In ([6], p.162) Burton and Mahfoud proved the following theorem. Brauer [2],
Jordan [13], and Krisztin [18] also investigated the same problem in one or higher
dimensions using the characteristic equation of (3.1).
Theorem B1 . Suppose that
A+
∞
Z
0
|
Z
Z
+∞
0
∞
t
C(u)du < 0,
(3.17)
C(u)du|dt < 1,
(3.18)
and
Z
∞
0
u|C(u)|du < +∞.
(3.19)
Then the zero solution of (3.1) is uniformly asymptotically stable.
Although (3.19) is weaker than (3.16) for C(u) with constant sign, one can show
that (3.19) implies (3.16) for α < 0. In fact, if (3.19) holds, we have
Z
0
+∞
u|Gα (u)|du ≤
Z
+∞
0
u
Z
+∞
u
|C(s)|eα(s−u) dsdu
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 12
=
Z
=
Z
+∞
0
s
Z
0
+∞
0
≤ α∗
Z
ue−αu du|C(s)|eαs ds
1 −αs
1
1
se
− 2 e−αs + 2 |C(s)|eαs ds
|α|
α
α
+∞
(s + 1)|C(s)|ds < +∞
0
1
where α∗ = max{ |α|
, α12 }.
Corollary 3.2. Suppose (3.19) holds and there exists a constant α < 0 such that
Z
+∞
0
|Gα (u)|du < 1
(3.20)
and
A+
Z
+∞
0
αu
C(u)e du + |α|
Z
+∞
0
|Gα (u)|du < 0,
(3.21)
then the zero solution of (3.1) is uniformly asymptotically stable.
It is known (see [10],[21]) that the zero solution of (3.1) is uniformly asymptotically
stable if and only if
|z − A − C ∗ (z)| =
6 0
for Re(z) ≥ 0, whereR z is a complex number and C ∗ is the Laplace transform of C
defined by C ∗ (z) = 0+∞ e−zt C(t)dt. Thus, condition (3.17) is necessary for uniform
asymptotic stability. The following example shows that (3.18) does not hold, but
(3.20) and (3.21) are satisfied for an appropriate α. Consider
′
x (t) = 0.06x(t) −
Z
0
Let A = 0.06 and C(t) = −e−0.9t . Then
Z
0
+∞
|
Z
t
+∞
C(u)du|dt =
t
Z
e−0.9(t−s) x(s)ds.
+∞
0
(3.22)
1
1 −0.9t
e
dt =
> 1.
0.9
(0.9)2
Thus, (3.18) fails. Letting α = −0.8, we have
1 −0.9t
e
,
1.7
t
Z
Z +∞
1
1 +∞ −0.9t
< 1,
e
dt =
|Gα (u)|du =
1.7 0
1.53
0
Gα (t) =
Z
+∞
C(u)eαu due−αt = −
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 13
and
A+
Z
+∞
0
= 0.06 −
Z
C(u)eαu du + |α|
+∞
0
e−1.7u du +
Z
+∞
|Gα (u)|du
0
0.8
1.53
< −0.005.
It is obvious that 0+∞ u|Gα(u)|du < +∞. By Theorem 3.1, the zero solution of (3.22)
is uniformly asymptotically stable.
R
Generalizations of Theorem B1 to systems and noncovolution type of equations
can be found in Burton and Mahfoud [6] and Zhang [24]. The author has not yet
obtained a satisfactory extension of Theorem 3.1 to systems and will pursue further
investigation. Finally we present a theorem on asymptotic stability.
Theorem 3.2. Suppose there exists a constant α ≤ 0 such that (3.20) and (3.21)
hold. Then the zero solution of (3.1) is asymptotically stable.
Proof. It is clear that the zero solution of (3.1) is stable by (3.13) and (3.14).
Let x(t) = x(t, t0 , φ) be a solution of (3.1). We will show that x(t) → 0 as t → +∞.
Indeed, by (3.14) and Cauchy’s inequality, we have
V ′ (t) ≤ −
β
x(t) +
2
Z
t
0
Gα (t − s)x(s)ds .
2
(3.23)
2
∈ L1 (R+ ).
(3.24)
This implies that
γ(t) =: x(t) +
Z
0
t
Gα (t − s)x(s)ds
By (3.4), we have
|γ ′ (t)| = 2 a1 x2 (t) + (a1 −α)x(t)
Z
0
t
Gα (t − s)x(s)ds − α
≤ (2|a1 | + |a1 − α|)x2 (t) + (2|α| + |a1 − α|)
Z
t
0
Z
0
t
2
Gα (t − s)x(s)ds
|Gα (t − s)|x2 (s)ds. (3.25)
Combining (3.24) Rand (3.25), we get |γ ′ (t)| ∈ L1 (R+ ). Therefore, limt→+∞ γ(t) = 0.
Let y(t) = x(t) + 0t Gα (t − s)x(s)ds. By Lemmas 2.1 and 2.2, we have
|x(t)| = |y(t) −
Z
0
t
Z(t − s)y(s)ds|
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 14
≤ |y(t)| + |
t
Z
Z(t − s)y(s)ds| → 0
0
since Z ∈ L1 (R+ ) and y(t) → 0 as t → +∞. This completes the proof.
