Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 613038, 14 pages
doi:10.1155/2012/613038
Research Article
A Generalized Nonuniform Contraction and
Lyapunov Function
Fang-fang Liao,1 Yongxin Jiang,2 and Zhiting Xie2
1
2
Nanjing College of Information Technology, Nanjing 210046, China
Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
Correspondence should be addressed to Fang-fang Liao, liaofangfang8178@sina.com
Received 19 November 2012; Accepted 1 December 2012
Academic Editor: Juntao Sun
Copyright q 2012 Fang-fang Liao et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
For nonautonomous linear equations x Atx, we give a complete characterization of general
nonuniform contractions in terms of Lyapunov functions. We consider the general case of
nonuniform contractions, which corresponds to the existence of what we call nonuniform D, μcontractions. As an application, we establish the robustness of the nonuniform contraction under
sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform
contraction persists under sufficiently small nonlinear perturbations.
1. Introduction
We consider nonautonomous linear equations
x Atx,
1.1
where A : R0 → BX is a continuous function with values in the space of bounded linear
operators in a Banach space X. Our main aim is to characterize the existence of a general
nonuniform contraction for 1.1 in terms of Lyapunov functions.
We assume that each solution of 1.1 is global, and we denote the corresponding
evolution operator by T t, s, which is the linear operator such that
T t, sxs xt,
t, s ∈ R0 ,
1.2
2
Abstract and Applied Analysis
for any solution xt of 1.1. Clearly, T t, t Id and
T t, τT τ, s T t, s,
t, τ, s ∈ R0 .
1.3
We shall say that an increasing function μ : R0 → 1, ∞ is a growth rate if
μ0 1,
lim μt ∞.
t → ∞
1.4
Given two growth rates μ, ν, we say that 1.1 admits a nonuniform μ, ν-contraction if there
exist constants K, α > 0 and ε ≥ 0 such that
T t, s ≤ K
μt
μs
−α
νε s,
t ≥ s ≥ 0.
1.5
We emphasize that the notion of nonuniform μ, ν-contraction often occurs under reasonably
weak assumptions. We refer the reader to 1 for details.
In this work, we mainly consider more general nonuniform contractions see 2.1
below and we give a complete characterization of such contractions in terms of Lyapunov
functions, especially in terms of quadratic Lyapunov functions, which are Lyapunov
functions defined in terms of quadratic forms. The importance of Lyapunov functions
is well established, particularly in the study of the stability of trajectories both under
linear and nonlinear perturbations. This study goes back to the seminal work of Lyapunov
in his 1892 thesis 2. For more results, we refer the reader to 3–6 for the classical
exponential contractions and dichotomies, 7–9 for the nonuniform exponential contractions
and nonuniform exponential dichotomies.
The proof of this paper follows from the ideas in 9, 10. As an application, we provide
a very direct proof of the robustness of the nonuniform contraction, that is, of the persistence
of the nonuniform contraction in the equation
x At Btx
1.6
for any sufficiently small linear perturbation Bt. We remark that the so-called robustness
problem also has a long history. In particular, the problem was discussed by Massera and
Schäffer 11, Perron 12, Coppel 3 and in the case of Banach spaces by Daletskiı̆ and Kreı̆n
13. For more recent work we refer to 14–16 and the references therein.
Furthermore, for a large class of nonlinear perturbations ft, x with ft, 0 0 for
every t, we show that if 1.1 admits a nonuniform contraction, then the zero solution of the
equation
x Atx ft, x
1.7
is stable. The proof uses the corresponding characterization between the nonuniform contractions and quadratic Lyapunov functions.
Abstract and Applied Analysis
3
2. Lyapunov Functions and Nonuniform Contractions
Given a growth rate μ and a function D : R0 → 0, ∞, we say that 1.1 admits a nonuniform
D, μ-contraction if there exists a constant α > 0 such that
T t, s ≤ Ds
μt
μs
−α
,
t ≥ s ≥ 0.
2.1
The nonuniform μ, ν-contraction is a special case of nonuniform D, μ-contraction with
Ds Kνε s.
Now we introduce the notion of Lyapunov functions. We say that a continuous
function V : 0, ∞ × X → R−0 is a strict Lyapunov function to 1.1 if
1 for every t > 0 and x ∈ X,
x ≤ |V t, x| ≤ Dtx,
2.2
2 for every t ≥ s > 0 and x ∈ X,
V s, x ≤ V t, T t, sx,
2.3
3 there exists a constant γ > 0 such that for every t ≥ s > 0 and x ∈ X,
|V t, T t, sx| ≤
μt
μs
−γ
|V s, x|.
