STABILIZATION OF WAVE EQUATIONS WITH VARIABLE

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 198, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

STABILIZATION OF WAVE EQUATIONS WITH VARIABLE COEFFICIENT AND DELAY IN THE DYNAMICAL BOUNDARY FEEDBACK DANDAN GUO, ZHIFEI ZHANG Communicated by Goong Chen

Abstract. In this article we consider the boundary stabilization of a wave equation with variable coefficients. This equation has an acceleration term and a delayed velocity term on the boundary. Under suitable geometric conditions, we obtain the exponential decay for the solutions. Our proof relies on the geometric multiplier method and the Lyapunov approach.

1. Introduction Let Ω be a bounded domain in RN (N ≥ 2) with smooth boundary Γ. We assume Γ = Γ0 ∪ Γ1 with Γ0 ∩ Γ1 = ∅. We consider the following wave equation with dynamical Neumann boundary condition utt − div A(x)∇u = 0, u(x, t) = 0,

in Ω × (0, ∞),

on Γ0 × (0, ∞),

m(x)utt (x, t) + ∂νA u(x, t) = C(t),

on Γ1 × (0, ∞),

u(x, 0) = u0 (x), ut (x, 0) = u1 (x),

(1.1)

in Ω,

where div X denote the divergence of the vector field X in the Euclidean metN ric, A(x) = (aij (x)) are symmetric and positive definite Pnmatrices for all x ∈ R N and aij (x) are smooth functions on R . ∂νA u = i,j=1 aij (x)∂xj uνi , where T ν = (ν1 , ν2 , . . . , νn ) denotes the outward unit normal vector of the boundary and νA = Aν. C(t) is the boundary feedback control. We call the equation (1.1) is with dynamical boundary conditions if m(x) 6= 0, which means the system has an acceleration term on part of the boundary. This is what happened in some physical applications when one has to take the acceleration terms into account on the boundary. Actually we need the models with dynamical boundary conditions, see [8, 10, 17] and many others. They are not only important theoretically but also have strong backgrounds for physical applications. There are numerous of these applications in the bio-medical domain [5, 20] as well as in 2010 Mathematics Subject Classification. 34H05, 35L05, 49K25, 93C20, 93D15. Key words and phrases. Exponential decay; wave equation with variable coefficient; dynamical boundary control; time delay. c

2017 Texas State University. Submitted June 13, 2017. Published August 9, 2017. 1

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DANDAN GUO, ZHIFEI ZHANG

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applications related to noise suppression and control of elastic structures [2, 3, 4, 21]. For the above reasons in this article we assume m(x) ∈ L∞ (Γ1 )

and m(x) > α > 0,

(1.2)

where α is a positive constant. Here we denote 1 (x) a nonnegative function as m(x) − 1 > 0, x ∈ Γ1 . (1.3) α On the other hand, time delay effects arise in many practical problems in science and engineering. Most phenomena depends on not only the present state but also the history of the system in a very complicated way. For instance the practical systems often suffer from the actuator saturation and sometimes the control input delay. It is well known that delay effects might turn a well-behaved system into a wild one by inducing some instabilities, see [9, 11, 18]. Recently boundary feedbacks are designed to overcome the negative effect of time delays and stabilize the system; see [1, 15, 16, 19, 22] and the references therein. In this article, we discuss the stabilization of wave equations subject to dynamical boundary conditions with acceleration terms(i.e. utt ) and time delayed velocity terms. We shall design a collocated boundary feedback with time delay effect in the velocity input to obtain the exponential stabilization of the system. That is, C(t) is a feedback law with input time delay 1 (x) =

C(t) = −αut (x, t) − βut (x, t − τ ) − ∂νA ut ,

(1.4)

where τ is the time delay, α is the positive constant given in (1.2) and the constant β > 0 denotes the effect of time delay in the velocity input. β = 0 means the absence of input delay. For the discussion of stabilization results in the case of absence of delay, see [6, 7, 8, 12, 14, 23] and our recent paper [28]. Here in this article we assume that, for x ∈ Γ1 , we have s 1 (x) β , (1.5) 0< < α 1 + 1 (x) 1 (x) ∈ R+ \ [

β2 β2 , ], 2α(α + β) 2α(α − β)