Corollary 3.3. If
A+
Z
+∞
C(u)du < 0
0
and
∞
Z
0
|
∞
Z
t
C(u)du|dt < 1,
then the zero solution of (3.1) is asymptotically stable.
Corollary 3.4. If
A+
+∞
Z
0
|C(u)|du < 0,
(3.26)
then the zero solution of (3.1) is asymptotically stable.
Proof. If (3.26) holds, we can choose α sufficiently large so that conditions of
Theorem 3.2 will be satisfied. In fact, for α < 0 we have
|α|
Z
+∞
0
|Gα (u)|du ≤ −α
=
Z
= −
This yields
Z
for α < 0. If
R +∞
0
+∞
0
Z
+∞
0
+∞
t
Z
A+
+∞
≤ A+
Z
+∞
≤ A+
Z
+∞
0
0
0
+∞
t
|C(u)|eαu due−αt dt
+∞
|C(u)|eαu du e−αt
+∞
0
|C(u)|eαu du +
|Gα (u)|du ≤
|C(u)|du < |α|, then
Z
Z
C(u)eαu du + |α|
C(u)eαu du −
Z
0
0
Z
+∞
0
|C(u)|du
|C(u)|du.
1 Z +∞
|C(u)|du
|α| 0
R +∞
0
Z
0
+∞
+
|Gα (u)|du < 1. By (3.26), we also have
Z
0
+∞
+∞
|Gα (u)|du
|C(u)|eαu du +
Z
0
+∞
|C(u)|du
|C(u)|du < 0.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 15
By Theorem 3.2, the zero solution of (3.1) is asymptotically stable.
Burton and Mahfoud ([6], p.146) showed that (3.26) yields uniform asymptotic
stability by proving that x(t) ∈ L1 (R+ , R).
References
[1] L. C. Becker, T. A. Burton, and S. Zhang, Functional differential equations and
Jenson’s inequality, J. Math. Anal. Appl. 138(1989), 137-156.
[2] F. Brauer, Asymptotic stability of a class of integrodifferential equations, J.
Differential Equations 28(1978), 180-188.
[3] T. A. Burton, Volterra Integral and Differential Equations, Academic Press, New
York, 1983.
[4] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, 1985.
[5] T. A. Burton and L. Hatvani, Stability theorems for nonautonomous functional
differential equations by Liapunov functionals, Tohoku Math. J. 41(1989), 65104.
[6] T. A. Burton and W. E. Mahfoud, Stability criteria for Volterra equations, Trans.
Amer. Math. Soc. 279(1983), 143-174.
[7] T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in
ordinary and functional differential equations, Ann. Mat. Pura Appl. 145(1986),
129-158.
[8] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New
York, 1977.
[9] G. Gripenberg, S. O. Londen, and O. Staffans, Volterra Integral and Functional
Equations, Cambridge University Press, New York, 1990.
[10] S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 19(1975), 142-166.
[11] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New
York, 1977.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 16
[12] Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with
Infinite Delay, Lecture Notes in Mathematics 1473, Springer-Verlag, New York,
1991.
[13] G. G. Jordan, Asymptotic stability of a class of integrodifferential equations, J.
Differential Equations 31(1979), 359-365.
[14] J. Kato, Liapunov functional vs Liapunov function, Proc. Internat. Symposium
on Functional Differential Equations, World Scientific, Singapore, 1991.
[15] L. B. Knyazhishche and V. A. Shcheglov, On the sign definiteness of Liapunov
functionals and stability of a linear delay equation, Electronic Journal of Qualitative Theory of Differential Equations, 8(1998), 1-13.
[16] V. B. Kolmanosvskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986.
[17] N. N. Krasovskii, Stability of Motion, Stanford University Press, Stanford, CA,
1963.
[18] T. Krisztin, Uniform asymptotic stability of a class of integrodifferential systems,
Journal of Integral Equations and Applications 4(1988), 581-597.
[19] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
[20] V. Lakshmikantham, L. Wen, and B. Zhang, Theory of Functional Differential
Equations with Unbounded Delay,
[21] R. K. Miller, Asymptotic stability properties of linear Volterra integrodifferential
equations, J. Differential Equations 10(1971), 485-506.
[22] I. P. Natanson, Theory of Functions of a Real Variable, Vol. II, Ungar, New
York, 1960.
[23] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc.
Japan, Tokyo, 1966.
[24] B. Zhang, Necessary and sufficient conditions for stability and convergence of
solutions of a Volterra integrodifferential equation, Dynamic Systems and Applications 9(1996), 45-58.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 17
Volterra Integrodifferential Equations and
Stability of Delay Systems
Bo Zhang
Department of Mathematics and Computer Science
Fayetteville State University
Fayetteville, NC 28301
Abstract Inspired by the construction of a new Liapunov functional for linear
Volterra integrodifferential equations, we prove some general stability theorems for
functional differential equations with infinite delay and weaken the usual requirement
for positive definiteness of Liapunov functionals involved in stability theory.
AMS (MOS) subject classification: 34K20, 45J05
Key words: stability of delay systems, construction of Liapunov functionals, linear
Volterra equations
1
Introduction.