2.4
The following result gives an optimal characterization of nonuniform D, μcontractions in terms of strict Lyapunov functions.
Theorem 2.1. 1.1 admits a nonuniform D, μ-contraction if and only if there exists a strict
Lyapunov function for 1.1.
Proof. We assume that there exists a strict Lyapunov function for 1.1. By 1 and 3, for
every t ≥ s > 0 and x ∈ X, we have
T t, sx ≤ |V t, T t, sx|
μt −γ
≤
|V s, x|
μs
μt −γ
≤ Ds
x.
μs
Therefore, 1.1 admits a nonuniform D, μ-contraction with α γ.
2.5
4
Abstract and Applied Analysis
Next we assume that 1.1 admits a nonuniform D, μ-contraction. For t > 0 and x ∈
X, we set
μτ α
:τ ≥t .
V t, x − sup T τ, tx
μt
2.6
By 2.1, we have |V t, x| ≤ Dtx. Moreover, setting τ t, we obtain |V t, x| ≥ x. This
establishes 1. Furthermore, for t ≥ s, we have
μτ α
:τ ≥t
|V t, T t, sx| sup T τ, tT t, sx
μt
α
μs
μτ α
sup T τ, sx
:τ ≥t
μt
μs
μs α
μτ α
≤
sup T τ, sx
:τ ≥s
μt
μs
−α
μt
|V s, x|.
μs
2.7
Therefore, V is a strict Lyapunov function for 1.1.
Next we consider another class of Lyapunov functions, namely, those defined in terms
of quadratic forms.
Let St ∈ BX be a symmetric positive-definite operator for each t > 0. A quadratic
Lyapunov function V is given as
Ht, x Stx, x
,
V t, x − Ht, x.
2.8
Given linear operators M, N, we write M ≤ N if Mx, x
≤ Nx, x
for x ∈ X.
Theorem 2.2. Assume that there exist constants c > 0 and d ≥ 1 such that
T t, s ≤ c
whenever μt ≤ dμs, t ≥ s > 0.
2.9
Then 1.1 admits a nonuniform D, μ-contraction (up to a multiplicative constant) if and only if
there exist symmetric positive definite operators St and constants C, K > 0 such that St is of class
C1 in t > 0 and
2.10
St ≤ CDt2 ,
S t A∗ tSt StAt ≤ −Id KSt
μ t
.
μt
2.11
Abstract and Applied Analysis
5
Proof. We first assume that 1.1 admits a nonuniform D, μ-contraction. Consider the linear
operators
St ∞
t
μτ
T τ, t T τ, t
μt
∗
2α−ρ
μ τ
dτ,
μτ
2.12
for some constant ρ ∈ 0, α. Clearly, St is symmetric for each t > 0. Moreover, by 2.8, we
have
Ht, x ∞
T τ, tx2
t
≤ Dt2 x2
∞ t
μτ
μt
μτ
μt
2α−ρ
−2ρ
μ τ
dτ
μτ
μ τ
dτ
μτ
2.13
Dt2
x2 .
2ρ
Since St is symmetric, we obtain
St sup
x/
0
|Ht, x|
x2
≤
Dt2
2ρ
2.14
and therefore 2.10 holds. Since
∂
T τ, t −T τ, tAt,
∂t
∂
T τ, t∗ −At∗ T τ, t∗ ,
∂t
2.15
we find that St is of class C1 in t with derivative
∞
μτ 2α−ρ μ τ
μ t
−
dτ
At∗ T τ, t∗ T τ, t
μt
μt
μτ
t
∞
μτ 2α−ρ μ τ
−
dτ
T τ, t∗ T τ, tAt
μt
μτ
t
μ t ∞
μτ 2α−ρ μ τ
−2 α−ρ
dτ,
T τ, t∗ T τ, t
μt t
μt
μτ
S t −
2.16
which implies that
S t −
μ t
μ t
− At∗ St − StAt − 2 α − ρ
St.
μt
μt
2.17
6
Abstract and Applied Analysis
Therefore,
S t At∗ St StAt −
μ t Id 2 α − ρ St ,
μt
2.18
which establishes 2.11 with K 2α − ρ.
Now we assume that conditions 2.9 and 2.10-2.11 hold. Set xt T t, τxτ. By
2.10, we have
Ht, xt ≤ St · xt2 ≤ CDt2 xt2 .
