Marwa Chafii
email: marwa.chafii@supelec.fr

Yahya J Harbi

Alister G Burr
email: alister.burr@york.ac.uk

Wavelet-OFDM vs. OFDM: Performance Comparison

Keywords: Wavelet-OFDM vs. OFDM: Performance Comparison Wavelet-OFDM, Discrete Wavelet Transform (DWT), Orthogonal Frequency Division Multiplexing (OFDM), Meyer Wavelet, Haar Wavelet

come

I. INTRODUCTION

Several wireline and wireless communication standards use orthogonal frequency division multiplexing (OFDM) system as a modulation technique for data transmission. OFDM has proved its worth due to its robustness against frequency selective channels compared with single carrier modulation, and also to its high spectral efficiency due to its use of orthogonal waveforms. However, OFDM suffers from some drawbacks such as high peak-to-average power ratio (PAPR), and synchronisation problems. To counter these disadvantages and to improve its performance in other respects, new multi-carrier modulation (MCM) systems have been designed, including Wavelet-OFDM [START_REF] Zhang | A novel Multicarrier Signal Transmission System over Multipath Channel of Low-voltage Power Line[END_REF], [START_REF] Galli | Recent Developments in the Standardization of Power Line Communications within the IEEE[END_REF]. Instead of using the fast Fourier transform (FFT) as in OFDM, Wavelet-OFDM is based on the discrete wavelet transform (DWT).

In this paper, the performance of Wavelet-OFDM is compared with OFDM, in terms of PAPR, bit error rate (BER) on an additive white gaussian noise (AWGN) channel, a flat fading channel and a frequency selective channel. In addition, the complexity of implementation and the power spectral density (PSD) are also analysed and compared. We show that the Wavelet-OFDM system performance depends on the number of scales considered in the wavelet transform. We also apply the frequency domain equalization to the Wavelet-OFDM system to improve its BER performance. Moreover, based on the study of the PSD, we show that the selection of the wavelet has a great impact on the bandwidth efficiency of the system. As a conclusion of our study, we propose the Wavelet-OFDM based on the Discrete Meyer (Dmey) wavelet as an alternative to OFDM, since it outperforms OFDM in terms of PAPR and BER at the cost of increased complexity.

The remainder of this paper is organized as follows. In Section II, the Wavelet-OFDM system is described and its different variants with the corresponding implementation are presented. The PAPR for different variants, the BER under different channel conditions, the implementation complexity as well as the PSD are investigated in Section III, IV, V, and VI respectively. Finally, Section VII concludes the paper and gives the perspectives of the work.

II. INTRODUCTION TO THE WAVELET-OFDM SYSTEM

A. Wavelet Basis

Let ψ and φ be two functions ∈ L 2 (R) of a finite support1 [0, T 0 ]. such that:

ψ 2 = φ 2 = 1, (1)
and

+∞ -∞ ψ(t) = 0. (2)
L 2 (R) is the space of square integrable functions. The norm . of a function g ∈ L 2 (R) is defined as:

g 2 = +∞ -∞ |g(t)| 2
dt. ψ and φ can be named the mother wavelet function and the mother scaling function respectively. Let J 0 be the first scale selected; we define contracted versions ψ j and φ j of the functions ψ and φ, for j ≥ J 0 ,

ψj(t) = 2 j/2 ψ(2 j t) (3)
φj(t) = 2 j/2 φ(2 j t). (4)
The contracted functions ψ j and φ j have a support of [0, T0 2 j ]. For every scale j, we define translated versions ψ j,k and φ j,k of the functions ψ j and φ j as follows:

ψ j,k (t) = ψj(t -2 -j kT0) = 2 j/2 ψ(2 j t -kT0), (5) φ j,k (t) = φj(t -2 -j kT0) = 2 j/2 φ(2 j t -kT0). (6)
The contracted translated functions ψ j,k and φ j,k have a support of [ kT0 2 j , (k+1)T0

2 j ]. For j ∈ [[J 0 , J -1]] and k ∈ [[0, 2 j -1]],
we define the wavelet basis as:

{φ J 0 ,k } k=2 J 0 -1 k=0 J-1 j=J 0 {ψ j,k } k=2 j -1 k=0 , ( 7
)
where J is the last scale considered.

B. Expression of the transmitted Wavelet-OFDM signal

Wavelets have been used in several wireless communication applications such as data compression, source and channel coding, signal denoising and channel modeling. Moreover, wavelets have been proposed as a modulation basis for multicarrier modulation systems. The resulting system is named Wavelet-OFDM [START_REF] Zhang | A novel Multicarrier Signal Transmission System over Multipath Channel of Low-voltage Power Line[END_REF], [START_REF] Galli | Recent Developments in the Standardization of Power Line Communications within the IEEE[END_REF] or also known as orthogonal wavelet division multiplexing (OWDM) [START_REF] Linfoot | Orthogonal Wavelet Division Multiplex: An Alternative to OFDM[END_REF]. In fact, the functions of the wavelet basis defined in [START_REF] Meyer | Principe d'incertitude, bases hilbertien nes et algebres d'operateurs[END_REF], can be considered as multi-carrier waveforms for data transmission [START_REF] Kumar | A Review of Wavelets for Digital Wireless Communication[END_REF], in the same way as the Fourier basis is considered in a conventional OFDM system. Wavelet-OFDM is the MCM system based on the wavelet basis. Instead of carrying the input symbols upon the exponential functions of Fourier system, the carriers are represented by the wavelet functions (ψ j,k ) j∈[[J0,J-1],k∈[[0,2 j -1]] and the scaling functions (φ J0,k ) k∈[[0,2 J 0 -1]] of the first scale. The transmitted Wavelet-OFDM signal can thus be expressed as follows 2 :

x(t) = n J-1 j=J 0 2 j -1 k=0 w j,k ψ j,k (t -nT0) + n 2 J 0 -1 k=0 a J 0 ,k φ J 0 ,k (t -nT0). (8)
• w j,k : wavelet coefficients located at k-th position from scale j, • a J0,k : approximation coefficients located at k-th position from the first scale J 0 , • ψ j,k and φ J0,k are the wavelet and the scaling functions defined in ( 5) and ( 6) respectively.