Consider a system of functional differential equations
x′ (t) = F (t, xt ),
(1.1)
in which F (t, φ), F (t, 0) = 0, is a functional defined for t ≥ 0 and φ ∈ C, where C
is the set of bounded continuous functions φ : R− → Rn , R− = (−∞, 0], with the
supremum norm k · k = sup{|φ(s)| : s ∈ R− }, where | · | is the Euclidean norm on
This paper is in final form and no version of it will be submitted for publication elsewhere
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 1
Rn . We assume that for each t0 ≥ 0 and each φ ∈ CH , CH = {φ ∈ C : kφk < H}
for some H > 0, there is at least one solution x = x(t, t0 , φ) of (1.1) defined on an
interval [t0 , α) with xt0 = φ. We may also denote the solution by xt = xt (t0 , φ). Here
xt (s) = x(t + s) for s ≤ 0. Moreover, if the solution remains bounded, then α = +∞.
We denote by C(X, Y ) the set of continuous functions φ : X → Y .
The object of this paper is to give conditions on Liapunov functionals to ensure
the asymptotic stability of the zero solution of (1.1) and to provide a new method of
constructing Liapunov functionals for the linear Volterra integrodifferential equation
x′ = Ax(t) +
Z
t
0
C(t − s)x(s)ds
(1.2)
where A is a constant and C ∈ C(R+ , R), R+ = [0, +∞). In fact, it was equation
(1.2) and its nonlinear perturbations which inspired this investigation. There are
many known results and applications on stability and basic theory of systems (1.1)
and (1.2) in the literature. For reference and history, we refer to the books of Burton
[3,4], Driver [8], Gripenberg, Londen, and Staffans [9], Hale [11], Hino, Murakami and
Naito [12], Kolmanovskii [16], Krasovskii [17], Kuang [19], Lakshmikantham, Wen,
and Zhang [20], Yoshizawa [24].
Let V : R × C → R+ be continuous. The upper right-hand derivative along a
solution of (1.1) is defined by
′
V(1.1)
(t, φ) = lim sup{V (t + δ, xt+δ (t, φ)) − V (t, φ)}/δ
δ→0+
Definition 1.1. The zero solution of (1.1) is said to be stable if for each ε > 0
and t0 ≥ 0, there is a δ = δ(ε, t0) > 0 such that [φ ∈ C, kφk < δ, t ≥ t0 ] imply
|x(t, t0 , φ)| < ε. The zero solution of (1.1) is uniformly stable if it is stable and δ is
independent of t0 .
Definition 1.2. The zero solution of (1.1) is said to be asymptotically stable if
it is stable and if for each t0 ≥ 0 there exists δ0 > 0 such that kφk < δ0 implies that
x(t, t0 , φ) → 0 as t → +∞. The zero solution is uniformly asymptotically stable if it
is uniformly stable and if there is δ1 > 0 and for each ε > 0 there exists T > 0 such
that [t0 ≥ 0, φ ∈ C, kφk < δ1 , t ≥ T + t0 ] imply that |x(t, t0 , φ)| < ε.
Definition 1.3. W : R+ → R+ is called a wedge if W is continuous and strictly
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 2
increasing with W (0) = 0. Throughout of this paper W and Wj (j = 1, 2, · · ·) will
denote the wedges.
Definition 1.4. A continuous function g : R+ → R+ is convex downward if
g[(t + s)/2] ≤ [g(t) + g(s)]/2 for all t, s ∈ R+ .
Jensen’s inequality.
Let g be convex downward and let x, p : [a, b] → R+ be
Rb
continuous with a p(s)ds > 0. Then
Z
b
a
p(s)ds g
hZ
b
a
p(s)x(s)ds
.Z
b
a
Z
i
p(s)ds ≤
b
a
p(s)g(x(s))ds.
For reference on Jensen’s inequality and its applications in stability theory, we refer
to Becker, Burton, and Zhang [1] and Natanson [22].
Lemma 1.1. Let W1 be a wedge. For any L > 0, define W0 (r) = 0r W1 (s)ds/L
on [0, L]. Then W0 is a convex downward wedge such that W0 (r) ≤ W1 (r) for all
r ∈ [0, L].
R
It is well-known (see [7]) that conditions
W1 (|φ(0)|) ≤ V (t, φ), V (t, 0) = 0,
′
V(1.1)
(t, φ) ≤ 0
(1.3)
guarantee the stability of the zero solution of (1.1) and
W1 (|x(t)|) ≤ V (t, xt ) ≤ W2 (|x(t)|) + W3 (
′
V(1.1)
(t, xt ) ≤ −W5 (|x(t)|)
Z
0
t
Φ(t − s)W4 (|x(s)|)ds),
(1.4)
yield the uniform asymptotic stability, where Φ : R+ → R+ is continuous, bounded,
and Φ ∈ L1 (R+ ).
The goal here is to weaken the lower bound of V (t, φ) to
W1 (|x(t) +
Z
0
t
G(t − s)x(s)ds|) ≤ V (t, xt )
(1.5)
and drop the boundedness condition on Φ(t). This will be done in section 2. As an
application of the main theorem, we construct a new Liapunov functional for (1.2) in
section 3.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 3
2
A General Stability Theorem
Liapunov functional of the form
V (t, xt ) = |x(t)|2 +
Z
t
Φ(t − s)|x(s)|2 ds, Φ(s) ≥ 0
0
may work for (1.2), but the stability condition may not be the best. Many concrete
examples and applications (see [5]) suggest that V (t, φ) should take the form
V (t, xt ) = V1 (t, xt ) + V2 (t, xt )
where
Z
V1 (t, xt ) = (x(t) +
t
0
G(t − s)x(s)ds)2
or a more general integral operator. A complete discussion on the construction of
such functionals for (1.2) will be given in Section 3. The following lemmas are needed
for our main theorem.