2.19
Lemma 2.3. There exists a constant η > 0 such that
Ht, xt ≥ ηxt2 .
2.20
Proof of Lemma 2.3. Note that
d
Ht, xt S txt, xt StAtxt, xt
Stxt, Atxt
dt
S t StAt At∗ St xt, xt .
2.21
Hence, by condition 2.11, and the fact that K > 0 we obtain
μ t
d
Ht, xt ≤ −
xt2 .
dt
μt
2.22
Now given τ > 0, take t > τ such that μt dμτ with d as in 2.9. Then
t
d
Hv, xvdv
dv
τ
t μ v
≤−
xv2 dv
τ μv
t μ v
−
T v, τxτ2 dv
μv
τ
t μ v
1
dv.
≤ −xτ2
2
τ μv T τ, v
Ht, xt − Hτ, xτ 2.23
Abstract and Applied Analysis
7
It follows from 2.9 that
Ht, xt − Hτ, xτ ≤ −
1
xτ2
c2
t
τ
μ v
dv
μv
2.24
log d
− 2 xτ2 .
c
Since Ht, xt ≥ 0, we have
Hτ, xτ ≥ Hτ, xτ − Ht, xt ≥
log d
xτ2
c2
2.25
which yields 2.20 with η log d/c2 > 0.
Lemma 2.4. For t ≥ τ, one has
Ht, xt ≤
μt
μτ
−K
Hτ, xτ.
2.26
Proof of Lemma 2.4. By conditions 2.11 and 2.21, we have
μ t
d
Ht, xt ≤ −K
Ht, xt.
dt
μt
2.27
Therefore,
t
d
Hv, xvdv
τ dv
t μ v
Hv, xvdv.
≤ −K
τ μv
Ht, xt − Hτ, xτ 2.28
It follows from Gronwall’s lemma that
Ht, xt ≤
which yields the desired result.
μt
μτ
−K
Hτ, xτ,
2.29
8
Abstract and Applied Analysis
By Lemmas 2.3 and 2.4 together with 2.19, we obtain
T t, τxτ2 xt2
≤ η−1 Ht, xt
≤ η−1
μt
μτ
−K
≤ η−1 CDτ2
Hτ, xτ
μt
μτ
−K
2.30
xτ2 ,
and therefore,
T t, τ2 ≤ η−1 CDτ2
μt
μτ
−K
,
2.31
which implies that 1.1 admits a nonuniform D, μ-contraction.
As an application of Theorem 2.2, we establish the robustness of nonuniform D, μcontractions. Roughly speaking, a nonuniform contraction for 1.1 is said to be robust if 1.6
still admits a nonuniform contraction for any sufficiently small perturbation Bt.
Theorem 2.5. Let A, B : R0 → BX be continuous functions such that 1.1 admits a nonuniform
D, μ-contraction with condition 2.9. Suppose further that Dt ≥ 1 for every t > 0 and
Bt ≤ δD−2 t
μ t
,
μt
t>0
2.32
for some δ > 0 sufficiently small. Then 1.6 admits a nonuniform D, μ-contraction.
Proof. Let Ut, s be the evolution operator associated to 1.6. It is easy to verify that
Ut, s T t, s t
T t, τBτUτ, sdτ.
2.33
s
For every t ≥ s > 0 with μt ≤ dμs, we have
Ut, s ≤ c t
s
≤ c cδ
cδD−2 τ
t
s
μ τ
Uτ, sdτ
μτ
μ τ
Uτ, sdτ.
μτ
2.34
Abstract and Applied Analysis
9
Using Gronwall’s inequality, we obtain
Ut, s ≤ c exp cδ
t
s
μ τ
dτ
μτ
≤ c exp cδ log d
2.35
for every t ≥ s > 0 with μt ≤ dμs. Therefore condition 2.9 also holds for the perturbed
equation 1.6.
Now we consider the matrices St in 2.12. Condition 2.10 can be obtained as in the
proof of Theorem 2.2. For condition 2.11, it is sufficient to show that
StBt Bt∗ St ≤ ϑ
μ t
Id
μt
2.36
for some constant ϑ < 1. Using 2.10 and 2.32, we have
StBt Bt∗ St ≤ 2St · Bt
≤ 2CDt2 δDt−2
2Cδ
μ t
μt
2.37
μ t
,
μt
and taking δ sufficiently small, we find that 2.36 holds with some ϑ < 1.