(1.6)

where 1 (x) is given in (1.3). If t < τ , then ut (x, t − τ ) is determined by the datum in the past and we need an initial value in the past. We thus give the initial condition ut (x, t − τ ) = f0 (x, t − τ ), (x, t) ∈ Γ1 × (0, τ ). (1.7) In equation (1.1) we adopt the feedback law given in (1.4) and the initial datum given in (1.7) to obtain the following closed loop system: utt − div A(x)∇u = 0, u(x, t) = 0,

in Ω × (0, ∞),

on Γ0 × (0, ∞),

m(x)utt (x, t) + ∂νA u(x, t) = −αut (x, t) − βut (x, t − τ ) − ∂νA ut , u(x, 0) = u0 (x), ut (x, 0) = u1 (x), ut (x, t − τ ) = f0 (x, t − τ ),

on Γ1 × (0, ∞),

in Ω,

(x, t) ∈ Γ1 × (0, τ ). (1.8)

We define g = A−1 (x),

x∈Ω

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STABILIZATION OF WAVE EQUATIONS

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as a Riemannian metric on Ω and consider the couple (Ω, g) as a Riemannian manifold. Let D denote the Levi-Civita connection of the metric g. For each x ∈ Ω, the metric g induces an inner product on Rxn by hX, Y ig = hA−1 (x)X, Y i,

|X|2g = hX, Xig ,

X, Y ∈ Rxn ,

where h·, ·i denotes the standard metric of the Euclidean space Rn . To obtain the stabilization of problem (1.8), the following geometric hypotheses are assumed: There exists a vector field H on Riemannian manifold (Ω, g) such that the following properties hold: (A1) DH(·, ·) is strictly positive definite on Ω: there exists a constant κ > 0 such that for all x ∈ Ω, for all X ∈ Mx (the tangent space at x): DH(X, X) ≡ hDX H, Xig ≥ κ|X|2 .

(1.9)

(A2) H ·ν ≤0

on Γ0 .

(1.10)

Remark 1.1. For any Riemannian manifold M , the existence of such a vector field H in (A1) has been proved in [24], where some examples are given. See also [26]. For the Euclidean metric, taking the vector field H = x − x0 and we have DH(X, X) = |X|2 , which means assumption (A1) always holds with κ = 1 for the Euclidean case. Before we go to the stabilization of the system, we should first define an energy connected with the natural energy of the hybrid system (1.1). We set η(x, t) = m(x)ut (x, t) + ∂νA u, x ∈ Γ1 .

(1.11)

Let u be a regular solution of system (1.8). Then we associate with system (1.8) the energy functional Z Z Z 1Z
1 E(t) = u2t + |∇g u|2g dx + η 2 dΓ +ξ u2t (x, t − ρτ )dΓdρ , m(x) − α Ω Γ1 0 Γ1 (1.12) for some constant ξ > 0 which will be determined later. The main result of this paper is the following: Theorem 1.2. Let the geometric assumptions (A1) and (A2) hold. Then there exist constants C > 0 and ω > 0 such that E(t) ≤ Ce−ωt E(0),

t ≥ 0.

(1.13)

This article is organized as follows. In the next section, we discuss the wellposedness of the nonlinear close-loop system by semigroup theory. Section 3 devotes to the proof of the exponential stability. We construct an appropriate Lyapunov functional to obtain the main result. 2. Well-posedness of the closed loop system In this section, we shall study well-posedness results for system (1.8) using semigroup theory. Let z(x, ρ, t) = ut (x, t − ρτ ),

x ∈ Γ1 , ρ ∈ (0, 1),

t > 0.

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Then the closed loop system (1.8) is equivalent to the system utt − div A(x)∇u = 0

in Ω × (0, ∞),

on Γ0 × (0, ∞),

u(x, t) = 0,

in Γ1 × (0, 1) × (0, ∞),

τ zt (x, ρ, t) + zρ (x, ρ, t) = 0,

ηt (x, t) = −η + (m − α)ut (x, t) − βut (x, t − τ ), u(x, 0) = u0 (x),

ut (x, 0) = u1 (x),

η(x, 0) = mu1 (x) + ∂νA u0 (x),

in Γ1 × (0, ∞), x ∈ Ω,

(2.1)

on Γ1 ,

on Γ1 × (0, ∞),

z(x, 0, t) = ut (x, t), z(x, ρ, 0) = f0 (x, −ρτ ),

on Γ1 × (0, 1) ,

where η is given by (1.11). We consider the unknown
T U = u, w = ut |Ω , η, z , in the state space, denoted by Υ := H 1 (Ω) × L2 (Ω) × L2 (Γ1 ) × L2 (Γ1 × (0, 1)) ,

