Math. J. Okayama Univ. 53 (2011), 173–183

THE UNIFORM EXPONENTIAL STABILITY OF LINEAR
SKEW-PRODUCT SEMIFLOWS ON REAL HILBERTSPACE
Pham Viet Hai and Le Ngoc Thanh
Abstract. The goal of the paper is to present some characterizations
for the uniform exponential stability of linear skew-product semiflows
on real Hilbert space.

1. Introduction:
In recent years, the classical ideas of exponential stability and other asymptotic properties concerning evolution equations in infinite dimensional
Banach space have witnessed significant development. The techniques usedin studying became very various and complex: input-output of characterization relative to integral equations and to difference equations have beenobtained in [1], [4], [9], ... ; discrete-time methods have been developedin [6], ... ; and also the theory of linear skew-product semiflow (LSPS) on
function spaces has been found.
This paper considers the concepts of LSPS and give conditions about
the uniform exponential stability of LSPS on real Hilbert space. Let X bethe Banach space, let (⊖, d) be the metric space. In what follows, we denoteby L(X) be the Banach algebra of all bounded linear operators acting on
X, R+ := [0, ∞), N := {0, 1, 2, ...}.
Definition 1.1. Continuous mapping σ : ⊖ × R+ → ⊖ is called a semiflow
on ⊖ if σ(θ, 0) = θ and σ(θ, t + s) = σ(σ(θ, s), t), for all (θ, s, t) ∈ ⊖ × R2+ .
Definition 1.2. π = (Φ, σ) is called a linear skew-product semiflow on
E = X × ⊖ if σ is a semiflow on ⊖ and Φ : ⊖ × R+ → L(X) satisfies thefollowing conditions:(1) Φ(θ, 0) = I, the identity operator on X, for all θ ∈ ⊖.
(2) Φ(θ, t + s) = Φ(σ(θ, t), s)Φ(θ, t), for all (θ, t, s) ∈ ⊖ × R2+ .
(3) lim Φ(θ, t)x = x, uniformly in θ.
t→0+Remark (See 11). If π = (Φ, σ) is a linear skew-product semiflow then there
are M, ω such that:kΦ(θ, t)xk ≤ M eωt kxk ,Mathematics Subject Classification. 34D05, 34E05.
Key words and phrases. stability, linear skew-product semiflow.
173

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P.V. HAI AND L.N. THANH

for all (θ, t, x) ∈ ⊖ × R+ × X.
The mapping Φ given Definition 1.2 is called the cocycle associated to thelinear skew-product semiflow π.
Example 1.1. It is easy to prove that C0 -semigroups, evolution families are
particular cases of linear skew-product semiflows.
Example 1.2 ( See 7). Let σ be a semiflow on the compact Hausdorff space⊖ and {T (t)}t≥0 a C0 -semigroup on the Banach space X. For every strongly
continuous mapping:F : ⊖ → L(X),there is a linear skew-product semiflow πF = (ΦF , σ) on E = X × ⊖ suchthat:Zt
ΦF (θ, t)x = T (t)x +T (t − s)F (σ(θ, s))ΦF (θ, s)x ds,0The linear skew-product semiflow πF = (ΦF , σ) is called the linear skewproduct semiflow generated by the triplet (T, F, σ).
Classical examples of cocycles appear as operator solutions for variationalequations.
Example 1.3. Let ⊖ be a compact metric space, σ a semiflow and A : ⊖ →L(X) a continuous map. If Φ(θ, t)x is the solution of the anstract Cauchy
problem:(u′ (t) = A(σ(θ, t))u(t),u(0) = x,then the pair π = (σ, Φ) is a linear skew-product semiflow.
The well-known theorem of Lyapunov states that if A is an n × n complex
matrix then A has all its characteristic roots with real parts negative if andonly if for any positive definite Hermitian H there exists an unique definite
Hermitian matrix B satisfying
A∗ B + BA = −H,(L).

Following Lyapunov’s idea, the paper extends in a natural way to linear
skew-product semiflows. Indeed, from the equation (L), we have:< A(σ(θ, t))x, W x > + < W x, A(σ(θ, t))x >= − kxk2 , (L∗ )Assume that (L∗ ) holds for some conditions, let f be the function definedbyf (t) =< W Φ(θ, t)x, Φ(θ, t)x > .

