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The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls

In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with 
boundary feedback controls. For the networks, there are some results on the exponential stability, 
but no result on estimate of the decay rate. The present work mainly estimates the decay rate for 
these systems, including signal wave equation, serially connected wave equations, and generic 
tree of 1-d wave equations. By defining the weighted energy functional of the system, 
and choosing suitable weighted functions, we obtain the estimation value of decay rate of the systems.

1. Introduction. In this paper, our aim is to estimate decay rate of some concrete 1-d wave network systems. Before going on, we introduce some notation. Let X be a Banach space and A : D(A) ⊂ X → X be a closed and densely defined linear operator. Let us consider the abstract differential equation in X: Suppose that A generates a C 0 semigroup T (t), then the solution to (1) is given by x(t) = T (t)x 0 , and there exist constants M > 0 and ω ∈ R such that ||x(t)|| ≤ M e ωt ||x 0 ||, where ω ≥ ω 0 (A) and the scalar defined by ω 0 (A) = lim t→∞ ln ||T (t)|| t is called the growth order of the semigroup T (t).
It is well known that for given A, to determine ω 0 (A) has been a difficult topic in mathematical system theory. Since A is known, we can calculate the spectrum of A, and determine the scalar 528 YARU XIE AND GENQI XU In general, it holds that s(A) ≤ ω 0 (A). If it holds equality, i.e., s(A) = ω 0 (A), then the system (1) is said to satisfy the spectrum determined growth assumption. When the system (1) satisfies the spectrum determined growth assumption, we can obtain ω 0 (A) by s(A). For example, if T (t) is eventually norm continuous semigroup, differentiable semigroup or analytic semigroup, it holds that s(A) = ω 0 (A)( see, [37]). In particular, if A is resolvent compact, and its eigenvectors system forms a uncondition basis for X, then system (1) satisfies the spectrum determined growth assumption (see, [11]). However, in practice, even if s(A) = ω 0 (A), we cannot calculate the exact value of ω 0 (A), this is because we cannot calculate the exact values of σ(A). In most case, we only obtain the asymptotical values of σ(A). So, to obtain the approximation of ω 0 (A) has been an interesting topic in practice. In this paper we concern with the estimate problem of ω 0 (A) for the 1-d wave networks with boundary controls.
Let us recall briefly research development of 1-d wave networks. The study of control problem on single 1-d wave equation or string started early in 1970's [39]. Rusell [38], Loins [30] obtained the stabilization result for it. In [41], Shubov studied the spectral property of the damping string system. Cox and Zuazua [12] gave the decay rate of the energy functional of a damped string. Xu et al. in [45] studied the stability of a string with feedback time delay, and in [46] they studied the general linear feedback on the boundary and calculated all spectrum of the closed loop system. Krstic et al. in [27] studied the output feedback of an unstable wave equation, and obtained the exponential stability. Ammari in [5] studied the large time behavior of the solutions and optimal location of a homogenous string equation.
Although single string is a simple system, under the boundary velocity feedback, the closed loop system is a more complex system. We can prove the system is exponentially stable, only a few systems can be determined the explicit decay rate.
For multi-link system, Liu et al. in [31] studied the stabilization problem of a serially connected strings and proved that the closed loop system is exponentially stable. Since then, the modelling and control for the wave equations, for instance, see [42,32,33,29,28], became gradually a hot topic in the world. Dager and Zuazua [15,14,13] studied the controllability of star-shaped and tree-shaped networks of strings; Ammari and Jellouli [1,2] studied the stabilization of star-shaped tree and generic tree of strings. Recently Ammari in [3] studied a chain of serially connected strings using a frequency domain method and a special analysis for the resolvent. In [4] the authors analysed the spectrum of the dissipative Schrodinger operator on binary tree-shaped networks, and proved the Riesz basis property of the system. Hence the system satisfies the spectrum determined growth assumption.
Jellouli [26] analyzed the spectrum of a degenerate tree and by the spectral decomposition. He proved the best decay rate identifying with the spectral abscissa of the system. However, they do not give the decay rate of the systems due to difficulty of spectrum exact calculation.
