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Uniform stabilization of a wave equation with partial Dirichlet delayed control

In this paper, we consider the uniform stabilization of some high-dimensional wave equations with partial Dirichlet delayed control. Herein we design a parameterization feedback controller to stabilize the system. This is a new approach of controller design which overcomes the difficulty in stability analysis of the closed-loop system. The detailed procedure is as follows: At first we rewrite the system with partial Dirichlet delayed control into an equivalence cascaded system of a transport equation and a wave equation, and then we construct an exponentially stable target system; Further, we give the form of the parameterization feedback controller. To stabilize the system under consideration, we choose some appropriate kernel functions and define a bounded inverse linear transformation such that the closed-loop system is equivalent to the target system. Finally, we obtain the stability of closed-loop system by the stability of target system.

1.1. Literature and motivation. In the last decades, controllability, observability and stabilization of systems described by PDEs have been studied extensively since this kind of systems can place the actuators and sensors on a part of the boundary of the spatial region. Due to the challenge in mathematical analysis, the boundary control problem of systems has been a hot topic in mathematical control field since 1970. For instance, Smyshlyaev et al. [30] considered the boundary stabilization of a 1-D wave equation with in-domain antidamping based on the backstepping method(similar method also appears in [3]). Guo et al. [11] solved the error feedback regulator problem for 1-D wave equation by using the adaptive control approach. In particular, the control problem of high-dimensional wave equations has been a difficult but hot topic in control field, here we refer to [4,5,13,16] and the references therein. If the control acts on the Neumann boundary, the collocated feedback control law leads to a dissipative closed-loop system. Note that when the control acts on the Dirichlet boundary, the wave equation has a solution only in the space L 2 (Ω)×H −1 (Ω) for each L 2 control u(ξ, t). In this case the expression of feedback operator becomes a main difficulty. To study the Dirichlet control, Lasiecka et al. in [14] and McMillan in [19] considered the feedback control of finite region, and obtained some results on stability and instability. Slightly later, Lasiecka and Triggiani [15] gave the expression of feedback control for the wave equation with Dirichlet control and proved the uniform stabilization, that is the case of Γ 1 = ∅. If Γ 1 = ∅ and the control acts only on the part Γ 0 , the controllability and stabilization of wave equation become more complicated, certain restrictive condition on Γ 0 is required [2,31]. Yao in [35] firstly gave the controllability condition on Γ 0 for UNIFORM STABILIZATION OF A WAVE EQUATION 511 the wave equation with variable coefficients. This condition was also useful for the controllability and stabilization of quasilinear wave equation in [36,38], and for the Cauchy problem with localized damping near infinity in [37].
According to earlier works e.g., see [2,31] and [35], for a wave equation with constant coefficients, Γ 1 and Γ 0 must satisfy the condition that there is a smooth vector field h on R n such that where n ξ is the outward unit normal at position ξ. Let L denote −∆ with zero Dirichlet boundary condition and Υ : L 2 (∂Ω) → L 2 (Ω) be the Dirichlet map on Ω. According to [15], the control operator can be expressed as Bu = ∆Υ(u) and the collocated observation is For a wave equation with constant coefficients, partial boundary Dirichlet control, Ammari in [1] extended the stabilization result of [15] from Γ 1 = ∅ to the case that Γ 1 = ∅ but satisfying (4). The well-posedness and regularity of (3) were proved in [9]. For a wave equation with variable coefficients, partial boundary Dirichlet control and collocated observation, Guo and Zhang in [10, Theorem 1.1] proved that the system is well-posed in the sense of Salamon and regular in the sense of Weiss via the Riemannian geometry method.
We observed that the papers mentioned above mainly considered the control systems without time delay. When the system has a time delay, Nicaise and Pignotti in [25] studied the stability of a wave equation with localized Kelvin-Voigt damping and mixed boundary condition with time delay. And they gave the exponential stability results by using a frequency domain approach. Other results can also be found in [21,23]. If there is a small delay in input, as shown by Datko [6][7][8], the collocated feedback control law may not stabilize the system. Xu et al. in [33] considered the stabilization problem of 1-d wave equation with difference-type control, i.e., αu(t) + βu(t − τ ), and proved under the collocated feedback control law that the closed-loop system is exponentially stable as α > β > 0 and instable as 0 < α < β. This is called the 1 2 -stability rule. Later many researchers verified the correctness of 1 2 -stability rule for different models, for example, see [20,22,24,27] and the references therein. Ning and Yan in [26] discussed a wave equation with variable coefficients and a port Neumann boundary with the difference-type control. And they verified the uniform stabilization of system under the 1 2 -rule assumption. Essentially speaking, these papers mentioned above mainly studied the stability of systems under the collocated feedback control law, which do not include a new feedback controller design. To overcome the restriction of parameters α and β, Shang and Xu et al. started dynamic feedback controller design for the system with difference-type control. The similar results for other systems can also be seen in [28,32,34]. Slight later, Shang and Liu et al. [17,18,29] extended such dynamic feedback controller design from the difference-type control to more complicated cases that include multi-delay-time and the integral term. Since the dynamic feedback controller design is based on full state of the system, Han et al. [12] studied output-based stabilization for an Euler-Bernoulli beam with timedelay in boundary input. Such an investigation shows that the state observer and dynamic feedback controller can settle the delay control problem.
However, by using the state observer and dynamic feedback controller, we get a more complicated closed-loop system. The stability analysis of closed-loop system becomes a challenge work in mathematics, although the closed-loop system is stable from the view point of controller design. This is because the multiplier technique and Riesz basis approach used in earlier works cannot be applied to the resulted system. Up to now the dynamic feedback controller design only apply to 1-d models and the equations with constant coefficients. How to overcome the difficulty in dimensional problem and variable coefficients is a main issue in the stabilization of system with delay in control.
The main goal of this paper is to find a feedback control u(ξ, t) such that the closed-loop system corresponding to (2) is exponentially stable.
1.2. The idea of research. The same as earlier works we assume that the system (1) or (2) has the collocated observation and Γ 0 , Γ 1 satisfy the condition (4). Then we can design a state observer for (2) as follows To show that (5) is a state observer for system (2), we set v(ξ, s, t) = z(ξ, s, t) − z(ξ, s, t), W (x, t) = w(x, t) − w(x, t) and consider the error system Note that the v-part of (6) has a solution v(ξ, s, t) = v 0 (ξ, t + s), so v(ξ, 0, t) = 0 as t > τ . In this case, the W -part satisfies the equation Under the condition (4), the result of [1] asserts that (5) is a state observer for system (2). Therefore, without loss of generality we assume that the full state of system (2) is known. Now we are in a position to find a feedback controller u(x, t) such the closed-loop system associated with (2) is exponentially stable. For this purpose we take the state space as H = L 2 (Γ 0 × [0, τ ]) × L 2 (Ω) × H −1 (Ω). For the system (2), we take the state feedback control u(ξ, t) of the form + Ω γ(ξ, τ, y)w(y, t)dy + where q(ξ, τ − r), γ(ξ, τ, y) and η(ξ, τ, y) are unknown functions that will be determined later. Then the closed-loop system corresponding to (2) is The control given by (7) is called parameterization feedback controller.
The main purpose of this paper is to provide a selecting approach of parameterization controller, i.e., choosing the functions q, γ and η, under which the solution of system (8) is exponentially stable in the sense of norm on space H.
2. Preliminaries, space setting up and operators. In this section we shall give some basic operators and their integral representation that will be used later.
To formulate the system into an abstract evolution equation, we define an operator called the Laplace operator on L 2 (Ω) by Clearly,

XIAORUI WANG AND GENQI XU
The following lemma gives the property of operator L.
Let us consider the boundary value problem where f ∈ L 2 (∂Ω).
The following lemma gives the solvability of (12).
where G D (y, ξ) is the Green's function defined as In particular, the map Υ is a bounded linear operator from L 2 (∂Ω) to L 2 (Ω).
Proof. For any ϕ n , it holds that By the orthonormal basis property of the family {ϕ n (x), n ≥ 1}, the solution of (12) is given by The desired result follows. and , Note that H 2s (Ω) equipped with norm are Hilbert spaces, and L s : H 2s (Ω) → L 2 (Ω) are the bounded linear operators. So we can define the negative fractional order spaces by Let H 2s (Ω) and H −2s (Ω) be defined as before for s ∈ (0, 1). Then the following statements are true: 1) H −2s (Ω) are Hilbert spaces; 2) L s can be extended the bounded linear operators from L 2 (Ω) to H −2s (Ω), i.e., According to Lemma 2.3, we have the following result.
Corollary 1. Let L and L 1 2 be defined as before. Then the following statements are true: 1) L is a bounded linear operator from H 1 0 (Ω) to H −1 (Ω), whose extension is a bounded linear operator from L 2 (Ω) to H −2 (Ω). Moreover,
Lemma 2.4. Let L and Υ be defined as before, and {ϕ n (x), n ≥ 1} be the eigenfunctions of L. We define functions and Then we have Proof. According to Lemma 2.2, for f ∈ L 2 (∂Ω), we have Υ(f ) ∈ L 2 (Ω) which implies By the definition of negative fractional order space, we have From the definition of (15) and (16), it holds that and . The desired result follows. Lemma 2.5. Let L and Υ be defined as before, and G D,1 (x, ξ) be defined as (15).
3. Stabilization of (8). In this section we shall choose functions q, γ and η such that (8) is exponentially stable. For simplicity, we choose the error system (6)(that is also the system (10)) as a reference system, which called the target system, i.e., where Γ 0 and Γ 1 satisfy the condition (4). To study the system (19) and (8), we take the state space as It is easy to know that H is a Hilbert space. In the sequel we always assume that H is a real space.
Theorem 3.1. The solution of target system (19) decays exponentially in the sense of norm of H. (8) to (19). We define an operator T on H by