Lemma 2.1 ([9], p.14). Let G, Z : R+ → Rn×n be continuous and satisfy
Z(t) = G(t) −
t
Z
G(t − s)Z(s)ds.
0
(2.1)
If y, x : R+ → Rn satisfy
t
y(t) = x(t) +
Z
t
x(t) = y(t) −
Z
then
G(t − s)x(s)ds,
0
0
Z(t − s)y(s)ds
for all t ∈ R+ .
Lemma 2.2. For all the functions given in Lemma 2.1, if 0+∞ |G(u)|du < 1, then
|Z(u)|du < +∞ and if y(t) → 0 as t → +∞, then x(t) → 0 as t → +∞.
R
R +∞
0
Proof. It follows from (2.1) that
|Z(t)| ≤ |G(t)| +
Z
0
t
|G(t − s)||Z(s)|ds.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 4
Integrate the above inequality from 0 to ∞ and interchange variables of the second
term to obtain
Z
+∞
0
|Z(t)|dt ≤
Z
+∞
0
|G(t)|dt +
Z
+∞
0
Z
|Z(s)|ds
0
+∞
|G(t)|dt.
This yields 0+∞ |Z(t)|dt < +∞ since 0+∞ |G(t)|dt < 1. The statement y(t) → 0 as
t → +∞ implies x(t) → 0 as t → +∞ is a known result ([3], p.48). The proof is
complete.
R
R
The following result has a general lower bound on V (t, φ) and removes the boundedness restriction on Φ(t) in ([7], p.144).
Theorem 2.1. Suppose that there exists a continuousR functional V : R+ ×
C → R+ , V (t, 0) = 0, functions G, Φ ∈ C(R+ , R) with 0+∞ |G(u)|du < 1 and
Φ ∈ L1 (R+ , R+ ), and wedges Wj such that the following conditions hold for t ≥ 0
and kxt k < H,
(i) W1 (|x(t) +
Rt
0
G(t − s)x(s)ds|) ≤ V (t, xt ),
(ii) V (t, xt ) ≤ W2 (|x(t)|) + W3 (
′
(iii) V(1.1)
(t, xt ) ≤ −W5 (|x(t)|).
Rt
0
Φ(t − s)W4 (|x(s)|)ds)
Then the zero solution of (1.1) is uniformly asymptotically stable.
Proof. Let J =
0 < δ < B and
R +∞
0
Φ(u)du. For each 0 < B < H, we choose δ > 0 with
W2 (δ) + W3 [JW4 (δ)] < W1 [B(1 −
Z
+∞
0
|G(u)|du)].
(2.2)
Let x(t) = x(t, t0 , φ) be a solution of (1.1) with kφk < δ and V (t) = V (t, xt ). Then
for t ≥ t0 , we have
W1 (|x(t) +
Z
0
t
G(t − s)x(s)ds|) ≤ V (t) ≤ V (t0 )
≤ W2 (δ) + W3 (JW4 (δ)).
We claim that |x(t)| < B for all t ≥ t0 . Note that |x(u)| < δ < B for all 0 ≤ u ≤ t0 .
If there exists the first tˆ > t0 such that |x(tˆ)| = B and |x(s)| < B for t0 ≤ s < tˆ, then
W1 [B(1 −
Z
0
+∞
|G(u)|du)] ≤ W1 (|x(tˆ) +
Z
0
tˆ
G(tˆ − s)x(s)ds|)
≤ W2 (δ) + W3 [JW4 (δ)],
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 5
which contradicts (2.2). Thus,
the zero solution of (1.1) is uniformly stable. By (iii),
R
we have V (t) − V (τ ) ≤ − τt W5 (|x(s)|)ds for t ≥ τ ≥ t0 . Since
V (τ ) ≤ W2 (B) + W3 (JW4 (B)) =: B ∗
for any τ ≥ t0 , we have
Z
t
τ
W5 (|x(s)|)ds ≤ V (τ ) ≤ B ∗ .
This implies for each σ > 0 there exists L > 0 such that W5 (|x(τ ∗ )|) < σ for some
τ ∗ ∈ [τ, τ + L]. Now let σ > 0 be fixed and find h > 0 so that
(B + W4 (B))
Z
+∞
h
Φ(u)du < σ.
(2.3)
By the definition of L, we can choose a sequence {tn } satisfying the following conditions:
tn−1 + h ≤ tn ≤ tn−1 + h + L
and
|x(tn )| < σ
for n = 1, 2, · · ·. Define K = sup{Φ(u) : 0 ≤ u ≤ h}. By (i), we have
V (tj ) ≤ W2 (|x(tj )|)
+W3
Z
tj −h
0
Φ(tj − s)W4 (|x(s)|)ds +
Z
Z
< W2 (σ) + W3 W4 (B)
≤ W2 (σ) + W3 σ + K
Since
V (tn ) − V (t0 ) ≤ −
we have
tn
Z
t0
+∞
X Z tj
j=1 tj −h
+∞
h
tj
tj −h
Z
Φ(s)ds + K
tj
tj −h
tj
Φ(tj − s)W4 (|x(s)|)ds
Z
tj −h
W4 (|x(s)|)ds
W4 (|x(s)|)ds .