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175

One can easily see that f ′ (t) = − kΦ(θ, t)xk2 . Integrating with respect to τon [s, t], we have
< W Φ(θ, t)x, Φ(θ, t)x > − < W Φ(θ, s)x, Φ(θ, s)x >= −ZtskΦ(θ, τ )xk2 dτ,which impliesΦ∗ (θ, t)W Φ(θ, t)x +ZtΦ∗ (θ, τ )Φ(θ, τ )x dτ = Φ∗ (θ, s)W Φ(θ, s)x.

s

In next section, we establish the uniform exponential stability of linear skewproduct semiflows and some equation.
Definition 1.3. A linear skew-product semiflow π = (Φ, σ) is said to beuniformly exponentially stable if and only if there exist K, ν > 0 such that:for all (θ, t) ∈ ⊖ × R+kΦ(θ, t)xk ≤ Ke−νt kxk ,2. Main Results:

2.1. Discrete characterizations for the uniform exponential stability of linear skew-product semiflows.
Definition 2.1. A map H : ⊖ × N → L(X) is called positive if< H(θ, m)x, x > ≥ 0,for all m ∈ N and x ∈ X; θ ∈ ⊖.

Let M be the set of all positive maps H defined in Definition 2.1 withthe property
sup kH(θ, 0)k < ∞.
θ∈ ⊖Definition 2.2. A map H : ⊖ × N → L(X) is called uniformly positive if
there exists the constant a > 0 such that
< H(θ, m)x, x > ≥ a kxk2 ,for all m ∈ N and x ∈ X; θ ∈ ⊖Let M∗ be the set of all positive maps H defined in Definition 2.2.

Lemma 2.1. If there are t0 > 0 and c ∈ (0, 1) such that:sup kΦ(θ, t0 )k ≤ c,θ∈ ⊖176

P.V. HAI AND L.N. THANH

then linear skew-product semiflow π = (Φ, σ) is uniformly exponentially stable.
Proof: For every t ∈ R+ , there are k ∈ N and s ∈ [0, t0 ) such that t =kt0 + s. An easy computation shows that
Φ(θ, kt0 ) =kY
i=1

Φ(θ, kt0 + s)==Φ(σ(θ, (i − 1)t0 ), t0 ),Φ(σ(θ, s), kt0 )Φ(θ, s)kY
i=1

Φ(σ(σ(θ, s), (i − 1)t0 ), t0 )Φ(θ, s).

Hence it follows that π is uniformly exponentially stable with ν = −and K = eωt0 −lnc M fromkΦ(θ, kt0 + s)xk

Lemma is proved.

≤ck M eωs kxk<e t0<eklnc M eωt0 kxk( t −1)lnc

M eωt0 kxk .

Lemma 2.2. If there exists C > 0 such that∞X
k=0kΦ(θ, k)xk2 ≤ C kxk2 < ∞,for all θ ∈ ⊖, then there is n0 such thatkΦ(θ, n0 )k ≤1,2for all θ ∈ ⊖.
Proof: By hypothesis, we get thatkΦ(θ, k)k2 ≤ C,for every θ ∈ ⊖ and k ∈ N. From this inequality, we obtain2(n + 1) kΦ(θ, n)xk=nX
k=0=nX
k=0kΦ(θ, n)xk2kΦ(σ(θ, k), n − k)Φ(θ, k)xk2

|lnc|
t0

LINEAR SKEW-PRODUCT SEMIFLOWS≤

Hence, it follows

CnX
k=0∞
X

177

kΦ(θ, k)xk2
kΦ(θ, k)xk2≤C≤C kxk .

k=022kΦ(θ, n)k ≤ √C.
n+1On the other handClim √= 0.
n→+∞
n+1
Then there exists n0 ∈ N such that
1kΦ(θ, n0 )k ≤ ,2for all θ ∈ ⊖. Lemma is proved.
Theorem 2.3. π = (Φ, σ) is uniformly exponentially stable if and only ifthere exist H ∈ M∗ and W ∈ M such that0 =−for every n ≥ 1, x, θ.

W (θ, 0)x − Φ∗ (θ, n)W (θ, n)Φ(θ, n)xn−1XΦ∗ (θ, k)H(θ, n)Φ(θ, k)x,(L)k=0

Proof:
Sufficiency:
Definition 1.3 guarantees that there are K, ν > 0 such thatkΦ(θ, t)k ≤ Ke−νt .