The number of carriers M is equal to 2 J . The mother wavelet and the mother scaling function correspond to ψ 0,0 and φ 0,0 respectively. For each scale j there are 2 j corresponding translated wavelet functions. From one scale to the next, the number of wavelet functions is then multiplied by two. For the first scale J 0 , there are 2 J0 scaling functions.

Several variants of Wavelet-OFDM can be constructed, depending on the first scale J 0 considered. Fig. 1 depicts the different variants of Wavelet-OFDM defined for different values of J 0 , for a total number of scales J = 3. The position of the functions in Fig. 1 refers to the time localization and the frequency localization characteristics of the functions. We can observe that these properties change from one variant to the other, and the selection of the variant depends on the application. In general, the wavelets of the same scale are translated in time and occupy the same bandwidth. They have the same time and frequency localization properties. From one scale to the next, the time localization is divided by two (contracted wavelets) and the frequency localization is multiplied by a factor of two.

C. Implementation

In order to implement the Wavelet-OFDM system expressed in [START_REF] Nikookar | Wavelet Radio: Adaptive and Reconfigurable Wireless Systems Based on Wavelets[END_REF], we apply the Mallat algorithm [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. For a Wavelet-OFDM signal based on the wavelets of L = J -J 0 scales and the scaling functions of the scale J 0 , the inverse discrete wavelet transform (IDWT) should be performed L times. L can also be interpreted as the number of decomposition levels. Let C n be a vector of M input complex symbols C m,n . The 2 J0 first C m,n symbols correspond to the 2 J0 scaling coefficients (a J0,k ) k∈[[0,2 J 0 -1]] . The second 2 J0 complex symbols correspond to the wavelet coefficients (w J0,k ) k∈[[0,2 J 0 -1]] of the first scale J 0 . First, one IDWT is performed, which gives in its output 2 J0+1 scaling coefficients. After that, the next 2 J0+1 coefficients from the vector C n are extracted and considered as wavelet coefficients, and the second IDWT is performed. The next symbols are processed in the same way until the last scale j = J -1 is reached. For example, let C n be a vector of M = 8 input symbols, and let us consider two cases: the first case:

• J 0 = 0, which means that we select the maximum number of decomposition levels L = 3.

The vector C n of input symbols for the n th period T 0 is expressed as:

Cn = {C1, C2, C3, C4, C5, C6, C7, C8} = {a0,0, w0,0, w1,0, w1,1, w2,0, w2,1, w2,2, w2,3.}

• J 0 = 1, which means that we consider L = 2 decomposition levels.

The vector C n of input symbols for the n th period T 0 is expressed as:

Cn = {C1, C2, C3, C4, C5, C6, C7, C8}
= {a1,0, a1,1, w1,0, w1,1, w2,0, w2,1, w2,2, w2,3.}

As depicted in Fig. 2, the IDWT consists of upsampling by a factor of two and filtering the approximation coefficients (scaling coefficients) and the detail coefficients (wavelet coefficients) respectively by a low-pass and a high-pass filter, whose responses are derived from the wavelet considered.

D. Dmey Wavelet

The Meyer wavelet is a frequency band limited orthogonal wavelet, which has been proposed by Yves Meyer in 1985 [START_REF] Meyer | Principe d'incertitude, bases hilbertien nes et algebres d'operateurs[END_REF], [START_REF] Nikookar | Wavelet Radio: Adaptive and Reconfigurable Wireless Systems Based on Wavelets[END_REF]. Meyer wavelets are indefinitely differentiable orthonormal wavelets, which are well localized and decay from their central peak faster than any inverse polynomial. Dmey is a discrete format approximation of the Meyer wavelet, and it can approximate the Meyer wavelet based on a finite impulse response (FIR) filter as depicted in Fig. 3

III. PAPR PERFORMANCE

In this section, we compare the PAPR performance of different variants of Dmey Wavelet-OFDM with that of OFDM. To evaluate the PAPR performance, the complementary cumulative distribution function (CCDF) of the PAPR is simulated, which is the probability that the PAPR exceeds a defined value γ. The simulations are performed for the 4-QAM (quadrature amplitude modulation) and the 16-QAM constellations, number of carriers M = 128 and for 10 6 iterations. The oversampling factor is fixed to 4. The first conclusion drawn from Fig. 4 is that the PAPR performance of the Dmey depends on the number of scales L of the wavelet transform. When L is high, the CCDF is shifted to the right and the PAPR performance is therefore degraded. This can be interpreted by the consideration that in every decomposition level, the multi-carrier system can be seen as a single carrier system, because the waveforms of the same scale j have the same bandwidth and are only shifted in time. As the single carrier system does not suffer from increased PAPR, the smaller L gets, the less the effect of the PAPR is observed. Moreover, the PAPR for the different variants of the Dmey Wavelet-OFDM 3 Generated using the MATLAB wavefun('dmey') command outperforms that of OFDM. The variant J 0 = J -1 achieves the best PAPR performance. The PAPR performance of this variant is compared with OFDM for different constellations in Fig. 5. As depicted in this figure, given that CCDF = 10 -3 , for 4-QAM (16-QAM respectively), the PAPR of the Dmey is less by 4.5 dB (3 dB respectively) than OFDM. The Dmey outperforms OFDM due to its well localized waveforms in the time domain, since Wavelet-OFDM has short time wavelets, especially for the small scales (j is high). We can also notice that, unlike OFDM, the PAPR performance of Dmey depends on the constellation used.