(2.2)

with the norm defined by kU k2 = k(u, w, η, z)T k2 Z Z
= hA(x)∇u, ∇ui + w2 dx + Ω

Γ1

1 η 2 dΓ + ξ m(x) − α

Z

1

(2.3) Z

2

z dρdΓ, 0

Γ1

where ξ > 0 is the constant in (1.12). The system (2.1) can be rewritten in the abstract form U 0 = AU,
T U0 = u0 , u1 , η0 , f0 (·, − · τ ) ,

(2.4)

where the operator A is defined by     w u w   div A(x)∇u   
A  η  = −η + m(x) − α w(x, t) − βz(x, 1, t) z −τ −1 zρ with domain n D(A) := (u, w, η, z)T ∈ H 1 (Ω) × L2 (Ω) × L2 (Γ1 ) × L2 (Γ1 ; H 1 (0, 1)) : div A(x)∇u ∈ L2 (Ω), η = m(x)w|Γ1 + ∂νA u, z(x, 0, t) = w(x, t) (2.5) o on Γ1 × (0, ∞) . We will show that A generates a C0 semigroup on Υ under the assumption (1.5). Now we state the well-posedness result. Theorem 2.1. For any initial datum U0 ∈ Υ, there exists a unique solution U ∈ C([0, ∞), Υ) of system (2.4). Moreover, if U0 ∈ D(A), then U ∈ C([0, ∞), D(A)) ∩ C 1 ([0, ∞), D(A)).

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Proof. Step 1. We prove that A is dissipative. We know Υ is a Hilbert space equipped with the adequate scalar product h·, ·iΥ and norm kU k defined by (2.3). For U ∈ D(A), a simple computation leads to hAU, U iΥ 1 d kU k2 2 dt Z 1Z Z Z 1 ηηt dΓ + ξ z(x, ρ, t)zt (x, ρ, t)dρdΓ = ut ∂νA udΓ + 0 Γ1 Γ Γ1 m(x) − α Z Z Z 1 = ut ∂νA udΓ − η 2 dΓ + ηut (x, t)dΓ Γ1 Γ1 m(x) − α Γ1 Z 1Z Z β z(x, ρ, t)zt (x, ρ, t)dρdΓ z(x, 1, t)ηdΓ + ξ − Γ1 0 Γ1 m(x) − α Z Z 1 = ut ∂νA udΓ − η 2 dΓ 2m(x) Γ1 Γ1 Z Z 1 1
2 − η dΓ + ηut (x, t)dΓ − 2m(x) Γ1 Γ1 m(x) − α Z Z 1Z β − z(x, 1, t)ηdΓ + ξ z(x, ρ, t)zt (x, ρ, t)dρdΓ Γ m(x) − α 0 Γ1 Z 1
m(x) 2 1 =− ut + ∂ν2A u dΓ 2 2m(x) Γ Z 1 Z 1 1
2 − − η dΓ + ηut (x, t)dΓ m(x) − α 2m(x) Γ Γ Z 1 Z 1
β ξ u2t (x, t) − z 2 (x, 1, t) dΓ , − z(x, 1, t)ηdΓ + m(x) − α 2τ Γ1 Γ1

=

(2.6)

where we used that Z

1

Z

ξ

z(x, ρ, t)zt (x, ρ, t)dρdΓ 0

Γ1

= −τ −1 ξ

Z

1

Z zρ (x, ρ, t)z(x, ρ, t)dρdΓ

0

Γ1

Z Z 1 2 dz 1 −1 dρdΓ =− τ ξ 2 Γ1 0 dρ Z
ξ = u2 − z 2 (x, 1, t) dΓ . 2τ Γ1 t

(2.7)

Now we handle the items in (2.6) by applying H¨older’s inequality Z Z 1 η2 k1 ηut (x, t)dΓ ≤ dΓ + m(x)u2t dΓ , k m(x) 4 1 Γ1 Γ1 Γ1 Z
β β 1 2 k2 2 z(x, 1, t)ηdΓ ≤ η + z (x, 1, t) dΓ , m(x) − α 2 Γ1 m(x) − α 2k2 Z

Z − Γ1

(2.8) (2.9)

