The goal of the paper is to present some characterizations for the uniform exponential stability of linear skew-product semiflows on real Hilbert space

Hai, Pham Viet

Thanh, Le Ngoc

Okayama University Scientific Achievement Repository

Math. J. Okayama Univ. 53 (2011), 173–183THE UNIFORM EXPONENTIAL STABILITY OF LINEARSKEW-PRODUCT SEMIFLOWS ON REAL HILBERTSPACEPham Viet Hai and Le Ngoc ThanhAbstract. The goal of the paper is to present some characterizationsfor the uniform exponential stability of linear skew-product semiflowson real Hilbert space.1. Introduction:In recent years, the classical ideas of exponential stability and other as-ymptotic properties concerning evolution equations in infinite dimensionalBanach space have witnessed significant development. The techniques usedin studying became very various and complex: input-output of characteri-zation 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) onfunction spaces has been found.This paper considers the concepts of LSPS and give conditions aboutthe 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 onX, R+ := [0,∞), N := {0, 1, 2, ...}.Definition 1.1. Continuous mapping σ : ×R+ →  is called a semiflowon  if σ(θ, 0) = θ and σ(θ, t+ s) = σ(σ(θ, s), t), for all (θ, s, t) ∈ × R2+.Definition 1.2. pi = (Φ, σ) is called a linear skew-product semiflow onE = 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) limt→0+Φ(θ, t)x = x, uniformly in θ.Remark (See 11). If pi = (Φ, σ) is a linear skew-product semiflow then thereare M,ω such that:‖Φ(θ, t)x‖ ≤ Meωt ‖x‖ ,Mathematics Subject Classification. 34D05, 34E05.Key words and phrases. stability, linear skew-product semiflow.173174 P.V. HAI AND L.N. THANHfor all (θ, t, x) ∈ × R+ ×X.The mapping Φ given Definition 1.2 is called the cocycle associated to thelinear skew-product semiflow pi.Example 1.1. It is easy to prove that C0-semigroups, evolution families areparticular 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 stronglycontinuous mapping:F :  → L(X),there is a linear skew-product semiflow piF = (ΦF , σ) on E = X ×  suchthat:ΦF (θ, t)x = T (t)x +t∫0T (t− s)F (σ(θ, s))ΦF (θ, s)x ds,The linear skew-product semiflow piF = (ΦF , σ) is called the linear skew-product 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 Cauchyproblem: {u′(t) = A(σ(θ, t))u(t),u(0) = x,then the pair pi = (σ,Φ) is a linear skew-product semiflow.The well-known theorem of Lyapunov states that if A is an n×n complexmatrix then A has all its characteristic roots with real parts negative if andonly if for any positive definite Hermitian H there exists an unique definiteHermitian matrix B satisfyingA∗B +BA = −H, (L).Following Lyapunov’s idea, the paper extends in a natural way to linearskew-product semiflows. Indeed, from the equation (L), we have:< A(σ(θ, t))x,Wx > + < Wx,A(σ(θ, t))x >= −‖x‖2 , (L∗)Assume that (L∗) holds for some conditions, let f be the function definedbyf(t) =< WΦ(θ, t)x,Φ(θ, t)x > .LINEAR SKEW-PRODUCT SEMIFLOWS 175One can easily see that f ′(t) = −‖Φ(θ, t)x‖2. Integrating with respect to τon [s, t], we have< WΦ(θ, t)x,Φ(θ, t)x > − < WΦ(θ, s)x,Φ(θ, s)x >= −t∫s‖Φ(θ, τ)x‖2 dτ,which impliesΦ∗(θ, t)WΦ(θ, t)x+t∫sΦ∗(θ, τ)Φ(θ, τ)x dτ = Φ∗(θ, s)WΦ(θ, s)x.In next section, we establish the uniform exponential stability of linear skew-product semiflows and some equation.Definition 1.3. A linear skew-product semiflow pi = (Φ, σ) is said to beuniformly exponentially stable if and only if there exist K, ν > 0 such that:‖Φ(θ, t)x‖ ≤ Ke−νt ‖x‖ ,for all (θ, t) ∈ × R+2. Main Results:2.1. Discrete characterizations for the uniform exponential stabil-ity 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 propertysupθ∈ ‖H(θ, 0)‖ < ∞.Definition 2.2. A map H :  × N → L(X) is called uniformly positive ifthere exists the constant a > 0 such that< H(θ,m)x, x > ≥ a ‖x‖2 ,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:supθ∈ ‖Φ(θ, t0)‖ ≤ c,176 P.V. HAI AND L.N. THANHthen linear skew-product semiflow pi = (Φ, σ) is uniformly exponentially sta-ble.Proof: For every t ∈ R+, there are k ∈ N and s ∈ [0, t0) such that t =kt0 + s. An easy computation shows thatΦ(θ, kt0) =k∏i=1Φ(σ(θ, (i− 1)t0), t0),Φ(θ, kt0 + s) = Φ(σ(θ, s), kt0)Φ(θ, s)=k∏i=1Φ(σ(σ(θ, s), (i− 1)t0), t0)Φ(θ, s).Hence it follows that pi is uniformly exponentially stable with ν = −|lnc|t0and K = eωt0−lncM from‖Φ(θ, kt0 + s)x‖ ≤ ckMeωs ‖x‖< eklncMeωt0 ‖x‖< e( tt0−1)lncMeωt0 ‖x‖ .Lemma is proved. Lemma 2.2. If there exists C > 0 such that∞∑k=0‖Φ(θ, k)x‖2 ≤ C ‖x‖2 < ∞,for all θ ∈ , then there is n0 such that‖Φ(θ, n0)‖ ≤ 12,for all θ ∈ .Proof: By hypothesis, we get that‖Φ(θ, k)‖2 ≤ C,for every θ ∈  and k ∈ N. From this inequality, we obtain(n+ 1) ‖Φ(θ, n)x‖2 =n∑k=0‖Φ(θ, n)x‖2=n∑k=0‖Φ(σ(θ, k), n− k)Φ(θ, k)x‖2LINEAR SKEW-PRODUCT SEMIFLOWS 177≤ Cn∑k=0‖Φ(θ, k)x‖2≤ C∞∑k=0‖Φ(θ, k)x‖2≤ C2 ‖x‖2 .Hence, it follows‖Φ(θ, n)‖ ≤ C√n+ 1.On the other handlimn→+∞C√n+ 1= 0.Then there exists n0 ∈ N such that‖Φ(θ, n0)‖ ≤ 12,for all θ ∈ . Lemma is proved. Theorem 2.3. pi = (Φ, σ) is uniformly exponentially stable if and only ifthere exist H ∈M∗ and W ∈M such that0 = W (θ, 0)x− Φ∗(θ, n)W (θ, n)Φ(θ, n)x−n−1∑k=0Φ∗(θ, k)H(θ, n)Φ(θ, k)x, (L)for every n ≥ 1, x, θ.Proof:Sufficiency:Definition 1.3 guarantees that there are K, ν > 0 such that‖Φ(θ, t)‖ ≤ Ke−νt.Let H,W : × N → L(X) be respectively given byH(θ,m) = I,W (θ,m) =∞∑k=0Φ∗(σ(θ,m), k)Φ(σ(θ,m), k).Here, it follows that W (θ,m) is well defined from the uniform convergencewith respect to θ ∈  and m ∈ N:‖W (θ, 0)‖ ≤∞∑k=0‖Φ(θ, k)‖2178 P.V. HAI AND L.N. THANH≤∞∑k=0K2e−2νk=K21− e−2ν .It is easy to check that< H(θ,m)x, x > ≥ ‖x‖2 .This says thatsupθ∈ ‖W (θ, 0)‖ < ∞.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 conditionW ∈M, we can putK := supθ∈ ‖W (θ, 0)‖ .Now, let a be the constant defined in Definition 2.2, this means:< H(θ,m)x, x > ≥ a ‖x‖2 ,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−1∑k=0‖Φ(θ, k)x‖2≤n−1∑k=01a< H(θ, n)Φ(θ, k)x,Φ(θ, k)x >=1a<n−1∑k=0Φ∗(θ, k)H(θ, n)Φ(θ, k)x, x >=1a< W (θ, 0)x, x >− 1a< Φ∗(θ, n)W (θ, n)Φ(θ, n)x, x >≤ 1a< W (θ, 0)x, x > ≤ Ka‖x‖2 .LINEAR SKEW-PRODUCT SEMIFLOWS 179Hence, it folows∞∑k=0‖Φ(θ, k)x‖2 ≤ Ka‖x‖2 < ∞.Applying Lemma 2.1, 2.2, we get that pi is uniformly exponentially stable.2.2. Continuous characterizations for the uniform exponential sta-bility 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 thepropertysup(θ, t)∈ ×R+‖H(θ, t)‖ < ∞.Definition 2.4. A map H : ×R+ → L(X) is called uniformly positive ifthere exists the constant a > 0 such that< H(θ, t)x, x > ≥ a ‖x‖2 ,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. pi is uniformly exponentially stable if and only if there is Lsuch that:∞∫t‖Φ(θ, τ)x‖2 dτ ≤ L ‖Φ(θ, t)x‖2 ,for all θ ∈  , x ∈ X, t ∈ R+.Proof: The necessity is obvious. Now, we prove the sufficiency. Putψ(t) = M2eω2t ;1L0=1∫01ψ(τ)dτ.Step 1. We prove there is L1 such that‖Φ(θ, t)x‖ ≤ L1 ‖Φ(θ, s)x‖ ,for all t ≥ s ≥ 0, x ∈ X and θ ∈ . Indeed, we consider two possibilities.If t ∈ [s; s+ 1], it follows easily‖Φ(θ, t)x‖ ≤ Meω(t−s) ‖Φ(θ, s)x‖180 P.V. HAI AND L.N. THANH≤ Meω ‖Φ(θ, s)x‖ .If t ≥ s+ 1‖Φ(θ, t)x‖2L0=1∫0‖Φ(θ, t)x‖2ψ(τ)dτ=t∫t−1‖Φ(θ, t)x‖2ψ(t− τ) dτ≤t∫s‖Φ(θ, t)x‖2ψ(t− τ) dτ≤t∫s‖Φ(θ, τ)x‖2 dτ≤∞∫s‖Φ(θ, τ)x‖2 dτ≤ L. ‖Φ(θ, s)x‖2 ,‖Φ(θ, t)x‖ ≤√L0L ‖Φ(θ, s)x‖ .Hence, Step 1 is proved by choosingL1 := max{Meω ;√L0L }.Step 2 We prove that Lemma 2.1 works. Indeed, notice that(t+ 1) ‖Φ(θ, t)x‖2 =t∫0‖Φ(θ, t)x‖2 dτ + ‖Φ(θ, t)x‖2≤ L21t∫0‖Φ(θ, τ)x‖2 dτ + L21 ‖x‖2≤ L21∞∫0‖Φ(θ, τ)x‖2 dτ + L21 ‖x‖2≤ L21L ‖x‖2 + L21. ‖x‖2= (L21L + L21) ‖x‖ ,LINEAR SKEW-PRODUCT SEMIFLOWS 181for all t ≥ 0, x ∈ X. Hence, it follows‖Φ(θ, t)x‖ ≤ L1√L+ 1√t+ 1‖x‖ .Since limt→∞L1√L+ 1√t+ 1=∞, we get that there are t0, c in Lemma 2.1. Lemmais proved. Theorem 2.5. pi = (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)x−t∫sΦ∗(θ, τ)H(θ, τ)Φ(θ, τ)x dτ, (L)for all θ ∈  and t ≥ s ≥ 0.Proof:Necessity.From Definition 1.3, we get that there are K, ν > 0 satisfying‖Φ(θ, t)‖ ≤ Ke−νt,for all (θ, t) ∈ × R+. Let H,W : × N → L(X) given byH(θ, t) = I,W (θ, t) =∞∫tΦ∗(σ(θ, t), τ − t)Φ(σ(θ, t), τ − t) dτ.It is easy to check that H,W satisfying the equation L. On the other hand,from the equality < H(θ, t)x, x > = ‖x‖2, we get the uniform positivity ofH. And the uniform boundedness of W (., .) is gotten from the inequalities‖W (θ, t)‖ ≤∞∫t‖Φ(σ(θ, t), τ − t)‖2 dτ≤∞∫tK2e−2ν(τ−t) dτ< ∞.Sufficiency.Assume that H ∈ M∗ and W ∈ M satisfying the equation L. Now, defineK := sup(θ, t)∈ ×R+‖W (θ, t)‖ .182 P.V. HAI AND L.N. THANHLet a be the constant defined in Definition 2.4, this means:< H(θ, t)x, x > ≥ a ‖x‖2 ,for all t ∈ R+, x ∈ X, θ ∈ . Using the uniform positivity of H, theequation (L) and the uniform boundedness of W , we get the inequalitiesat∫s‖Φ(θ, τ)x‖2 dτ≤ <t∫sΦ∗(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 >≤ K ‖Φ(θ, s)x‖2 .Thus∞∫s‖Φ(θ, τ)x‖2 dτ ≤ Ka|| Φ(θ, s)x ||2 .Applying Lemma 2.4, we get that pi 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 CollegeOf Science, Viet Nam National University, Ha Noi.References[1] N.V. Minh, F. Rabiger and R. Schnaubelt,(1998) Exponential stability, Exponentialexpanssiveness and exponential dichtomy of evolution equations on the half line ,Inte-gral Equations Operator Theory (332-353).[2] B. Aulbach, N.V. Minh, P.P. 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Differential Equations, 102 (429-477).Pham Viet HaiFaculty 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.comLe Ngoc ThanhBasic 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 )

THE UNIFORM EXPONENTIAL STABILITY OF LINEAR SKEW-PRODUCT SEMIFLOWS ON REAL HILBERT SPACE

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