IV. BER PERFORMANCE

The BER performance of the Wavelet-OFDM and the OFDM systems is evaluated using the parameters illustrated in Table I. The 4-QAM and the 16-QAM constellations are used in the simulations of this section.

A. AWGN and Flat Fading Channels

In this part, a comparison between the BER of OFDM and the Dmey Wavelet for AWGN and flat fading channels, is presented in Fig. 6. Since these are all orthonormal waveform sets, it is only to be expected that the performance of all of them will be the same. The simulations confirm that the Dmey is as good as OFDM in terms of BER performance. In a frequency selective fading channel, a cyclic prefix is added to the Wavelet-OFDM transmitted signal in the time domain, and a frequency domain equalization is performed in the receiver side using either the zero forcing (ZF) equalizer or the minimum mean-square error (MMSE) equalizer as presented in Fig. 7.

The extended typical urban (ETU) model for LTE multipath channel standard [START_REF][END_REF], defined by the channel delay and power profile in Table II, is used in this section. As is shown in Fig. 8, OFDM outperforms the Dmey in terms of BER under frequency selective channel conditions when using ZF equalizer. However, when using MMSE equalizer as depicted in Fig. 9, the Dmey reaches a gain of 6.5 dB in terms of SNR for a BER of 10 -3 and the 4-QAM constellation, compared with OFDM. For higher constellations (16-QAM) and for low SNR values, the performance of OFDM is comparable to that of the Dmey. Starting from SNR = 25 dB, the Dmey outperforms OFDM. Notice that, unlike in Wavelet-OFDM, the MMSE receiver does not change the unbiased SNR in the case of OFDM, and hence does not change the BER of the OFDM system. It is important to highlight that when using channel coding techniques, the gain in terms of SNR will be less significant and will depend also on the efficiency of the coding technique implemented. We have chosen not to use coding to evaluate only the effect of the modulation scheme on BER performance.

V. COMPLEXITY OF IMPLEMENTATION

Since Wavelet-OFDM needs two more blocks (IDWT,DWT) compared with OFDM as presented in Fig. 7, its complexity is therefore higher than OFDM. Let us compute the complexity of the wavelet modulation (IDWT) and the wavelet demodulation (DWT) blocks. According to the Mallat Algorithm [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF], IDWT consists of up-sampling by a factor of two and filtering the approximation coefficients a j,k (scaling coefficients) and the detail coefficients w j,k (wavelet coefficients) respectively by a low-pass filter g and a high-pass filter h. Let K be the length of the filters h and g (K non-zero coefficients). The wavelet modulation is calculated with

J j=J 0 +1 2 j K ≤ J j=1 2 j K = 2M K. (9)
The complexity order in terms of the number of additions and multiplications is therefore O(M K). Knowing that the complexity order of the FFT or the IFFT is O(M log 2 (M )), the complexity increase order is about O( K log2(M ) ), which is affordable since K is bounded, and the number of carriers M is usually large.

VI. PSD PERFORMANCE

The PSD is an interesting characteristic of the system, since it gives a measure of the bandwidth efficiency, the side lobe rejection and the adjacent channel interference. We believe that it may be the most important constraint in many applications. In order to compare OFDM and the Dmey wavelet, we simulate in Fig. 10 their corresponding PSD. The simulation is performed using Matlab, and the PSD is estimated via the periodogram method with a rectangular window. Before applying the wavelet modulation based on IDWT, zero padding by a factor of 8 is performed on the input signal in the frequency domain. Based on the observation of Fig. 10, the width of the main lobe of the PSD of the Dmey is comparable to that of OFDM. In fact, among the commonly used wavelets, Dmey is the most frequency compacted wavelet after Shannon, while this latter is not compatible with the discrete transform. Dmey achieves therefore the best spectrum characteristics, unlike the Haar wavelet for example, which is characterized by a PSD of main lobe larger by around 40% than that of OFDM as presented in Fig. 10. The Dmey wavelet does not have this downside and represents then an alternative waveform which provides good spectral characteristics.

VII. CONCLUSION

The Wavelet-OFDM system based on the Dmey wavelet for different variants has been investigated in this paper and compared with OFDM. We have shown that the Dmey wavelet is at least as good as OFDM in terms of BER performance for AWGN channel and flat fading channel. In addition, the selected variant of Dmey outperforms OFDM in frequency selective channel by up to 6.5 dB, at BER of 10 -3 , for MMSE equalizer. Furthermore, the Dmey reaches a gain up to 4.5 dB in terms of PAPR compared with OFDM. However, the Dmey Wavelet-OFDM requires more implementation complexity.

Our future work is to study a suitable channel coding for wavelets to compare its performance with Coded-OFDM under frequency selective channel.

Figure 1 :

 1 Figure 1: Variants of Wavelet-OFDM.

Figure 2 :

 2 Figure 2: Wavelet-OFDM implementation for J 0 = 0 and J 0 = 1.

Figure 3 :

 3 Figure 3: Dmey wavelet and scaling functions.

Figure 4 :

 4 Figure 4: PAPR performance for different variants of Dmey wavelet.

Figure 5 :

 5 Figure 5: PAPR performance comparison for the 4-QAM and the 16-QAM constellations.

Figure 6 :Figure 7 :

 67 Figure 6: BER performance comparison in AWGN and flat fading channels.

Figure 8 :

 8 Figure 8: BER performance comparison for ZF equalizer in a frequency selective channel.

Figure 9 :

 9 Figure 9: BER performance comparison for MMSE equalizer in a frequency selective channel.

Figure 10 :

 10 Figure 10: PSD comparison.

Table II :

 II Channel delay and power profile.

	Discrete delay (ns)	0	50	120	200	230	500	1600	2300	5000
	Average path gains (dB)	-1.0	-1.0	-1.0	0.0	0.0	0.0	-3.0	-5.0	-7.0

Table I :

 I Simulation parameters.