3. Stability of Nonlinear Perturbations
Before stating the result, we fist prove an equivalent characterization of property 3. Given
matrices St ∈ BX for each t ∈ R0 , we consider the functions
Ḣt, x d
Ht h, T t h, hx |h0 ,
dh
d
V t h, T t h, hx |h0 ,
V̇ t, x dh
3.1
whenever the derivatives are well defined and H, V are given as 2.8.
Lemma 3.1. Let V, μ be C1 functions. Then property 3 is equivalent to
V̇ t, T t, τx ≥ −γV t, T t, τx
μ t
,
μt
t > τ.
3.2
10
Abstract and Applied Analysis
Proof. Now we assume that property 3 holds. If t > τ and h > 0, then
V t h, T t h, τx V t h, T t h, tT t, τx
≥
μt h
μt
−γ
V t, T t, τx,
−γ
−1
μt h/μt
V t h, T t h, τx − V t, T t, τx
lim
≥ V t, T t, τx lim
h → 0
h→0
h
h
−γV t, T t, τx
3.3
μ t
.
μt
Similarly, if h < 0 is such that t h > τ, then
V t h, T t h, τx ≤
μt h
μt
−γ
V t, T t, τx,
V t h, T t h, τx − V t, T t, τx
h→0
h
−γ
−1
μt h/μt
≥ V t, T t, τx lim−
h→0
h
lim−
−γV t, T t, τx
3.4
μ t
.
μt
This establishes 3.2.
Next we assume that 3.2 holds. We rewrite 3.2 in the form
μ t
V̇ t, T t, τx
≥ −γ
,
V t, T t, τx
μt
t > τ,
3.5
which implies that
V t, T t, τx
log
V τ, x
and hence property 3 holds.
t
V̇ v, T v, τx
dv
τ V v, T v, τx
t μ v
dv
≥ −γ
τ μv
μt −γ
log
,
μτ
3.6
Abstract and Applied Analysis
11
Theorem 3.2. Assume that 1.1 admits a nonuniform D, μ-contraction satisfying 2.9. Suppose
further that there exists a constant l > 0 such that l < α and
ft, x ≤ l μ t x,
μt
3.7
t > 0, x ∈ X.
Then for each k > −α l, there exists C > 0 such that
k
yt ≤ CDs μt ys,
μs
3.8
t≥s
for every solution yt of 1.7.
Proof. For St as in 2.12 and Ht, xt as in 2.8, we have, for every t ≥ s,
Ht, T t, sxs ∞
2
T v, sxs
t
≤
μt
μs
μt
μs
μt
μs
−2α−ρ ∞
μv
μt
2α−ρ
μ v
dv
μv
2
T v, sxs
t
−2α−ρ ∞
2
T v, sxs
s
μv
μs
μv
μs
2α−ρ
2α−ρ
μ v
dv
μv
μ v
dv
μv
3.9
−2α−ρ
Hs, xs.
Since V t, x − Ht, x, we have
V t, T t, sxs ≥
μt
μs
−α−ρ
V s, xs,
t ≥ s.
3.10
Applying Lemma 3.1, we obtain
μ t
V t, T t, sxs,
V̇ t, T t, sxs ≥ − α − ρ
μt
t ≥ s.
3.11
In particular, for t s,
μ s
V̇ s, xs ≥ − α − ρ
V s, xs.
μs
3.12
From the identity Ḣ 2V V̇ that for every s > 0 and x ∈ X, we have
μ s
Ḣs, x ≤ −2 α − ρ
Hs, x.
μs
3.13
12
Abstract and Applied Analysis
On the other hand,
Ḣs, x S s SsAs As∗ Ss x, x .
3.14
Therefore,
μ s
0 ≥ Ḣs, x 2 α − ρ
Hs, x
μs
μ s
S s SsAs As∗ Ss 2 α − ρ
Ss x, x ,
μs
3.15
μ t
S t StAt At∗ St 2 α − ρ
St ≤ 0.
μt
3.16
and hence
Therefore, if yt is a solution of 1.7, then
d H t, yt S tyt, yt StAtyt, yt Styt, Atyt
dt
Stf t, yt , yt Styt, f t, yt
S t StAt At∗ St yt, yt
St St∗ yt, f t, yt
2 μ t
≤ −2 α − ρ
St · yt St St∗ yt, f t, yt
μt
3.17
2
μ t
≤ −2 α − ρ
St · yt 2St · f t, yt · yt
μt
2
2
μ t
μ t
≤ −2 α − ρ
St · yt 2l
St · yt
μt
μt
2
μ t
−2 α − ρ − l
St · yt .