W5 (|x(s)|)ds ≤ −
n Z
X
tj
j=1 tj −h
W5 (|x(s)|)ds,
W5 (|x(s)|)ds ≤ V (t0 ) ≤ B ∗ .
This implies there exists a positive integer N such that
Z
tN
tN −h
W5 (|x(s)|)ds < σ.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 6
By Lemma 1.1 , there exists a convex downward wedge W6 with W6 (r) ≤ W5 [W4−1 (r)]
for 0 ≤ r ≤ W4 (B). This yields
tN
Z
tN −h
W5 (|x(s)|)ds =
Z
tN
tN −h
W5 (W4−1)[W4 (|x(s)|)]ds
≥
Z
tN
tN −h
W6 [W4 (|x(s)|)ds.
Apply Jensen’s inequality to obtain
Z
tN
tN −h
W6 [W4 (|x(s)|)]ds ≥ hW6
Thus,
Z
tN
h1 Z
h
tN
tN −h
W4 (|x(s)|)ds].
W4 (|x(s)|)ds < hW6−1 (σ/h)
tN −h
and
V (t) ≤ V (tN ) ≤ W2 (σ) + W3 σ + KhW6−1 (σ/h)
for all t ≥ tN . Therefore,
W1 (|x(t) +
Z
0
t
G(t − s)x(s)ds|) ≤ W2 (σ) + W3 σ + KhW6−1 (σ/h)
and
|x(t) +
Z
t
0
G(t − s)x(s)ds| ≤ W1−1 W2 (σ) + W3 [σ + KhW6−1 (σ/h)] =: σ ∗ .
Let y(t) = x(t) +
Rt
0
G(t − s)x(s)ds. By Lemmas 2.1 and 2.2, we have
Z
|x(t)| = |y(t) −
≤ σ∗ +
∗
Z
t
tN
0
≤ σ + 2B
Z(t − s)y(s)ds|
0
|Z(t − s)||y(s)|ds +
Z
t
tN
|Z(u)|du + σ
∗
Z
0
Z
t
tN
+∞
|Z(t − s)||y(s)|ds
|Z(u)|du
for t ≥ tN . Note that tN ≤ t0 + N(h + L). We also choose N large enough so that
2B
Z
+∞
N (h+L)
|Z(u)|du < σ ∗ .
Define Γ = 0+∞ |Z(u)|du. Then |x(t)| < (2 + Γ)σ ∗ for all t ≥ tN . For any ε > 0,
choose σ > 0 so that σ ∗ = ε/(2 + Γ). Letting T = N(h + L), we get |x(t)| < ε for all
t ≥ T + t0 . This proves the zero solution of (1.1) is uniformly asymptotically stable.
R
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 7
Remark 2.1. It is clear from the proof that Theorem 2.1 can be extended to
nonconvolution type with modifications on G and Φ. For more discussions on such
extension, we refer the reader to Lakshmikantham, Wen, and Zhang [20].
3
Construction of a Liapunov functional
Consider the linear scalar Volterra integrodifferential equation
′
x (t) = Ax(t) +
t
Z
C(t − s)x(s)ds
0
(3.1)
where A is a constant and C : R+ → R is continuous. We will construct a Liapunov
functional that is more general than those exist in the literature (see [6], [14]) and
provides a stability region that is closer to the one given by the characteristic equation
of (3.1). Our method here is inspired by the work of Knyazhishche and Shcheglov
[15] who investigated the linear scalar differential equations with finite delay.
Theorem
3.1. Suppose there exists aRconstant α ≤ 0 such that
R
R
Gα (t) = t+∞ C(u)eαu due−αt exists with 0+∞ |Gα (u)|du < 1 and 0+∞ u|Gα(u)|du <
+∞. If
A+
Z
+∞
0
αu
C(u)e du + |α|
Z
0
+∞
|Gα (u)|du < 0,
(3.2)
then the zero solution of (3.1) is uniformly asymptotically stable.
Proof. Let x(t) = x(t, t0 , φ) be a solution of (3.1) and define
V1 (t) =
1
x(t) +
2
Z
t
0
Gα (t − s)x(s)ds
2
(3.3)
for t ≥ t0 . Taking the derivative along the solution, we obtain
V1′ (t) = (x(t) +
+
Z
0
+∞
Z
t
Gα (t − s)x(s)ds) Ax(t) +
0
C(u)eαu du x(t) −
= (x(t) +
Z
Z
0
t
0
Z
0
C(t − s)x(s)ds − α
Gα (t − s)x(s)ds) a1 x(t) − α
= a1 x2 (t) + (a1 − α)x(t)
0
C(t − s)x(s)ds
t
Z
t
Z
0
Z
t
Gα (t − s)x(s)ds
0
t
Gα (t − s)x(s)ds
t
Gα (t − s)x(s)ds − α
(3.4)
Z
0
t
Gα (t − s)x(s)ds
2
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 8
where a1 = A +
Z
t
0
R +∞
0
C(u)eαu du. It follows from Cauchy’s inequality
Gα (t − s)x(s)ds
2
t
Z
≤
|Gα (t − s)|ds
0
t
Z
0
|Gα (t − s)|x2 (s)ds.