	Parameters	Definition	Values
	M	Number of carriers	128
	S	Number of frames	100
	SNR	Signal to Noise Ratio in dB	0 : 5 : 25 and 0 : 5 : 35
	nloop	Number of iteration loops	100000
	∆F	Intercarrier spacing	15 KHz
	J0	First scale considered	J -1 = 6

The support of a function means here the interval outside which the function is equal to zero

In[START_REF] Chafii | Power Spectral Density Limitations of the Wavelet-OFDM System[END_REF], we include some material from this Section to introduce the Wavelet-OFDM system in a general way, it appears also here for completeness.

ACKNOWLEDEGMENTS

The authors would like to thank Jacques Palicot from CentraleSupélec and Rémi Gribonval from Inria for their contributions to this study. This work has received a French state support granted to the CominLabs excellence laboratory and managed by the National Research Agency under reference Nb.ANR-10-LABX-07-01. The authors would also like to thank the Region Bretagne, France, and the Matisse Doctoral School for their support of this work. The work in this paper was supported in part by UK EPSRC under grant EP/K000406.


Paola Loreti

Bopeng Rao

Optimal energy decay rate for partially damped systems by spectral compensation

Keywords: weaker damping, spectral compensation, optimal energy decay rate, Riesz basis in higher dimension space

We study the stability of weakly coupled and partially damped systems by means of Riesz basis approach in higher dimension spaces. We propose a weaker distributed damping that compensates the behaviour of the eigenvalues of the system, therefore gives the optimal polynomial energy decay rate for smooth initial data.

AMS subject classifications 93D20 §1. Introduction. The aim of this paper, the results of which were announced in [START_REF] Loreti | Compensation spectrale et taux de décroissance optimal de l'energie de systèmes partiellement amortis[END_REF], is to investigate the energy decay rate of the following weakly coupled and partially damped system (1.1) y tt + Ay + By t + au = 0, u tt + Au + ay = 0 where a is real number, A is a self-adjoint coercive operator and B a linear bounded positive operator in a separated Hilbert space H. Assume furthermore that the resolvent of A is compact in H. Then there exists an increasing sequence µ 2 n → +∞ and an orthonormal sequence e n ∈ H such that (1.2) Ae n = µ 2 n e n , ∀n ≥ 1.

Write (1.1) as

(1.3) d dt     y z u v     =     z -Ay -Bz -au v -Au -ay     =: A     y z u v     .
If a is small enough, it was shown in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] that the equation (1.3) generates a C 0 -semigroup of contractions on the Hilbert space

H = D(A 1 2 ) × H × D(A 1 2 ) × H.
Moreover, let

w n = 1 √ 2     0 0 e n iµ n e n     .
Then using (1.2) a straightforward computation gives

w n H = 1, (iµ n -A)w n 2 H = a 2 2µ 2 n → 0.
This shows that the resolvent of A is not uniformly bounded on the imaginary axis. Following [START_REF] Guo | Riesz basis approach to the stabilization of a flexible beam with a tip mass[END_REF] and [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF] (see also [START_REF] Liu | Semigroup associated with dissipative systems[END_REF] for applications) the system (1.3) is not uniformly stable in H. Now assume that, and this is true in the case B = I, the spectrum of A has asymptotic expansions

(1.4) λ ± 1,n ∼ i ± µ n - 1 2 , λ ± 2,n ∼ i ± µ n - a 2 2µ 2 n .
Then the energy corresponding to the first branch of eigenvalues decays exponentially and that one corresponding to the second branch of eigenvalues decays only at the rate 1/t. Therefore the total energy decays at the rate 1/t. Inspired by this remark, we look for a weaker damping operator for example B = A γ with γ < 0. In that case, we have the following asymptotic expansions of the eigenvalues (1.5)

   λ ± 1,n ∼ ±iµ n - µ 2γ n 2 , λ ± 2,n ∼ ±iµ n -a 2 2µ 2γ+2 n , 2γ + 1 > 0, λ ± 1,n ∼ ±iµ n - µ 2γ n 2 , λ ± 2,n ∼ ±iµ n - µ 2γ n 2 , 2γ + 1 < 0
and the eigenvectors

e ± 1,n ∼ 1 √ 2     e n ±iµ n e n 0 0     , e ± 2,n ∼ 1 √ 2     0 0 e n ±iµ n e n     2γ + 1 > 0, (1.6) e ± 1,n ∼ 1 2
If 2γ + 1 > 0, then the real part of λ ± 1,n is of order µ 2γ n and λ ± 2,n is of order 1 µ 2γ+2 n . Therefore the energy corresponding to the first branch decays at the rate t 1 γ , and that one corresponding to the second branch decays at the rate t -1 γ+1 . Since -1 γ+1 > 1 γ , the total energy decays only at the rate t -1 γ+1 . In this case the eigenvectors e ± 1,n , e ± 2,n are asymptotically decoupled, and the two equations of (1.1) are very weakly coupled or almost independent.

If 2γ + 1 < 0, then the real parts of λ ± 1,n and λ ± 2,n are of the same order µ 2γ n . The total energy decays at the rate t 1 γ and achieves the maximum decay 1/t 2 for 2γ + 1 = 0. In that case the eigenvectors e ± 1,n are well involved with the eigenvectors e ± 2,n , and the two equations are really coupled. The paper is organized as follows. In section 2, we establish a general result on polynomial decay rate of energy by a spectral approach. Section 3 was devoted to the study of the optimal decay rate of system (1.1) with a weaker damping B = A γ for γ < 0. Under a suitable framework, we can establish the asymptotic expansions for the eigenvalues and the eigenvectors as in (1.6)-(1.7). But these expansions couldn't give, except in one-dimension, the desired quadratical convergence for the usual Riesz basis approach. We will construct a sequence of pairwise orthogonal subspaces. By this way, we reduced the quadratical convergence to the Parseval equality. In order to obtain a Riesz basis we only need some weaker asymptotic expansions on the subspaces. In section 4, we give some examples of application.