To study the decay rate, Xu, Guo et al in [20,47] studied the Riesz basis property of the closed loop system. Under certain conditions, they proved that the closed loop systems have Riesz basis property, and hence the systems satisfy the spectrum determine growth assumption, i.e., s(A) = ω 0 (A). Since then, there are many papers studying the Riesz basis property of the closed loop system for different 1-d wave networks, for example, see, [34,50,48,17,25,24,23]. These results show that the closed loop systems satisfy s(A) = ω 0 (A). However, due to the difficulty of spectrum calculation of the wave networks, they do not give the estimate of the decay rate of the systems.
Moreover, Nicaise and Valein in [36] studied the stabilization of the 1-d wave networks with a delay in the feedbacks. Under certain conditions, they proved that the networks system is exponentially stable, but have no estimate of the decay rate. More recent results on stabilization and supper-stability of the 1-d wave networks, we refer to [53,54,52], [19,18]. About research development for the general 1-d wave network, we refer to literatures [51] and [49].
Stabilization is one of the most important problems in the research of 1-d wave networks. Under suitable feedback control laws, we can use the different approach, such as multiplier method [51], spectral analysis method [46], as well as resolvent estimate method [22], to prove the exponential stability of the closed loop systems. But there is no result on the estimate of the decay rate.
The estimate problem of the decay rate of the system appears not only in the 1-d wave networks, but also in the other networks, for instance, the first hyperbolic systems, [16,10,9,8] for 1-d linear hyperbolic systems, [43] for thermo-elastic networks, [21] for gas networks and others [6,35]. Based on the reasons above, in this paper, we concentrate our attention on the estimate problem of the 1-d wave networks. Our approach is inspired by the works [16,10]. The most important thing is that we find out the conditions which make the inequalities hold and hence get the decay rate estimate.
The rest of this paper is organized as follows: In section 2, we discuss the decay rate of a serially connected 1-d wave equations. At first we discuss a single 1-d wave equation, and from it we will obtain some information about the decay rate of the weighted energy functional and the spectrum of the system. Using this information we can assert the decay rate of the serially connected wave system. In section 3, we discuss the decay rate of the generic tree of 1-d wave networks. Herein we will extend the approach used in section 2 to the generic tree of 1-d wave networks. At first, we discuss the simple tree of 1-d wave network. From this simple model, we will find out some rule of the parameter choices. After then we use this rule to get the estimate of the decay rate of the generic tree of 1-d wave networks. Finally, in section 4, we conclude this paper.
2. The decay rate of serially connected wave equations. In this section we estimate the decay rate of serially connected 1-d wave equations. At first, we study a control problem of 1-d wave equation. By defining a weighted energy functional of the system, we get a feedback control law. Furthermore, we study the decay rate of the closed loop system. For a single wave equation, Loins [30] proved the exponential stability and Xu [46] proved the system has the Riesz basis by the spectral analysis. Next, we study the serially connected strings, and determine its decay rate. This model was studied early in 1989 by Liu et al in [31], they proved the exponential stability, but they had not given the decay rate of the system.
Although some results of this section are known, we hope to find a general approach which can be apply to more complex system.
2.1. The decay rate of single wave equation. In this subsection, we study the decay rate of signal 1-d wave equation. Although it is a simple model and has been studied in [46] by the spectral analysis method, we hope one can find a general approach from it.
We begin with recalling a control problem of 1-d wave equation: where x ∈ (0, 1) is the space variable, and t > 0 is the time variable. c is a positive real number, which presents the wave-speed. The function u(t) is exterior force (control).
In the sequel, we always use the abbreviations w t , w tt , w x and w xx to represent ∂w ∂t , ∂ 2 w ∂t 2 , ∂w ∂x and ∂ 2 w ∂x 2 , respectively. For (2), we introduce new functions Using (2), we can find out Now we define a weighted energy functional of (2) by where the weighting functions p(x) and q(x) is defined as follows: where γ, p 1 and q 1 are positive constants, they are determined later. Obviously, Differentiating V (t) leads tȯ Integration by parts, we obtaiṅ

