EXPONENTIAL STABILITY OF 1-D WAVE EQUATION WITH THE BOUNDARY TIME DELAY BASED ON THE INTERIOR CONTROL

. In this paper, the stability problem of 1-d wave equation with the boundary delay and the interior control is considered. The well-posedness of the closed-loop system is investigated by the linear operator. Based on the idea of Lyapunov functional technology, we give the condition on the relationship between the control parameter α and the delay parameter k to guarantee the exponential stability of the system.


1.
Introduction. Time delays often appear in many systems such as biological systems, electrical engineering and mechanical applications [2,22], in particular, in distributed parameter systems [3,4,5,6]. In real control systems, time delays in control actions are usually caused by acquisition of response, excitation data, online data processing and computation of control forces. Efforts have been devoted to eliminating the negative influence of time delays. However, time delays cannot be eliminated due to its inherent nature. It is well known that an arbitrarily small delay can destabilize the system. For example, Datko in [3,4,5,6] showed that the occurrence of delays in the control could destabilize hyperbolic control system with the boundary feedback. So it is important to study the stability problems of systems with delays. However, an appropriate time-delay effect can sometimes improve the performance of the system. Abdallah et al. [2], Kwon et al. [10] and Suh et al. [22] proposed some controllers to stabilize the systems with time delays. By now there are many methods to design control laws for systems, such as the Lyapunov function approach, Linear Matrix Inequality(LMI), Pole assignment and so on.
Concerning wave equations with the boundary or interior delays, many scholars have made great efforts to discuss the stability of these systems. For example, Li et al. [11] estimated the decay rates of the nonlinear wave equation with a time-varying delay, based on the method of the Riemannian geometry. Researchers [14,16,18,24,25,23,9] obtained the exponential stability of wave equations with the input delays under certain conditions by spectral analysis. Scholars [1,8,12,19] used the Lyapunov method to analyze the exponential stability of wave equations with the time-varying delays in the feedback controls. Nicaise [17] scrutinized the stability of wave equations with the time dependent delays by using the observability inequality. Pignotti [21] showed the exponential stability of locally damped wave equations with the interior time delays as the coefficients of the delay terms are sufficiently small.
In this paper, we mainly consider the interior control problem of the wave equation with the boundary time delay. The wave equation with the boundary time delay was first introduced [7]. The study of wave equations is of practical significance. For example, Lutzen [13], Miranker [15] used the wave equations to describe the vibration problem of the instruments and physical phenomena, respectively. Let us first recall the control problem of the following wave equation with the boundary delay where x ∈ (0, 1) is the space variable, t > 0 is the time variable, w(x, t) is the displacement deviating from its equilibrium position, k ∈ R is the delay parameter, τ is the delay time and u(x, t) is the control.
It is well known that system (1) can be stabilized exponentially under the feedback control law: Note that if k > 0, α = 0, and τ = 0, system (1) is exponentially stable, and if α > 0 and k = 0, system (1) is also exponentially stable. Here, if α > 0, k ∈ R and τ > 0, we will further discuss the relationship between α and k to guarantee the exponential stability of system (1).
The rest of the paper is organized as follows. In Section 2, we study the wellposeness of the closed loop system and show that the system operator generates a C 0 semigroup. In Section 3, we prove that the closed loop system is exponentially stable by establishing the relationship between the control parameter and the delay parameter. Finally, in Section 4, a brief conclusion is given.
2. The well-poseness. In this section, we shall study the well-posedness of system (1). To this end, we first formulate the system in an appropriate Hilbert space H.
Then system (1) can be rewritten as Let H k ((0, 1)) be an usual Sobolev space, denoted by A direct verification shows that (H, · H ) is a Hilbert space.
Define the operator A in H as follows, System (2) or (3) can be written as where For the operator A, we have the following result.
Lemma 2.1. Let A and H be defined as above. Then 0 ∈ ρ(A), and A −1 is compact on H. Hence, σ(A) consists of all isolated eigenvalues of the finite multiplicity.

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Clearly In what follows, we discuss the generation property of the semigroup C 0 . To this end, we define a new inner product in H, for W j = (w j , v j , z j ) ∈ H, i = 1, 2, where γ, δ and β are positive real numbers.
Observe that Obviously, there exists a positive constant M , such that A − M I is a dissipative operator in the sense of the new inner product.
For any real W = (w, v, z) ∈ D(A),

Now we estimate the boundary term
Using the boundary conditions We get the following result.
Proof. If e γ > 1 + e δ , we get k 2 e γ + e β − k 2 e δ > 0 and Summarizing the discussion above we get the following result.

Theorem 2.3. Let
A and H be defined as above. Let γ, δ and β be positive constants and satisfy Then A −M I is a dissipative operator in the space (H, (·, ·) new ). Hence A generates a C 0 semigroup of the bounded linear operators in H. 3. Exponential stability of the system. In this section, the Lyapunov function approach is used to discuss the stability result of system (1). Here the key point is to construct an appropriate Lyapunov function of (1).
At first, we consider some basic functions. Let w(x, t) be a solution of system (1). We define Lemma 3.1. Let E 0 (t) be defined as (6). Then it is easy to estimate The inequality (7) follows.
We calculatė The desired result follows.
We define Then we have the following result .
Using (8) and (11), we geṫ In order to study the stability of system (1), we define

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Thus This is impossible. So there is no solution of inequalities (20). Based on the above discussion, we know So we define Thus using (17) we havė Theorem 3.5. Let V 0 , V 1 , V 2 (t) and V (t) be defined as above. If α, k τ , ρ and λ satisfy then V 0 (t) decays exponentially at the rate of 2λ.
Proof. Suppose that α, k τ , ρ and λ satisfy inequalities (21). Theṅ Further, the solvability of inequalities (21) is considered. Note that inequalities (21) is equivalent to Obviously, the necessary conditions on the solvability of inequalities (21) are that α, k and ρ satisfy We need to establish the relationship among α, k and ρ.
Then f (2) = 0, lim x→0 f (x) = +∞ and i.e., f (x) is a decreasing function. So for each α > 0, there exists a unique x(α) Multiplying αx on both sides of the above equation, we get So it holds that .
The desired result follows.
All inequalities in (21) are verified.
Observe that the stability result in Theorem 3.9 depends strongly upon the additional parameter ρ. However, in equation (1), we only have three parameters α, k and τ . Hence, we shall further consider how to give a stability result by balancing the relationship between α and k. Theorem 3.10. Let α and k be given as in (1). If α and |k| satisfy (2 + α)|k| (1 + k 2 ) < min 2α(1 − |k|) 1 + 2α , then (1) is exponentially stable.