Transformation from
The following theorem gives a choice of kernel functions under which the system (8) is stable.
Then the transformation T defined by (20) maps the solution of (8) to a solution of (19).
So v(x, s, t) satisfies the differential equation and boundary condition in (19).
Then w(x, t) satisfies the differential equation and boundary condition in (19).
The proof is complete. (19) to (2). We consider the transformation of the following form

(26)
Then S maps the solution of (19) to a solution of (2) with control Proof. Let (v(ξ, s, t), w(y, t), w t (x, t)) be a solution to (19) and S be defined as (24). Thus, (Thanks to (27) and the boundary condition of v) = u(ξ, t).
In what follows, we shall check the differential equations satisfied by z and w. We begin with checking w. Note that w tt (x, t) = ∆w(x, t) holds in (19) and (2), we only need to check the boundary conditions. Since then the boundary condition of w in (19) becomes (using the equality (17) (using the initial value condition in (26)) = w(ξ, t) − z(ξ, 0, t).
Assume that H = L 2 (Ω) × H −1 (Ω) and U = L 2 (Γ 0 ). We recall the following wave equation with partial Dirichlet control and collocated observation: The following result comes from [1] and [9], which will be used later.

The boundedness of transformations
The following theorems give the desired results.
The proof is complete.
Summarizing discussions above, we have the following results.
Theorem 3.8. Let u(x, t) be defined as (7) where the kernel functions satisfy equation (21). Then the closed-loop system (8) is exponentially stable in the sense of H.
The result follows directly from Theorem 3.1.

Conclusion.
In this paper, we consider the uniform stabilization of a wave equation with partial Dirichlet delayed control. We present a kind of new feedback controller that is called parameterization state feedback controller to stabilize the system. Different from the existing dynamic feedback controller, this approach of controller design overcomes the difficulty in stability analysis of the closed-loop system in dimensional problem. By choosing a suitable target system and giving the form of parameterization state feedback controller, we can focus on the selection of kernel functions and then define a bounded inverse linear transformation that establishes the feedback equivalence between the system under consideration and the target system of exponential stability. In the whole process of controller design, the important issue is the selection of kernel functions. By the detailed discussion, we find that the selection of kernel functions depends only on the original system and target system. From the present paper we see that the kernel function γ(ξ, s, y) is a solution to the original system without control, satisfying certain initial value conditions, and the kernel function γ(ξ, s, y) is a solution to the w-part of target system as t > τ , satisfying other certain initial value conditions. Further, thanks to the boundedness and invertibility of transformation, we conclude that the closedloop system we considered is equivalent to the target system. Hence we assert that the closed-loop system is exponentially stable.
On the other hand, we shall note that in order to prove the boundedness and invertibility of the transformation, we do not need an explicit expression of the kernel function, we only need the L 2 well-posedness of the partial differential equation satisfied by the kernel function with partial Dirichlet control and collocated observation. This gives a method for selecting the target system.
Although such a new approach of feedback controller design proposed in this paper is carried out for a high-dimensional wave equation with partial Dirichlet delayed control, it fits more general case theoretically. We hope it can be applied more complicated control form, for example, difference-type control or including a distributed term. In the future, we shall use this approach to study more complicated high-dimensional models.