Using (3) (4) and (5) we geṫ
Obviously, if we can choose u(t) and p 1 , q 1 and γ such that the boundary parts B(t) satisfy then the system (2) can be exponentially stable.
For example, we take u(t) = w t (1, t), p 1 = q 1 , then the boundary parts have the form Therefore, the γ satisfying 0 < γ ≤ ln 1+α 1−α is desired. In this case, we havė V (t) ≤ −βV (t) and hence The decay rate of the weighted functional of the system (2) is at least β = γc.
Note that according to above choice of u(t), the closed loop system corresponding to (2) is Note that the weighted energy functional satisfies the following inequality So the decay rate of the system (6) is β 2 . Summarizing above discussion, we have proved the following result.
Proof. If α = 1, we can take p 1 > q 1 . In this case, the boundary parts always satisfy that holds for all β = γc. Therefore, it must exist a τ such that

Remark 2.
We can prove that τ = 2 c . Remark 3. The super-stability problem of a system was studied by [7]. The similar questions for the 1-d wave network were studied in [46,40,54].

2.2.
The decay rate of serially connected 1-d wave equations. Let us recall the model studied in [31]. Under suitable change, the model can be written as follows: Similarly, we take transform of the variable: Under the transform the equations become We define the weighted energy functional of the system (7) as and where Using the boundary and connection conditions we get The following theorem gives the conditions for p i , q i and γ i , i = 1, 2, · · · , n. Theorem 2.3. Suppose that the parameters p i , q i and γ i , i = 1, 2, · · · , n, satisfy the following conditions: then the decay rate of the system (7) is at least β 2 , where Proof. If p 1 = q 1 = p, and c i+1 q i+1 = c i q i e −γi , c i+1 p i+1 = c i p i e γi for i = 1, 2, · · · , n − 1, then we have recursion relationship

YARU XIE AND GENQI XU
Therefore, In this case, we have Taking and In particular, when γ i = The desired result follows.

Remark 4.
By the spectral analysis, we can show that the asymptote of the spectrum is 1 3. The decay rate of the generic tree of 1-d wave equations. In this section we will extend the approach used in section 2 to more complex model. Here we mainly study the decay rate of the generic tree of 1-d wave equations. It is well known that if a network is of tree-shaped, and all boundary vertices (but one) are acted on control, then the system is exactly controllable, and under the feedback control laws, then the closed loop system is exponentially stable, see [13,15,1,2,50]. But there is no the estimate of decay rate. In this section we will estimate the decay rate of the generic tree wave networks.

THE EXPONENTIAL DECAY RATE OF GENERIC TREE 535
3.1. A simple tree-shaped network of 1-d wave equations. In this subsection we consider a simple tree-shaped network of wave equations, which is governed by the following partial differential equations: where c i > 0, α i > 0, i = 1, 2, 3. The functions u i (t), i = 2, 3, are controls.
For the equations (11), we let Using (11) we can find out Similarly, we define the weighted energy functional by the weight functions p i (x) and q i (x) are defined as follows: For simplicity, we denote the boundary parts by Using the boundary conditions in (11), we have Our aim is to prove that we can select parameters p i , q i and γ i and control u i (t) such that B(t) ≤ 0 for all t ≥ 0.
The simple selections for u i (t), i = 2, 3 are Hence the closed loop system associated with (11) is The following theorem gives the selection condition of the parameters. Theorem 3.1. Let the boundary control laws u 2 , u 3 be defined by (15). Suppose that p i , q i , i = 1, 2, 3 satisfy the following conditions: then B(t) ≤ 0, and hence for β i = γ i c i , and it holds thatV (t) ≤ −βV (t), i.e.,the exponential decay rate of the closed loop systems (16) is at leat β 2 . Proof. Note thatV So, we only need to prove that under the conditions (17), it holds B(t) ≤ 0.
Substituting the last inequality we get If we take From above we get The following theorem gives the selection conditions of the parameters.

THE EXPONENTIAL DECAY RATE OF GENERIC TREE 541
Theorem 3.2. Let the boundary control laws u i , i = 2, · · · , m, be defined by (15). Suppose that p i , q i , i = 1, · · · , m, satisfy the following conditions: it holds thatV (t) ≤ −βV (t), i.e., the exponential decay rate of the closed loop systems (19) is at leat β 2 ,and β 2 = min c j 4 ln (1+α j ) 2 (2m−3)(1−α j ) 2 ,j = 2, 3,· · ·, m 4. Conclusions. In this paper, we use the weighted energy functional of the systems to estimate the decay rate of the wave networks. Under the boundary velocity feedback control laws, we calculate the decay rate β of the weighted energy functional, and hence the system has decay rate estimate β 2 . The key point of this approach is to select the parameters such that boundary parts and the connection parts are less or equal to zero. This will lead to solve an inequality group for the parameters. By a suitable choices of parameters, we can get the estimate of decay rate of the systems. Usually, if the parameters γ j are selected suitable large, we can get more accurate estimate for the decay rate of the system. But according to the current method, we notice that the parameters γ j are more and more small, and may tend to zero. Because we can't get accurate decay rate for complicated treeshaped wave networks by spectral methods, in next step, we will analyze whether the parameters γ j are smaller and smaller with the increase of m. This problem is worth thinking.