There are many results concerning the polynomial decay rate. The majority was obtained by spectral approaches ( [START_REF] Jaffard | Singular inter stabilization of the wave equation[END_REF], [START_REF] Huang | Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces[END_REF], [START_REF] Wehbe | Boundary stabilization of wave equations with dynamical control[END_REF], [START_REF] Zuazua | Polynomial decay and control of a 1 -d hyperbolicparabolic coupled system[END_REF]) or frequency domain method ( [START_REF] Bátkai | Polynomial stability of operator semigroups[END_REF], [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF], [START_REF] Liu | Exponential stability of an abstract non-dissipative linear system[END_REF]). Others results were obtained by multiplier method ( [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF], [START_REF] Rao | Stabilization of a plate equation with dynamical boundary control[END_REF], [START_REF] Rao | Polynomial energy decay rate and strong stability of Kirchhoff plates with non-compact resolvent[END_REF]). Also we mention [START_REF] Russell | A general framework for the study of indirect damping mechanisms in elastic systems[END_REF] for a general formulation for partially damped systems and [START_REF] Lions | Contrôlabilité exacte et stabilisation de systèmes distribués[END_REF] for exact controllability and observability for coupled distributed systems. §2. Optimal polynomial energy decay rate by spectral approach. We give a spectral approach for the polynomial energy decay rate of C 0 -semigroups. Theorem 2.1. Let S(t) be a C 0 semigroup of contractions generated by the operator A on a Hilbert space H. Let λ k,n (1 ≤ k ≤ K) denote the k-th branch of eigenvalues of A and {e k,n } 1≤k≤K,n≥1 the system of eigenvectors which forms a Riesz basis in H. Assume that for each 1 ≤ k ≤ K there exist a positive sequence µ k,n → +∞ as n → +∞ and two positive constants

α k ≥ 0, β k > 0 such that (2.1) Reλ k,n ≤ - β k µ α k k,n and Imλ k,n ≥ µ k,n , ∀n ≥ 1.
Then for any u 0 ∈ D(A θ ) with θ > 0, there exists a constant M > 0 independent of u 0 such that

(2.2) S(t)u 0 2 H ≤ A θ u 0 2 H M t 2θδ , ∀t > 0
where the decay rate δ is given by

(2.3) δ := min 1≤k≤K 1 α k = 1 α l .
Moreover if there exists a constant

c 1 > 0, c 2 > 0 such that (2.4) Reλ l,n ≥ - c 1 µ α l l,n and Imλ l,n ≤ c 2 µ l,n ∀n ≥ 1,
then the decay rate δ given in (2.3) is optimal.

Proof. Since {e k,n } 1≤k≤K,n≥1 is a Riesz basis in H, any u 0 ∈ D(A θ ) can be written as

(2.5) u 0 = K k=1 +∞ n=1 a k,n e k,n .
Moreover there exist two constants

C 1 > 0, C 2 > 0 such that (2.6) C 1 K k=1 +∞ n=1 |a k,n | 2 ≤ u 0 2 H ≤ C 2 K k=1 +∞ n=1 |a k,n | 2 .
Using the expansion (2.5) and the continuity of the semigroup S(t), we get

(2.7) S(t)u 0 = K k=1 +∞ n=1
a k,n e λ k,n t e k,n .

Then using the first conditions of (2.1) and the second inequality of (2.6) into (2.7), we obtain

(2.8) S(t)u 0 2 H ≤ sup 1≤k≤K sup n≥1 C 2 µ 2θ k,n exp 2β k t µ α k k,n K k=1 +∞ n=1 |a k,n | 2 µ 2θ k,n .
On the other hand, the conditions of (2.1) imply that there exists a positive constant C 3 > 0 such that (2.9)

K k=1 +∞ n=1 µ 2θ k,n |a k,n | 2 ≤ C 3 K k=1 +∞ n=1 |λ k,n | 2θ |a k,n | 2 ≤ C 3 C 1 A θ u 0 2 H .
Then inserting (2.9) into (2.8), we get (2.10)

S(t)u 0 2 H ≤ sup 1≤k≤K sup n≥1 C 2 C 3 C 1 µ 2θ k,n exp 2β k t µ α k k,n A θ u 0 2 H . If α k = 0 for some 1 ≤ k ≤ K, since inf n≥1 µ k,n > 0, then it is easy to find a constant M k > 0 such that (2.11) 1 µ 2θ k,n exp 2β k t ≤ M k t 2θδ , ∀n ≥ 1. If α k > 0 for some 1 ≤ k ≤ K, then putting t µ α k k,n = 1 x , f k (x) = x 2θ α k e 2β k x
we rewrite (2.12)

µ 2θ k,n exp 2β k t µ α k k,n = f k (x)t 2θ α k .
Then a straightforward computation gives

f k (x) = 2e β k x x 2θ α k -2 θ α k x -β k > 0, x > α k β k θ , < 0, x < α k β k θ . It follows that (2.13) inf x>0 f k (x) = f k α k β k θ = α k β k θ 2θ α k e 2θ α k := M -1 k > 0.
Inserting (2.13) into (2.12) gives that (2.14) 1

µ 2θ k,n exp 2β k t µ α k k,n ≤ M k t 2θ α k , n ≥ 1.
Finally combining the cases (2.11) and (2.14), we get the polynomial energy decay rate (2.2) with the constant M given by (2.15)

M = C 2 C 3 C 1 max 1≤k≤K M k .
Now we consider the optimality of δ. To simplify the notations, we write

α l = α, µ l,n = µ n , λ l,n = λ n , e l,n = e n .
Since µ n → +∞, then for any > 0, we can find a subsequence, still denoted by µ n , such that (2.16)