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where k1 > 0 and k2 > 0 can be chosen later. Substituting (2.7), (2.8) and (2.9) in (2.6) yields Z m(x) k1 m(x) ξ
2 hAU, U iΥ ≤ − u dΓ − − 2 4 2τ t Γ Z 1 1 β 1 1
2 − (1 − )− − η dΓ (2.10) 2k2 2m(x) k1 m(x) Γ1 m(x) − α Z Z k2 ξ
2 β 1 + − z (x, 1, t)dΓ − ∂ν2A udΓ. m(x) − α 2 2τ 2m(x) Γ1 Γ1 Next we find positive numbers k1 , k2 and ξ that guarantee the negativity of hAU, U iΥ . That is, we need, for all x ∈ Γ1 , m(x) k1 m(x) ξ − − > 0, 2 4 2τ 1 β 1 1 (1 − )− − > 0, m(x) − α 2k2 2m k1 m(x) k2 ξ β ( )− < 0. m(x) − α 2 2τ

(2.11) (2.12) (2.13)

Inequalities (2.11) and (2.13) imply β k2 m(x) k1 m(x) ( )< − , m(x) − α 2 2 4

(2.14)

that is, (2 − k1 )m(x) m(x) − α . 2 β We can easily show that inequality (2.12) is equivalent to 1 k1 + 2 β k2 ( − > 0, )− m(x) − α 2m(x)k1 2(m(x) − α) from which we find that
2m(x)k1 − (k1 + 2) m(x) − α > 0 , k2
2 and k2 >

for all x ∈ Γ1 . Now we can determine the constants k1 and k2 according to inequalities (2.15), (2.18) and (2.19). Firstly we aim to get k1 . The inequalties (2.15) and (2.19) yield
(2 − k1 )m(x) m(x) − α βm(x)k1
< , 2β 2m(x)k1 − (k1 + 2) m(x) − α that is,  


 k12 m(x) m(x) − α m(x) + α + k1 (2m(x)) β 2 − 2m(x) m(x) − α
2 + 4m(x) m(x) − α < 0 .

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For the quadratic form of k1 , we compute the discriminant
2 ∆(x) = 4m2 β 2 − 2m(m − α) − 16m2 (m − α)2 (m2 − α2 )
= 4m2 β 4 + 4(m − α)(mα2 − α3 − mβ 2 ) > 0 , for all x ∈ Γ1 , where we used assumption (1.5). Thus we get a possible candidate  2m(x) β2 k1 = min − . (2.20) 2 2 x∈Γ1 m(x) + α m(x) − α It’s easy to verify that 0 < k1 < 2 and inequality (2.18) holds. Secondly we should determine the constant k2 . Now inequalities (2.15) and (2.19) become

m(x) m(x) − α 2m(x)α m(x) − α + m(x)β 2
, (2.21) k2 < (2 − k1 ) = 2β 2β m(x) + α and
βm(x)k1 βm(x) 2m(x) m(x) − α − β 2
=
k2 > , m(x) + α 2α m(x) − α − β 2 2m(x)k1 − (k1 + 2) m(x) − α that is,

2m(x)α m(x) − α + m(x)β 2 βm(x) 2m(x) m(x) − α − β 2

. (2.22) < k2 < m(x) + α 2α m(x) − α − β 2 2β m(x) + α To obtain k2 we only have to verify that

2m(x)α m(x) − α + m(x)β 2 βm(x) 2m(x) m(x) − α − β 2

− m(x) + α 2α m(x) − α − β 2 2β m(x) + α 



= 2m(x)α m(x) − α + m(x)β 2 2α m(x) − α − β 2 




 − 2β 2 m(x) 2m(x) m(x) − α − β 2 2β m(x) + α 2α m(x) − α − β 2 > 0. In fact, we have


2mα(m − α) + mβ 2 2α(m − α) − β 2 − 2β 2 m 2m(m − α) − β 2 = 4m3 α2 + 4mα4 − 8m2 α3 + mβ 4 − 4β 2 m3 + 4β 2 m2 α
= (1 + 1 ) 4α3 21 (α2 − β 2 ) − 4α3 β 2 1 + αβ 4 > 0 , where we used assumption (1.6). Thus by (2.22) we can take
2m(x)α m(x) − α + m(x)β 2 1
min k2 = 2 x∈Γ1 2β m(x) + α
βm(x) 2m(x) m(x) − α − β 2 
+ max . x∈Γ1 m(x) + α 2α m(x) − α − β 2

(2.23)

Finally, we can take a constant ξ to satisfy βk2 ξ m(2 − k1 ) < < , m−α τ 2 after k1 and k2 are given in (2.20) and (2.23). Thus we show that the operator A is dissipative.