μt
If ρ is small enough such that α − ρ − l > 0, then
μ t d H t, yt ≤ −2 α − ρ − l
H t, yt ,
dt
μt
3.18
and hence
H t, yt − H s, ys ≤ −2 α − ρ − l
t
s
μ τ H τ, yτ dτ.
μτ
3.19
Abstract and Applied Analysis
13
It follows from Gronwall’s inequality that
μt −2α−ρ−l
,
H t, yt ≤ H s, ys
μs
t ≥ s.
3.20
Now given s > 0, take t > s such that μt dμs with d as in 2.9. Then
H s, ys ∞
s
2α−ρ μ τ
T τ, sys2 μτ
dτ
μs
μτ
t
2α−ρ μ τ
T τ, sys2 μτ
dτ
μs
μτ
s
2 t μτ 2α−ρ μ τ
1
≥ 2 ys
dτ
μs
μτ
c
s
2α−ρ
2
μt
1
2
−1
ys
μs
2c α − ρ
≥
3.21
2 1
ys d2α−ρ − 1 .
2c α − ρ
2
Taking
κ
1
d2α−ρ − 1 > 0,
2c α − ρ
3.22
2
H s, ys ≥ κys .
3.23
2
then
It follows from 2.13 and 3.20 that
yt ≤ κ1/2 H t, yt
μt −α−ρ−l
≤ κ1/2 H s, ys
μs
μt −α−ρ−l 1
ys.
Ds
≤ κ1/2
2ρ
μs
Now the proof is finished.
3.24
14
Abstract and Applied Analysis
Acknowledgments
The authors would like to deliver great thanks to Professor Jifeng Chu for his valuable
suggestions and comments. Y. Jiang was supported by the Fundamental Research Funds for
the Central Universities.
References
1 L. Barreira, J. Chu, and C. Valls, “Robustness of nonuniform dichotomies with different growth rates,”
São Paulo Journal of Mathematical Sciences, vol. 5, pp. 1–29, 2011.
2 A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, 1992.
3 W. A. Coppel, Dichotomies in Stability Theory, vol. 629 of Lecture Notes in Mathematics, Springer, Berlin,
Germany, 1978.
4 W. Hahn, Stability of Motion, Grundlehren der mathematischen Wissenschaften 138, Springer, 1967.
5 J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method with Applications, vol. 4 of Mathematics
in Science and Engineering, Academic Press, 1961.
6 Y. A. Mitropolsky, A. M. Samoilenko, and V. L. Kulik, Dichotomies and Stability in Nonautonomous Linear
Systems, vol. 14 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, 2003.
7 L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, vol. 1926 of Lecture Notes in
Mathematics, Springer, Berlin, Germany, 2008.
8 L. Barreira and C. Valls, “Quadratic Lyapunov functions and nonuniform exponential dichotomies,”
Journal of Differential Equations, vol. 246, no. 3, pp. 1235–1263, 2009.
9 L. Barreira, J. Chu, and C. Valls, “Lyapunov functions for general nonuniform dichotomies,” Preprint.
10 L. Barreira and C. Valls, “Lyapunov functions versus exponential contractions,” Mathematische
Zeitschrift, vol. 268, no. 1-2, pp. 187–196, 2011.
11 J. L. Massera and J. J. Schäffer, “Linear differential equations and functional analysis. I,” Annals of
Mathematics, vol. 67, pp. 517–573, 1958.
12 O. Perron, “Die Stabilitätsfrage bei Differentialgleichungen,” Mathematische Zeitschrift, vol. 32, no. 1,
pp. 703–728, 1930.
13 J. Daletskiı̆ and M. Kreı̆n, Stability of Solutions of Differential Equations in Banach Space, Translations of
Mathematical Monographs 43, American Mathematical Society, 1974.
14 L. Barreira and C. Valls, “Robustness via Lyapunov functions,” Journal of Differential Equations, vol.
246, no. 7, pp. 2891–2907, 2009.
15 S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product
semiflow in Banach spaces,” Journal of Differential Equations, vol. 120, no. 2, pp. 429–477, 1995.
16 V. A. Pliss and G. R. Sell, “Robustness of exponential dichotomies in infinite-dimensional dynamical
systems,” Journal of Dynamics and Differential Equations, vol. 11, no. 3, pp. 471–513, 1999.
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