∞ n=1 1 µ α n < +∞.
Then putting (2.17)

u 0 = ∞ n=1 C 4 µ (θ+α /2) n e n ,
we see that u 0 ∈ D(A θ ) due to the convergence (2.16). On the other hand, thank to the second condition of (2.4), we can choose a constant C 4 > 0 such that A θ u 0 H = 1. Now using the first condition of (2.4), we get

(2.18) S(t)u 0 2 H = ∞ n=1 C 4 e λ n t e n µ (θ+α /2) n 2 H ≥ C 1 C 2 4 µ 2θ+α m exp 2c 1 t µ α m , ∀m ≥ 1.
Finally setting t m = µ α m in (2.18), we obtain that

(2.19) S(t m )u 0 2 H ≥ C 1 C 2 4 e 2c 1 µ 2θ+α m = C 1 C 2 4 e 2c 1 t 2θδ+ m , ∀m ≥ 1.
This means that the trajectory S(t)u 0 decays slower than 1 t 2θδ+ on the time sequence t m → +∞. Then for any > 0, we can't expect the decay rate 1 t 2θδ+ for all initial data u 0 ∈ D(A θ ) and for all t > 0. The proof is thus complete. Remark 2.1. If λ k,n is an eigenvalue of algebraic multiplicity d > 1, then the corresponding factor in (2.8) and (2.14) will be replaced by

|p(t)| 2 µ 2θ k,n exp 2β k t µ α k k,n ≤ M k t 2θδ ,
where p(t) is a polynomial of degree d -1 and M k > 0 is a constant. Therefore, Theorem 2.1 remains valid if the operator A admits a finite number of algebraically multiple eigenvalues and the system of root vectors forms a Riesz basis in H.

Remark 2.2. Theorem 2.1 is valid for all initial data u 0 ∈ D(A θ ) with θ > 0. This is different from an earlier result of Littman-Markus in [START_REF] Littman | Some recent results on control and stabilization of flexible structures[END_REF] where the initial data u 0 should satisfy some stronger conditions. §3. Spectral compensation for weakly damped systems. Let A be a densely defined closed self-adjoint operator in a Hilbert space H such that for some positive constant c we have

(3.1) (Au, u) ≥ c u 2 H , ∀u ∈ D(A).
Let γ ≤ 0, and a be a small real number. We consider the following weakly coupled equations

y tt + Ay + A γ y t + au = 0, (3.2) u tt + Au + ay = 0. (3.3)
In order to curry the explicit computation of the eigenvalues, we have chosen the same operator A in the two equations of the system (3.2)- (3.3). The results of this section complete and improve a recent work of Alabau-Cannarsa-Komornik [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF].

Define the Hilbert space

H = D(A 1 2 ) × H × D(A 1
2 ) × H equipped with the equivalent inner product

(y, z, u, v), (f, g, p, q) H =(A 1 2 y, A 1 2 f ) + (z, g) + (A 1 2 u, A 1 2 p) + (v, q) + a (y, p) + (u, f ) . Setting D(A) = {w =(y, z, u, v) T ∈ H z, v ∈ D(A 1 2 ), y, u ∈ D(A)}, A     y z u v     =     z -Ay -A γ z -au v -Au -ay     we can write the system (3.2)-(3.3) into an evolution equation (3.4) d dt w = Aw, w(0) = w 0 .
Under the condition (3.1), we can prove easily that A is a maximal dissipative operator on H provided that a is small enough. Therefore A generates a C 0semigroup of contractions on H (Theorem 1.4.3 in [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]). Moreover, setting the energy by

E(t) = 1 2 A 1 2 y 2 + y t 2 + A 1 2 u 2 + u t 2 + a(y, u) + a(u, y)
then a straightforward computation gives that

d dt E(t) = -A γ y t 2 ≤ 0.
Assume furthermore that the resolvent of A is compact in H. Then there exists an increasing sequence µ 2 n → +∞ and a orthonormal sequence e n ∈ H such that (3.5) Ae n = µ 2 n e n , ∀n ≥ 1.

Moreover the system {e n } n≥1 forms a Hilbert basis on H. Now let λ be an eigenvalue and (y, z, u, v) be the associated eigenvector of the operator A. Then we have

(3.6)        z = λy, -Ay -A γ z -au = λz, v = λu, -Au -ay = λv.
We will see in Proposition 3.3 that all eigenvectors of A are of the following form

(3.7) y = α n e n , z = λα n e n , u = β n e n , v = λβ n e n .
Inserting (3.7) into (3.6) we get

(3.8) (λ 2 + µ 2 n + λµ 2γ n )α n + aβ n = 0, aα n + (λ 2 + µ 2 n )β n = 0 which has non trivial solution (α n , β n ) = (0, 0) if and only if λ is a solution of the equation (3.9) (λ 2 + λµ 2γ n + µ 2 n )(λ 2 + µ 2 n ) = a 2 . Proposition 3.1. For each µ n > 0, the corresponding eigenvalues λ ± 1,n , λ ± 2,n of the system (3.2)-(3.
3) satisfy the following asymptotic expansions.

I. If -1/2 < γ ≤ 0, λ ± 1,n = ±iµ n - µ 2γ n 2 + O 1 µ n , (3.10) λ ± 2,n = ±iµ n - a 2 2µ 2γ+2 n + O 1 µ 6γ+4 n . (3.11) II. If γ < -1/2, λ ± 1,n = ±i µ 2 n + a - µ 2γ n 2 + O(µ 4γ+1 n ), (3.12) λ ± 2,n = ±i µ 2 n -a - µ 2γ n 2 + O(µ 4γ+1 n ). (3.13) III. If γ = -1/2, λ ± 1,n = ±iµ n - 1 ± √ 1 -4a 2 4µ n + O 1 µ 2 n , (3.14) λ ± 2,n = ±iµ n - 1 ∓ √ 1 -4a 2 4µ n + O 1 µ 2 n . (3.15)
Proof. Firstly, let λ n be one of the four solutions of the equation (3.9). If

|λ 2 n + µ 2 n | ≤ 1, then it follows that (3.16) λ n µ n = ±i + O(µ -2 n ). If |λ 2 n + µ 2 n | ≥ 1
, then from the equation (3.9) we get