Thus, for any constant δ > 0 we have
V1′ (t) ≤ a1 x2 (t) + (a1 − α)x(t)
+|α|
Z
0
t
|Gα (t−s)|ds
Z
Z
t
0
t
0
Gα(t−s)x(s)ds
|Gα (t−s)|x2 (s)ds
(a1 − α)2 2
x (t)
4δ
Z
≤ a1 x2 (t) +
t
+(|α| + δ)
|Gα (t − s)|ds
0
Z
t
|Gα (t − s)|x2 (s)ds.
0
(3.5)
Next, define
V2 (t) = p
Z tZ
0
+∞
t−s
|Gα (u)|dux2(s)ds
(3.6)
where p is a positive constant and set
V (t) = V1 (t) + V2 (t).
Observe that
V2′ (t)
=p
Z
+∞
0
2
|Gα (u)|dux (t) − p
Z
0
t
|Gα (t − s)|x2 (s)ds.
Combining with (3.5), this yields
′
V (t) ≤
(a1 − α)2
+p
a1 +
4δ
+ [(|α| + δ)
Z
+∞
0
Z
0
+∞
|Gα (u)|du x2 (t)
|Gα (u)|du − p]
Z
0
t
(3.7)
|Gα (t − s)|x2 (s)ds.
We will choose the best δ and p so that
Z +∞
(a1 − α)2
a1 +
|Gα (u)|du < 0
+p
4δ
0
(3.8)
and
(|α| + δ)
Z
0
+∞
|Gα (u)|du − p ≤ 0.
(3.9)
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 9
Notice that if
R +∞
0
(|α| + δ)
|Gα (u)|du 6= 0, then (3.8) and (3.9) hold if and only if
Z
+∞
0
|Gα (u)|du
2
≤p
Z
+∞
0
|Gα (u)|du < |a1 | −
(a1 − α)2
.
4δ
This implies that we choose δ > 0 so that
(|α| + δ)
Z
+∞
0
|Gα (u)|du
2
< |a1 | −
(a1 − α)2
.
4δ
(3.10)
Let Q = ( 0+∞ |Gα (u)|du)2. We may assume that Gα (u) 6≡ 0 since Gα (u) ≡ 0 for all
u ∈ R+ implies C(u) ≡ 0 for all u ∈ R+ . Thus we assume that Q > 0. It is clear that
(3.10) is equivalent to
R
4Qδ 2 + 4(|α|Q − |a1 |)δ + (a1 − α)2 < 0.
(3.11)
The quadratic function of δ on the left-hand side of (3.11) has its minimum at
|a1 | − |α|Q
.
2Q
δ∗ =
Since
R +∞
0
|Gα (u)|du < 1, we have 0 < Q < 1 and
|a1 | − |α|Q = −A −
Z
≥ − A+
+∞
0
Z
0
C(u)eαu du − |α|Q
+∞
C(u)eαu du + |α|
Z
0
+∞
|Gα (u)|du > 0
by (3.2). Thus δ∗ > 0 and
4Qδ∗2 + 4(|α|Q − |a1 |)δ∗ + (a1 − α)2
(|a1 | − |α|Q)2 2(|a1 | − |α|Q)2
−
+ (a1 − α)2
Q
Q
Q−1
=
[|a1 |2 − Q|α|2 ]
Q
q
q
Q−1
=
(|a1 | + Q|α|)(|a1| − Q|α|)
Q
Z +∞
Z +∞
q
1 − Q
αu
=
|a1 | + Q|α|)(A +
|Gα (u)|du < 0.
C(u)e du + |α|
Q
0
0
=
Let
γ1 =: (|α| + δ∗ )Q = (|α| +
|a1 | − |α|Q
|a1 | + |α|Q
)Q =
2Q
2
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 10
and
γ2
(a1 − α)2
|a1 |2 − α2 Q + (1 − Q)|a1 |2
=: |a1 | −
=
.
4δ∗
2(|a1 | − |α|Q)
p=
γ1 + γ2
(1 − Q)|a1 |2 + 2|a1 |2 − Q(1 + Q)|α|2
√
√
=
.
2 Q
4(|a1 | − |α|Q) Q
Define
We now obtain
a1 +
and
(3.12)
q
(a1 − α)2
(1 − Q)(|a1 |2 − α2 Q)
+p Q=−
4δ∗
4(|a1 | − |α|Q)
q
(|α| + δ∗ )Q − p Q = −
Define
(1 − Q)(|a1 |2 − α2 Q)
.
4(|a1 | − |α|Q)
(1 − Q)(|a1 |2 − α2 Q)
1
min{1, √ }.
4(|a1 | − |α|Q)
Q
Combining (3.3), (3.6), and (3.7), we have
β=
Z t
Z t Z +∞
2
1
|Gα (u)|dux2 (s)ds
x(t) +
Gα (t − s)x(s)ds + p
2
0
0 t−s
V (t) =
≤ x2 (t) +
Z t
=: x2 (t) +
Z
0
0
t
|Gα (t − s)| + p
+∞
Z
t−s
|Gα (u)|du x2 (s)ds
Φ(t − s)x2 (s)ds
(3.13)
and
′
2
V (t) ≤ −βx (t) − β
Z
t
0
|Gα (t − s)|x2 (s)ds
(3.14)
where p is given in (3.12) and
Φ(t) = |Gα (t)| + p
Z
+∞
t
|Gα (u)|du.