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Step 2. We will show that λI −A is surjective for a fixed λ > 0. Given (¯ a, ¯b, c¯, f¯)T ∈ T Υ, we seek a solution U = (u, w, η, z) ∈ D(A) of     a ¯ u w  ¯b     (λI − A)   η  =  c¯ f¯ z that is, satisfying λu − w = a ¯,

in Ω, λw − div A(x)∇u = ¯b, in Ω, λη + η − (m − α)w(x, t) + βz(x, 1, t) = c¯, λz + τ −1 zρ = f¯, on Γ1 × (0, 1),

on Γ1 ,

η = mw|Γ1 + ∂νA u(x, t), z(x, 0, t) = w(x), z(x, ρ, 0) = f0 (x, −ρτ )

(2.24) on Γ1 .

where we take t as a parameter. Suppose that we have found u with the appropriate regularity, then from equation (2.24) we have w := λu − a ¯. Therefore we have the initial value problem for z, λz + τ −1 zρ = f¯, z(x, 0) = w(x)

on Γ1 .

We can easily see that z(x, ρ) = w(x)e−λρτ + τ e−λρτ

Z

ρ

f¯(x, θ)eλθτ dθ .

0

In particular we have −λτ

z(x, 1) = (λu − a ¯)e

+ τe

−λτ

Z

1

f¯(x, θ)eλθτ dθ =: λue−λτ + z0 ,

(2.25)

0

R1 where z0 = −¯ ae−λτ + τ e−λτ 0 f¯(x, ρ)eλρτ dρ. So once we know u,we can get w = λu − a ¯,and we already know z(x, 1), so we obtain c¯ m−α β + w− z(x, 1) (2.26) λ+1 λ+1 λ+1 from (2.24)(3). Now we lay the emphasis on how to get u. Eliminating w we find that the function u satisfies λ2 u = div A(x)∇u + λ¯ a + ¯b, in Ω η=

u = 0, on Γ0 (2.27) c¯ λm + α β ∂νA u = − (λu − a ¯) − z(x, 1), in Γ1 λ+1 λ+1 λ+1 where z(x, 1) is given in (2.25). We obtain a weak formulation of system (2.27) by multiplying the equation by ψ and using Green’s formula Z Z Z λ(λm + α) λβe−λτ (λ2 uψ + hA(x)∇u, ∇ψi)dx + uψ dΓ + uψ dΓ λ+1 λ+1 Ω Γ1 Γ1 Z Z Z Z c¯ a ¯(λm + α) βz0 = (λ¯ a + ¯b)ψdx + ψ dΓ + ψ dΓ − ψ dΓ , λ + 1 λ + 1 λ +1 Ω Γ1 Γ1 Γ1 (2.28)

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for any ψ ∈ H 1 (Ω), where z0 is given in (2.25). As the left hand side of (2.28) is coercive on H 1 (Ω), Lax-Milgram Theorem guarantees the existence and uniqueness of a solution u ∈ H 1 (Ω) of (2.27). Step 3. Finally, the well-posedness result follows from Lummer-Phillips Theorem.  3. Energy decay In this section, we obtain the exponential stability of the system by energy perturbed approach. At first we rewrite the energy E(t) defined in (1.12) as E(t) = E(t) + Eτ (t) , where E(t) is the standard energy regardless of the time delay Z Z
1 η 2 dΓ , E(t) = u2t + |∇g u|2g dx + Ω Γ1 m(x) − α Z 1Z Eτ (t) = u2t (x, t − ρτ )dΓdρ . 0

Γ1

To obtain the estimate of the energy E(t) we define three auxiliary functions: Z V1 (t) = H(u)ut dx , Z Ω 1 V2 (t) = (div H − κ)uut dx , 2 Ω Z Z 1 V3 (t) = ξ e−2τ ρ u2t (x, t − ρτ )dρdΓ . Γ1