λ n µ n 2 + λ n µ n µ 2γ-1 n + 1 = O(µ -2 n ).
It follows that (3.17)

λ n µ n = ±i + O(µ 2γ-1 n ).
Combining (3.16) and (3.17), we get

(3.18) λ n µ n = ±i + O(µ 2γ-1 n ) + O(µ -2 n ).
Next, solving the equation (3.9), we get

(3.19) 2(λ 2 + µ 2 n ) = -λµ 2γ n ∓ λ 2 µ 4γ n + 4a 2 .
The symbol a ∼ b means that a -b → 0. In the following computations we use many times the expansion

√ 1 + z = 1 + z/2 + O(z 2 ).
Case I : -1/2 < γ ≤ 0. From (3.18), we have

(3.20) |λ n µ 2γ n | ∼ µ 1+2γ n → +∞.
Using the asymptotic expansion (3.20) into (3.19), we get

(3.21)        λ 2 1,n = -µ 2 n -λ 1,n µ 2γ n + O 1 µ 2γ+1 n , λ 2 2,n = -µ 2 n + a 2 λ 2,n µ 2γ n + O 1 µ 6γ+3 n . It follows from (3.21) that (3.22)          λ ± 1,n = ±iµ n ± iλ 1,n µ 2γ-1 n 2 + O 1 µ n , λ ± 2,n = ±iµ n ∓ ia 2 2λ 2,n µ 2γ+1 n + O 1 µ 6γ+4 n .
In particular, we get

(3.23)    λ ± 1,n = ±iµ n + O(µ 2γ n ), λ ± 2,n = ±iµ n + O 1 µ 2γ+2 n .
Inserting (3.23) into (3.22) gives that

λ ± 1,n = ±iµ n - µ 2γ n 2 + O 1 µ n , (3.24) λ ± 2,n = ±iµ n - a 2 2µ 2γ+2 n + O 1 µ 6γ+4 n . (3.25) Case II : γ < -1/2. From (3.18) we have (3.26) |λ n µ 2γ n | ∼ |µ 1+2γ n | → 0.
Using the asymptotic expansion (3.26) into (3.19), we have

(3.27)        λ 2 1,n = -(µ 2 n + a) - λ 1,n µ 2γ n 2 + O µ 4γ+2 n , λ 2 2,n = -(µ 2 n -a) - λ 1,n µ 2γ n 2 + O µ 4γ+2 n .
It follows from (3.27)

(3.28)            λ ± 1,n = ±i µ 2 n + a ± iλ 1,n µ 2γ n 2 µ 2 n + a + O(µ 4γ+1 n ), λ ± 2,n = ±i µ 2 n -a ± iλ 1,n µ 2γ n 2 µ 2 n -a + O(µ 4γ+1 n ).
In particular, we get

(3.29)    λ ± 1,n = ±i µ 2 n + a + O(µ 2γ n ), λ ± 2,n = ±i µ 2 n -a + O(µ 2γ n ).
Inserting (3.29) into (3.28) we have

(3.30)        λ ± 1,n = ±i µ 2 n + a - µ 2γ n 2 + O(µ 4γ+1 n ), λ ± 2,n = ±i µ 2 n -a - µ 2γ n 2 + O(µ 4γ+1 n ).
Case III : γ = -1/2. From (3.18) we have

(3.31) λ ± 1,n µ n ∼ ±i + O 1 µ 2 n , λ ± 2,n µ n ∼ ±i + O 1 µ 2 n .
Using the asymptotic expansion (3.31) into (3.19), we have

(3.32)        λ 2 1,n = -µ 2 n ∓ i 2 (1 ± 1 -4a 2 ) + O 1 µ 2 n , λ 2 2,n = -µ 2 n ∓ i 2 (1 ∓ 1 -4a 2 ) + O 1 µ 2 n . It follows from (3.32) that λ ± 1,n = ±iµ n - 1 ± √ 1 -4a 2 4µ n + O 1 µ 3 n , (3.33) λ ± 2,n = ±iµ n - 1 ∓ √ 1 -4a 2 4µ n + O 1 µ 3 n . (3.34)
The proof is thus complete.

If λ ±

1,n , λ ± 2,n are simple eigenvalues, then setting

β ± 1,n = - a (λ ± 1,n ) 2 + µ 2 n α ± 1,n , (3.35) α ± 2,n = - (λ ± 2,n ) 2 + µ 2 n a β ± 2,n (3.36) in (3.8) we get (3.37) e ± 1,n = α ± 1,n         e n λ ± 1,n e n - ae n λ ± 1,n ((λ ± 1,n ) 2 +µ 2 n ) - ae n (λ ± 1,n ) 2 +µ 2 n         , e ± 2,n = β ± 2,n          - ((λ ± 2,n ) 2 +µ 2 n )e n aλ ± 2,n - ((λ ± 2,n ) 2 +µ 2 n )e n a e n λ ± 2,n e n         
. Now let us denote by V n the eigen-space corresponding to the four eigenvalues λ ± 1,n , λ ± 2,n :

V n = Sp{e + 1,n , e + 2,n , e - 1,n , e - 2,n }.
From the expressions in (3.37), the subspaces {V n } n≥1 are clearly pairwise orthogonal in H. It is possible that the equation (3.9) has multiple solutions for some special parameters µ n , a. However for a small enough, the Rouché theorem shows that the equation (3.9) admits at most a double solution for the first branch of eigenvalues λ ± 1,n = λ 1,n , e ± 1,n = e 1,n which satisfies the following equation of derivation