One can show by interchanging the order of integration that
Z
0
+∞
Z
+∞
s
|Gα (u)|duds =
Z
0
+∞
u|Gα (u)|du < +∞.
(3.15)
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 11
Thus, Φ ∈ L1 (R+ ). By Theorem 2.1, the zero solution of (3.1) is uniformly asymptotically stable. This completes the proof.
When α = 0, Theorem 3.1 takes the following form.
Corollary 3.1. Suppose that
A+
∞
Z
0
∞
Z
0
∞
+∞
Z
s
0
+∞
t
and
Z
Z
s
C(u)du < 0,
C(u)du dt < 1,
C(u)du ds < +∞.
(3.16)
Then the zero solution of (3.1) is uniformly asymptotically stable.
In ([6], p.162) Burton and Mahfoud proved the following theorem. Brauer [2],
Jordan [13], and Krisztin [18] also investigated the same problem in one or higher
dimensions using the characteristic equation of (3.1).
Theorem B1 . Suppose that
A+
∞
Z
0
|
Z
Z
+∞
0
∞
t
C(u)du < 0,
(3.17)
C(u)du|dt < 1,
(3.18)
and
Z
∞
0
u|C(u)|du < +∞.
(3.19)
Then the zero solution of (3.1) is uniformly asymptotically stable.
Although (3.19) is weaker than (3.16) for C(u) with constant sign, one can show
that (3.19) implies (3.16) for α < 0. In fact, if (3.19) holds, we have
Z
0
+∞
u|Gα (u)|du ≤
Z
+∞
0
u
Z
+∞
u
|C(s)|eα(s−u) dsdu
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 12
=
Z
=
Z
+∞
0
s
Z
0
+∞
0
≤ α∗
Z
ue−αu du|C(s)|eαs ds
1 −αs
1
1
se
− 2 e−αs + 2 |C(s)|eαs ds
|α|
α
α
+∞
(s + 1)|C(s)|ds < +∞
0
1
where α∗ = max{ |α|
, α12 }.
Corollary 3.2. Suppose (3.19) holds and there exists a constant α < 0 such that
Z
+∞
0
|Gα (u)|du < 1
(3.20)
and
A+
Z
+∞
0
αu
C(u)e du + |α|
Z
+∞
0
|Gα (u)|du < 0,
(3.21)
then the zero solution of (3.1) is uniformly asymptotically stable.
It is known (see [10],[21]) that the zero solution of (3.1) is uniformly asymptotically
stable if and only if
|z − A − C ∗ (z)| =
6 0
for Re(z) ≥ 0, whereR z is a complex number and C ∗ is the Laplace transform of C
defined by C ∗ (z) = 0+∞ e−zt C(t)dt. Thus, condition (3.17) is necessary for uniform
asymptotic stability. The following example shows that (3.18) does not hold, but
(3.20) and (3.21) are satisfied for an appropriate α. Consider
′
x (t) = 0.06x(t) −
Z
0
Let A = 0.06 and C(t) = −e−0.9t . Then
Z
0
+∞
|
Z
t
+∞
C(u)du|dt =
t
Z
e−0.9(t−s) x(s)ds.
+∞
0
(3.22)
1
1 −0.9t
e
dt =
> 1.
0.9
(0.9)2
Thus, (3.18) fails. Letting α = −0.8, we have
1 −0.9t
e
,
1.7
t
Z
Z +∞
1
1 +∞ −0.9t
< 1,
e
dt =
|Gα (u)|du =
1.7 0
1.53
0
Gα (t) =
Z
+∞
C(u)eαu due−αt = −
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 13
and
A+
Z
+∞
0
= 0.06 −
Z
C(u)eαu du + |α|
+∞
0
e−1.7u du +
Z
+∞
|Gα (u)|du
0
0.8
1.53
< −0.005.
It is obvious that 0+∞ u|Gα(u)|du < +∞. By Theorem 3.1, the zero solution of (3.22)
is uniformly asymptotically stable.
R
Generalizations of Theorem B1 to systems and noncovolution type of equations
can be found in Burton and Mahfoud [6] and Zhang [24]. The author has not yet
obtained a satisfactory extension of Theorem 3.1 to systems and will pursue further
investigation. Finally we present a theorem on asymptotic stability.
Theorem 3.2. Suppose there exists a constant α ≤ 0 such that (3.20) and (3.21)
hold. Then the zero solution of (3.1) is asymptotically stable.
Proof. It is clear that the zero solution of (3.1) is stable by (3.13) and (3.14).
Let x(t) = x(t, t0 , φ) be a solution of (3.1). We will show that x(t) → 0 as t → +∞.
Indeed, by (3.14) and Cauchy’s inequality, we have
V ′ (t) ≤ −
β
x(t) +
2
Z
t
0
Gα (t − s)x(s)ds .
2
(3.23)
2
∈ L1 (R+ ).
(3.24)
This implies that
γ(t) =: x(t) +
Z
0
t
Gα (t − s)x(s)ds
By (3.4), we have
|γ ′ (t)| = 2 a1 x2 (t) + (a1 −α)x(t)
Z
0
t
Gα (t − s)x(s)ds − α
≤ (2|a1 | + |a1 − α|)x2 (t) + (2|α| + |a1 − α|)
Z
t
0
Z
0
t
2
Gα (t − s)x(s)ds
|Gα (t − s)|x2 (s)ds. (3.25)
Combining (3.24) Rand (3.25), we get |γ ′ (t)| ∈ L1 (R+ ). Therefore, limt→+∞ γ(t) = 0.