Let H, W : ⊖ × N → L(X) be respectively given byH(θ, m) = I,W (θ, m) =∞XΦ∗ (σ(θ, m), k)Φ(σ(θ, m), k).

k=0Here, it follows that W (θ, m) is well defined from the uniform convergencewith respect to θ ∈ ⊖ and m ∈ N:∞
XkΦ(θ, k)k2
kW (θ, 0)k ≤k=0

178

P.V. HAI AND L.N. THANH≤=∞X

K 2 e−2νkk=0

K2.
1 − e−2ν

It is easy to check that
< H(θ, m)x, x > ≥ kxk2 .
This says thatsup kW (θ, 0)k < ∞.

θ∈ ⊖The uniform positivity of H and the positivity of W follow directly fromtheir individual definitions. Therefore, we can conclude that H ∈ M∗ andW ∈ M.
Necessity:
Assume that H ∈ M∗ and W ∈ M satisfying (L). From the condition
W ∈ M, we can put
K := sup kW (θ, 0)k .
θ∈ ⊖Now, let a be the constant defined in Definition 2.2, this means:< H(θ, m)x, x > ≥ a kxk2 ,for all m ∈ N, x ∈ X, θ ∈ ⊖. Using the uniform positivity of H, theequation (L) and the uniform boundedness of W (., 0), we obtain:n−1X≤=k=0n−1X
k=0kΦ(θ, k)xk2
1< H(θ, n)Φ(θ, k)x, Φ(θ, k)x >
an−1

1 X ∗
Φ (θ, k)H(θ, n)Φ(θ, k)x, x >
<ak=0

=−≤

1
< W (θ, 0)x, x >a1< Φ∗ (θ, n)W (θ, n)Φ(θ, n)x, x >aK
1< W (θ, 0)x, x > ≤kxk2 .
aaLINEAR SKEW-PRODUCT SEMIFLOWS

179

Hence, it folows∞X
k=0kΦ(θ, k)xk2 ≤Kkxk2 < ∞.
aApplying Lemma 2.1, 2.2, we get that π is uniformly exponentially stable.
2.2. Continuous characterizations for the uniform exponential stability of linear skew-product semiflows.
Definition 2.3. A map H : ⊖ × R+ → L(X) is called positive if:< H(θ, t)x, x > ≥ 0,for all t ∈ R+ and x ∈ X; θ ∈ ⊖.

Let M be the set of positive maps defined in Definition 2.3 with theproperty
supkH(θ, t)k < ∞.
(θ, t)∈ ⊖×R+Definition 2.4. A map H : ⊖ × R+ → L(X) is called uniformly positive if
there exists the constant a > 0 such that
< H(θ, t)x, x > ≥ a kxk2 ,for all t ∈ R+ and x ∈ X; θ ∈ ⊖Let M∗ be the set of positve maps H defined in Definition 2.4.

Lemma 2.4. π is uniformly exponentially stable if and only if there is L
such that:
Z∞kΦ(θ, τ )xk2 dτ ≤ L kΦ(θ, t)xk2 ,tfor all θ ∈ ⊖ , x ∈ X, t ∈ R+ .
Proof: The necessity is obvious. Now, we prove the sufficiency. Put1ψ(t) = M 2 eω2t ;=L0Z11dτ.
ψ(τ )0Step 1. We prove there is L1 such thatkΦ(θ, t)xk ≤ L1 kΦ(θ, s)xk ,for all t ≥ s ≥ 0, x ∈ X and θ ∈ ⊖. Indeed, we consider two possibilities.
If t ∈ [s; s + 1], it follows easilykΦ(θ, t)xk≤

M eω(t−s) kΦ(θ, s)xk

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P.V. HAI AND L.N. THANH

M eω kΦ(θ, s)xk .

≤If t ≥ s + 1

Z1kΦ(θ, t)xk2
dτ
ψ(τ )

Zt

t−1kΦ(θ, t)xk2dτ
ψ(t − τ )≤ZtkΦ(θ, t)xk2dτ
ψ(t − τ )≤ZtkΦ(θ, τ )xk2 dτ≤Z∞kΦ(θ, τ )xk2 dτ

≤L. kΦ(θ, s)xk2 ,kΦ(θ, t)xk ≤pL0 L kΦ(θ, s)xk .

kΦ(θ, t)xk2
L0=0=sssHence, Step 1 is proved by choosing

L1 := max{M eω ;pL0 L }.