0

We need some lemmas from [26] and [13]. Lemma 3.1 ([26, Theorem 2.1.]). Suppose that u(x, t) is a solution of the utt − div A∇u = 0. We have V˙ 1 (t) = B1 + I1 , where the boundary term is Z Z 1 B1 (Γ) = ∂νA uH(u)dΓ + (u2t − |∇g u|2g )H · νdΓ, (3.1) 2 Γ Γ and the internal term Z Z 1 I1 = − Dg H(∇g u, ∇g u)dx − (u2t − |∇g u|2g ) div Hdx . 2 Ω Ω Lemma 3.2 ([26, Theorem 2.2.]). Suppose that u(x, t) is a solution of the equation utt − div A∇u = 0. Then V˙ 2 (t) = B2 + I2 , where the boundary term is Z Z 1 1 B2 (Γ) = − u2 ∂νA (div H)dΓ + (div H − κ)u∂νA udΓ, 4 Γ 2 Γ and the internal term Z Z 1 1 I2 = (div H − κ)(u2t − |∇g u|2g )dx + u2 div A(x)∇(div H)dx . 2 Ω 4 Ω Lemma 3.3 ([18, Prosition 3.1.]). Let u solve problem (1.8). We have Z Z Z Z 1 ξ ξ V˙3 (t) = u2t dΓ − e−2τ z 2 (x, 1, t)dΓ − 2ξ e−2ρτ z 2 dρdΓ . τ Γ1 τ Γ1 Γ1 0 From Lemmas 3.1, 3.2 and 3.3, we obtain the following result.

(3.2)

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Lemma 3.4. Suppose that the geometric assumptions (A1) and (A2) hold. Let u solves problem (1.8). There exist constants c1 , c2 , c3 , c4 , c5 > 0 such that κ κ E(t) + V˙ 1 (t) + V˙ 2 (t) + e2τ V˙ 3 (t) 2 4ξ Z Z Z (3.3) κ 1 ≤ η 2 dΓ + c1 (u2t + |∇g u|2g )dΓ + c2 u2 dx , 2 Γ1 m(x) − α Γ1 Ω |V1 (t)| ≤ c3 E(t), |V2 (t)| ≤ c4 E(t), |V3 (t)| ≤ c5 Eτ (t) . (3.4) Proof. Obviously the estimate (3.4) is true. Now we prove inequality (3.3). First we estimate the boundary terms B1 , B2 given in Lemma 3.1 and Lemma 3.2. Since u|Γ0 = 0, we have on Γ0 that νA , ∇g u = ∂νA u |νA |2g which implies H(u) = hH, ∇g uig = ∂νA u

1 H · ν. |νA |2g

(3.5)

Substituting equality (3.5) in (3.1) yields Z 1 B1 (Γ0 ) = (u2 + |∇g u|2g )H · νdΓ ≤ 0 , 2 Γ0 t where we noticed the geometric assumption (A2) is true. It is obvious that Z c1 (u2 + |∇g u|2g )dΓ . B1 (Γ) = B1 (Γ0 ) + B1 (Γ1 ) ≤ 4 Γ1 t Since u|Γ0 = 0, we have B2 (Γ0 ) = 0. Also we have Z c1 |∇g u|2g dΓ. B2 (Γ) = B2 (Γ0 ) + B2 (Γ1 ) ≤ 4 Γ1

(3.6)

(3.7)

Next, we estimate the internal terms I1 and I2 . From to the geometric assumption (A1), we have Z Z 1 (u2 − |∇g u|2g ) div Hdx. (3.8) I1 ≤ −κ |∇g u|2g dx − 2 Ω t Ω It is obvious that Z Z κ I1 + I2 ≤ − (u2t + |∇g u|2g )dx + c2 u2 dx. (3.9) 2 Ω Ω Combining inequalities (3.6), (3.7), (3.8) and (3.9) we obtain κ E(t) + V˙ 1 (t) + V˙ 2 (t) 2 Z Z Z (3.10) κ 1 c1 ≤ η 2 dΓ + (u2t + |∇g u|2g )dΓ + c2 u2 dx . 2 Γ1 m − α 2 Γ1 Ω Using Lemma 3.3 we know that κ κ Eτ (t) + e2τ V˙ 3 (t) 2 4ξ Z Z Z 1 κ κ 2τ κ 2 ≤ Eτ (t) + e2τ −2ρτ z 2 dρdΓ e ut dΓ − 2 4τ 2 Γ1 0 Γ1 Z κ 2τ ≤ e u2t dΓ . 4τ Γ1

(3.11)

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Combining inequalities (3.10) and (3.11) we complete the proof.



To elinate the tangential part of the derivative ∇g u we need the following lemma from [13]. Lemma 3.5. p[13, Lemma 7.2.]] Let ε > 0 be given small. Let u solves the problem (1.8). Then Z

T −ε

Z

ε

|∇g u|2g dΓdt

Γ1

≤ CT,ε

nZ

T

(3.12)

Z

2

(∂νA u) +

0

u2t




dΓdt + kuk

Γ1

o 1

H 2 +ε (Ω×(0,T ))

.