Let y(t) = x(t) + 0t Gα (t − s)x(s)ds. By Lemmas 2.1 and 2.2, we have
|x(t)| = |y(t) −
Z
0
t
Z(t − s)y(s)ds|
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 14
≤ |y(t)| + |
t
Z
Z(t − s)y(s)ds| → 0
0
since Z ∈ L1 (R+ ) and y(t) → 0 as t → +∞. This completes the proof.
Corollary 3.3. If
A+
Z
+∞
C(u)du < 0
0
and
∞
Z
0
|
∞
Z
t
C(u)du|dt < 1,
then the zero solution of (3.1) is asymptotically stable.
Corollary 3.4. If
A+
+∞
Z
0
|C(u)|du < 0,
(3.26)
then the zero solution of (3.1) is asymptotically stable.
Proof. If (3.26) holds, we can choose α sufficiently large so that conditions of
Theorem 3.2 will be satisfied. In fact, for α < 0 we have
|α|
Z
+∞
0
|Gα (u)|du ≤ −α
=
Z
= −
This yields
Z
for α < 0. If
R +∞
0
+∞
0
Z
+∞
0
+∞
t
Z
A+
+∞
≤ A+
Z
+∞
≤ A+
Z
+∞
0
0
0
+∞
t
|C(u)|eαu due−αt dt
+∞
|C(u)|eαu du e−αt
+∞
0
|C(u)|eαu du +
|Gα (u)|du ≤
|C(u)|du < |α|, then
Z
Z
C(u)eαu du + |α|
C(u)eαu du −
Z
0
0
Z
+∞
0
|C(u)|du
|C(u)|du.
1 Z +∞
|C(u)|du
|α| 0
R +∞
0
Z
0
+∞
+
|Gα (u)|du < 1. By (3.26), we also have
Z
0
+∞
+∞
|Gα (u)|du
|C(u)|eαu du +
Z
0
+∞
|C(u)|du
|C(u)|du < 0.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 15
By Theorem 3.2, the zero solution of (3.1) is asymptotically stable.
Burton and Mahfoud ([6], p.146) showed that (3.26) yields uniform asymptotic
stability by proving that x(t) ∈ L1 (R+ , R).
References
[1] L. C. Becker, T. A. Burton, and S. Zhang, Functional differential equations and
Jenson’s inequality, J. Math. Anal. Appl. 138(1989), 137-156.
[2] F. Brauer, Asymptotic stability of a class of integrodifferential equations, J.
Differential Equations 28(1978), 180-188.
[3] T. A. Burton, Volterra Integral and Differential Equations, Academic Press, New
York, 1983.
[4] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, 1985.
[5] T. A. Burton and L. Hatvani, Stability theorems for nonautonomous functional
differential equations by Liapunov functionals, Tohoku Math. J. 41(1989), 65104.
[6] T. A. Burton and W. E. Mahfoud, Stability criteria for Volterra equations, Trans.
Amer. Math. Soc. 279(1983), 143-174.
[7] T. A. Burton and S. Zhang, Unified boundedness, periodicity, and stability in
ordinary and functional differential equations, Ann. Mat. Pura Appl. 145(1986),
129-158.
[8] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New
York, 1977.
[9] G. Gripenberg, S. O. Londen, and O. Staffans, Volterra Integral and Functional
Equations, Cambridge University Press, New York, 1990.
[10] S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations, 19(1975), 142-166.
[11] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New
York, 1977.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 16
[12] Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with
Infinite Delay, Lecture Notes in Mathematics 1473, Springer-Verlag, New York,
1991.
[13] G. G. Jordan, Asymptotic stability of a class of integrodifferential equations, J.
Differential Equations 31(1979), 359-365.
[14] J. Kato, Liapunov functional vs Liapunov function, Proc. Internat. Symposium
on Functional Differential Equations, World Scientific, Singapore, 1991.
[15] L. B. Knyazhishche and V. A. Shcheglov, On the sign definiteness of Liapunov
functionals and stability of a linear delay equation, Electronic Journal of Qualitative Theory of Differential Equations, 8(1998), 1-13.
[16] V. B. Kolmanosvskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986.
[17] N. N. Krasovskii, Stability of Motion, Stanford University Press, Stanford, CA,
1963.
[18] T. Krisztin, Uniform asymptotic stability of a class of integrodifferential systems,
Journal of Integral Equations and Applications 4(1988), 581-597.
[19] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
[20] V. Lakshmikantham, L. Wen, and B. Zhang, Theory of Functional Differential
Equations with Unbounded Delay,
[21] R. K. Miller, Asymptotic stability properties of linear Volterra integrodifferential
equations, J. Differential Equations 10(1971), 485-506.
[22] I. P. Natanson, Theory of Functions of a Real Variable, Vol. II, Ungar, New
York, 1960.
[23] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc.
Japan, Tokyo, 1966.
[24] B. Zhang, Necessary and sufficient conditions for stability and convergence of
solutions of a Volterra integrodifferential equation, Dynamic Systems and Applications 9(1996), 45-58.
EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 17