Step 2 We prove that Lemma 2.1 works. Indeed, notice that(t + 1) kΦ(θ, t)xk2=

ZtkΦ(θ, t)xk2 dτ + kΦ(θ, t)xk2

L21

Zt

kΦ(θ, τ )xk2 dτ + L21 kxk2

L21Z∞kΦ(θ, τ )xk2 dτ + L21 kxk20≤≤≤=002L1 L kxk2 + L21 . kxk2
(L21 L + L21 ) kxk ,LINEAR SKEW-PRODUCT SEMIFLOWS

181

for all t ≥ 0, x ∈ X. Hence, it follows√
L1 L + 1kΦ(θ, t)xk ≤ √kxk .
t+1√
L1 L + 1
Since lim √= ∞, we get that there are t0 , c in Lemma 2.1. Lemmat→∞t+1
is proved.
Theorem 2.5. π = (Φ, σ) is uniformly exponentially stable if and only ifthere exist H ∈ M∗ and W ∈ M such that0=−Φ∗ (θ, s)W (θ, s)Φ(θ, s)x − Φ∗ (θ, t)W (θ, t)Φ(θ, t)xZtΦ∗ (θ, τ )H(θ, τ )Φ(θ, τ )x dτ,(L)sfor all θ ∈ ⊖ and t ≥ s ≥ 0.
Proof:Necessity.
From Definition 1.3, we get that there are K, ν > 0 satisfyingkΦ(θ, t)k ≤ Ke−νt ,for all (θ, t) ∈ ⊖ × R+ . Let H, W : ⊖ × N → L(X) given byH(θ, t) = I,W (θ, t) =Z∞Φ∗ (σ(θ, t), τ − t)Φ(σ(θ, t), τ − t) dτ.

t

It is easy to check that H, W satisfying the equation L. On the other hand,from the equality < H(θ, t)x, x > = kxk2 , we get the uniform positivity ofH. And the uniform boundedness of W (., .) is gotten from the inequalities
Z∞
kW (θ, t)k ≤kΦ(σ(θ, t), τ − t)k2 dτ≤tZ∞

K 2 e−2ν(τ −t) dτt<∞.

Sufficiency.
Assume that H ∈ M∗ and W ∈ M satisfying the equation L. Now, defineK :=sup(θ, t)∈ ⊖×R+kW (θ, t)k .

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P.V. HAI AND L.N. THANH

Let a be the constant defined in Definition 2.4, this means:< H(θ, t)x, x > ≥ a kxk2 ,for all t ∈ R+ , x ∈ X, θ ∈ ⊖. Using the uniform positivity of H, theequation (L) and the uniform boundedness of W , we get the inequalities
aZtsThuskΦ(θ, τ )xk2 dτZtΦ∗ (θ, τ )H(θ, τ )Φ(θ, τ )x dτ, x >≤<=−≤< Φ∗ (θ, s)W (θ, s)Φ(θ, s)x, x >< Φ∗ (θ, t)W (θ, t)Φ(θ, t)x, x >< Φ∗ (θ, s)W (θ, s)Φ(θ, s)x, x >s≤K kΦ(θ, s)xk2 .

Z∞kΦ(θ, τ )xk2 dτ ≤sK|| Φ(θ, s)x ||2 .
aApplying Lemma 2.4, we get that π is uniformly exponentially stable.
Acknowledgement.
The authors are grateful to the referee for carefully reading and valuablesuggestions, which led to the improvement of the paper.
The first author was partially supported from the grant TN-10-08 of College
Of Science, Viet Nam National University, Ha Noi.
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Pham Viet Hai
Faculty of Mathematics, Mechanics and Informatics,
College Of Science, Viet Nam National University, Ha Noi,334, Nguyen Trai Street, Thanh Xuan, Ha Noi, Viet Nam.
e-mail address: phamviethai86@gmail.com
Le Ngoc Thanh
Basic Science, Hoa Binh University,CC2, My Dinh II Urban, Tu Liem Dist., Ha Noi, Viet Nam.
e-mail address: thanhln2008@yahoo.com(Received August 21, 2009 )(Revised July 13, 2010 )