The following is the observability inequality for the system (1.8). Lemma 3.6. Suppose that the geometric assumptions (A1) and (A2) hold. Let u solve problem (1.8). Then for any given ε > 0, there exists a time T0 > 0 and a positive constant CT,ε,ρ such that nZ

T

Z


u2t + (∂νA u)2 + z 2 (x, 1, t) + η 2 dΓdt 0 Γ1 o + kukH 1/2+ε (Ω×(0,T )) ,

E(0) ≤ CT,ε,ρ

(3.13)

for all T > T0 . Proof. For a given ε small enough, integrating inequality (3.3) on the interval (ε, T − ε) yields κ 2

T −ε

Z

E(t)dt + V1 (T − ε) − V1 (ε) + V2 (T − ε) − V2 (ε) ε

κe2τ κe2τ V3 (T − ε) − V3 (ε) 4ξ 4ξ Z Z Z T −ε Z Z T −ε Z κ T −ε 1 ≤ η 2 dΓ + c1 (u2t + |∇g u|2g )dΓ + c2 u2 dx . 2 ε m − α Γ1 ε Γ1 ε Ω

+

Then we use inequality (3.12) in Lemma 3.5 and inequality (3.4) in Lemma 3.4 to obtain Z

T −ε

ε

nZ

T

Z

(η 2 + u2t + |∂νA u|2g )dΓ o
+ kukH 1/2+ε (Ω×(0,T )) dx + c0 E(T − ε) + E(ε) ,

E(t)dt ≤ CT,ε,ρ

0

Γ1

where the constant CT,ε,κ depends on c1 , κ, given in inequality (3.12).

1 m−α ,

(3.14)

meas(Ω) and the constant CT,ε

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We notice that
E(0) + c0 E(T − ε) + E(ε) Z 2c0 +ε+1 Z 2c0 +ε+1
= E(t)dt + E(0) − E(t) dt ε
ε
+ c0 E(ε) − E(0) + c0 E(T − ε) − E(0) Z 2c0 +ε+1 Z 2c0 +ε+1  Z t Z  ˙ )dτ dt + c0 = E(t)dt − E(τ ε

ε

Z

˙ )dτ E(τ

0

T −ε

+ c0 Z

0

ε

(3.15)

˙ )dτ E(τ

0 2c0 +ε+1

≤

E(t)dt ε

Z

2c0 +ε+1

Z


u2t + (∂νA u)2 + z 2 (x, 1, t) + η 2 dΓdt ,

+c 0

Γ1

˙ where we used E(t) ≤ 0 and the fact which is known from (2.6) that Z
˙ − E(t) = −hAU, U iΥ ≤ c u2t + (∂νA u)2 + z 2 (x, 1, t) + η 2 dΓdt .

(3.16)

Γ1

Now we shall take T0 = 2c0 + 2ε + 1 to guarantee that T − ε > 2c0 + ε + 1, for all T > T0 . Substituting (3.15) to inequality (3.14) completes the proof.  In what follows we use the compactness-uniqueness argument to absorb the lower order term in (3.13). We list the lemma and omit the proof, which could be found in [14, 19, 25, 27] and many others. Lemma 3.7. Suppose that the geometric assumptions (A1) and (A2) hold. Let u solve problem (1.1). Then for any T > T0 , there exists a positive constant C depending on T , ε, ρ, meas(Ω) such that Z TZ
E(0) ≤ C u2t + (∂νA u)2 + z 2 (x, 1, t) + η 2 dΓdt . 0

Γ1

Proof of Theorem 1.2. From (2.10), we have Z E˙ ≤ −C1 [u2t + η 2 + z(x, 1, t)2 + ∂ν2A u]dx , Γ1

where we denote  m k m ξ
1 β 1 1
1 − ,( (1 − )− − , C1 = min ( − 2 4 2τ m−α 2k2 2m k1 m β k2 ξ  −( − ) . m−α 2 2τ Thus from Lemma 3.7 we have that for all T > T0 , Z TZ
E(0) ≤ C η 2 + ∂ν A2 u + u2t + z 2 (x, 1, t) dΓdt 0

≤−

C C1

Z 0

Γ1 T

˙ = − C (E(T ) − E(0)) ; Edt C1

(3.17)

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that is, C − C1 E(0) , C from which the exponential decay result follows. E(T ) ≤



Acknowledgements. This work was supported by the National Natural Science Foundation of China, Grants No. 61473126 and by the Fundamental Research Funds for the Central Universities. References [1] W. Aernouts, D. Roose, R. Sepulchre; Delayed control of a Moore-Greitzer axial compressor model, Intern. J. Bifurcation Chaos, 10 (2000), 115-1164. [2] K.T. Andrews, K. L. Kuttler, M. Shillor; Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (3) (1996), 781-795. [3] J. T. Beale, S. I. Rosencrans; Acoustic boundary conditions, Bull. Amer. Math. Soc., 80 (1974), 1276-1278. [4] B. M. Budak, A. A. Samarskii, A. N. Tikhonov; A Collection of Problems on Mathematical Physics (A.R.M. Robson, Trans.), The Macmillan Co, New York, 1964. ˇ [5] S. Cani´ c and A. Mikeli´ c; Effective equations modeling the ow of a viscous incompressible fluid through a long elastic tube arising in the study of blood ow through small arteries, SIAM J. Appl. Dyn. Syst., 2 (2003), 431-463. [6] G. Chen; Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl., 58 (1979), 249-274. [7] B. Chentouf, A. Guesmia; Neumann-boundary stabilization of the wave equation with damping control and applications, Communications in Applied Analysis, 14 (2010), no. 4, 541-566. [8] F. Conrad, G. O’Dowd, F. Saouri; Asymptotic behaviour for a model of flexible cable with tip masses, Asymptotic Analysis, 30 (2002), 313-330. [9] R. Datko; Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713. [10] N. Fourrer, I. Lasiecka; Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Eqns. and Control Th. (EECT), pp. 631-667, 2013. [11] P. Freitas, D. Krejcirik; Instability results for the damped wave equation in unbounded domains, J. Diff. Equ., 211 (2005), 168-186. [12] J. Lagnese; Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182. [13] I. Lasiecka, R. Triggiani; Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim., 25 (1992) 189224. [14] I. Lasiecka, R. Triggiani; Exact controllability of the wave equation with Neumann boundary control, Appl. Math. and Optimiz., 19 (1989), 243-290. [15] J. Li, S. G. Chai; Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback. Nonlinear Anal., 112 (2015), 105-117. [16] W. Littman, L.Markus; Some recent results on control and stabilization of flexible structures, Proc.COMCON Workshop, Montpellier 1987. [17] T. Meurer, A. Kugi; Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator, International Journal of Robust and Nonlinear Control, 21 (2011), 542-562. [18] S. Nicaise, C. Pignotti; Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585. [19] Z. H. Ning, C. X. Shen, X. P. Zhao, H. Li, C. S. Lin, Y. M. Zhang; Nonlinear boundary stabilization of the wave equations with variable coefficients and time dependent delay, Applied Mathematics and Computation, 232 (2014), 511-520. [20] R. P. Vito, S. A. Dixon; Blood Vessel Constitutive Models, Annual Review of Biomedical Engineering, 5 (2003), 41-439. [21] T. J. Xiao, J. Liang; Second order parabolic equations in Banach spaces with dynamic boundary conditions, Trans. Amer. Math. Soc., 356 (2004), 4787-4809.

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[22] G. Q. Xu, S. P. Yung, L. K. Li; Stabilization of wave systems with input delay in the boundary control, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 770-785. [23] Y. C. You, E. B. Lee; Dynamical boundary control of two-dimensional wave equations: vibrating membrane on general domain, IEEE Transactions on Automatic Control, 34 (1989), 1181-1185. [24] P. F. Yao; On the observatility inequality for exact controllability of wave equations with variable coefficients, SIAM J. Contr. and Optim., Vol. 37, No.5 (1999), 1568-1599. [25] P. F. Yao; Energy decay for the cauchy problem of the linear wave equation of variable coefficients with dissipation, Chinese Annals of Mathematics, Series B, Vol31, no.1, (2010), 59-70. [26] P. F. Yao; Modeling and Control in Vibrational and Structural Dynamics. A Differential Geometric Approach. Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL, 2011. xiv+405 pp. ISBN: 978-1-4398-3455-8. [27] Z. F. Zhang, P. F. Yao; Global smooth solutions of the quasilinear wave equation with an internal velocity feedback, SIAM J. Control Optim., 47 (2008), no. 4, 2044-2077. [28] Z. F. Zhang; Stabilization of the wave equation with variable coefficients and a dynamical boundary control, Electron. J. Diff. Equ., Vol. 2016 (2016), No. 27, pp. 1-10. Dandan Guo School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China E-mail address: [email protected] Zhifei Zhang (corresponding author) School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China E-mail address: [email protected]