(3.38) (2λ + µ 2γ n )(λ 2 + µ 2 n ) + 2λ(λ 2 + λµ 2γ n + µ 2 n ) = 0.
In that case, we look for the corresponding root vector e 1,n = (

y n , z n , u n , v n ) such that (I -λ 1,n A) e 1,n = e 1,n
which involves that

(3.39)                  z n -λ 1,n y n = e n λ 1,n , -A y n -A γ z n -a u n -λ z n = e n , v n -λ 1,n u n = - ae n λ 1,n (λ 2 1,n +µ 2 n ) -A u n -a y n -λ v n = - ae n (λ 2 1,n + µ 2 n ) . Setting (3.40) y n = α 1,n e n , u n = β 1,n e n in (3.39) we get (3.41)        z n = α 1,n λ 1,n e n + e n λ 1,n , v n = β 1,n λ 1,n e n - ae n λ 1,n (λ 2 1,n + µ 2 n )
where the constants α 1,n and β 1,n satisfy

(3.42)          -(µ 2 + λ 1,n µ 2γ + λ 2 1,n )α 1,n -aβ 1,n = 2λ 1,n + µ 2γ λ 1,n , -aα 1,n -(µ 2 + λ 2 1,n )β 1,n = - 2a λ 2 1,n + µ 2 n .
Since λ 1,n satisfies (3.9) and (3.38), the first equation of (3.42) can be reduced to the second one. Therefore choosing

(3.43) α 1,n = 2 λ 2 1,n + µ 2 n , β 1,n = 0.
in (3.40)-(3.41) we get the corresponding root vector

(3.44) e 1,n =          2 λ 2 1,n +µ 2 n e n 2aλ 1,n λ 2 1,n +µ 2 n + 1 λ 1,n e n 0 - a λ 1,n (λ 2 1,n +µ 2 n ) e n         
.

Accordingly, we modify the subspace V n as

V n = Sp{e 1,n , e + 2,
n , e 1,n , e - 2,n }. Once again, the expression (3.44) shows that the subspaces {V n } n≥1 are still pairwise orthogonal in H. Proposition 3.2. Let E ± 1,n , E ± 2,n be the eigenvectors of the decoupled system (corresponding to a = 0)

(3.45) E ± 1,n = 1 √ 2     e n ±iµ n e n 0 0     , E ± 2,n = 1 √ 2     0 0 e n ±iµ n e n     .
Then the following relationship holds

(3.46) (e + 1,n , e + 2,n , e - 1,n , e - 2,n ) = (E + 1,n , E + 2,n , E - 1,n , E - 2,n )L n
where L n is a 4 × 4 matrix such as

L n =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     + O 1 µ 2γ+1 n 2γ + 1 > 0, (3.47) L n = 1 √ 2     1 -1 0 0 1 1 0 0 0 0 1 -1 0 0 1 1     + O 1 µ min{2,-(2γ+1)} n 2γ + 1 < 0, (3.48) L n = √ 2      √ a + √ a - 0 0 ia √ a + ia √ a - 0 0 0 0 √ a - √ a + 0 0 -ia √ a - -ia √ a +      + O 1 µ 2 n 2γ + 1 = 0 (3.49)
where we have put

(3.50) a ± = 1 ± √ 1 -4a 2 2 .
Proof. For 2γ + 1 > 0, using the asymptotic expansions (3.10)-(3.11), we have

1 (λ ± 1,n ) 2 + µ 2 n = O 1 µ 2γ+1 n , 1 λ ± 1,n = 1 ±iµ n + O(µ 2γ-2 n ), (3.51) (λ ± 2,n ) 2 + µ 2 n = O 1 µ 2γ+1 n , 1 λ ± 2,n = 1 ±iµ n + O 1 µ 2γ+4 n . (3.52) Inserting (3.51)-(3.52) into (3.37) gives e ± 1,n = 1 √ 2     e n ±iµ n e n 0 0     + O 1 µ 2γ+1 n , (3.53) e ± 2,n = 1 √ 2     0 0 e n ±iµ n e n     + O 1 µ 2γ+1 n .
(3.54) For 2γ + 1 < 0, using (3.12)-(3.13), we have

1 (λ ± 1,n ) 2 + µ 2 n = - 1 a + O(µ 2γ+1 n ), 1 λ ± 1,n = 1 ±iµ n + O 1 µ 3 n (3.55) 1 λ ± 2,n = 1 ±iµ n + O 1 µ 3 n , (λ ± 2,n ) 2 + µ 2 n = a + O(µ 2γ+1 n ). ( 3
(λ ± 1,n ) 2 + µ 2 n = ∓ia ± + O 1 µ 2 n , 1 λ ± 1,n = 1 ±iµ n + O 1 µ 3 n , (3.59) (λ ± 2,n ) 2 + µ 2 n = ∓ia ∓ + O 1 µ 2 n , 1 λ ± 2,n = 1 ±iµ n + O 1 µ 3 n . (3.60) Inserting (3.59)-(3.60) into (3.37) gives e ± 1,n = √ a ±     e n ±iµ n e n -ae n a ± µ n ae n ±ia ±     + O 1 µ 2 n , (3.61) e ± 2,n = √ a ∓     e n ±iµ n e n -ae n a ∓ µ n ae n ±ia ∓     + O 1 µ 2 n . (3.62)
The proof is thus complete. Remark 3.1. If 2γ + 1 > 0, the leading term of the eigenvectors e ± 1,n and e ± 2,n are decoupled. The system (3.2)-(3.3) is over-damped and "essentially decoupled". While if 2γ + 1 < 0, the leading term of e ± 1,n and e ± 2,n are well well coupled. The system (3.2)-(3.3) is right-damped in that case. In particular, when 2γ + 1 = 0 the parameter a appears explicitly in the leading term of the matrix L n . Lemma 3.1. Let {X n } n≥1 be a Riesz basis of subspaces in a Hilbert space X and {Y n } n≥1 a Riesz sequence of subspace in X. Assume that there exist a sequence of isomorphisms {L n } n≥1 from X n onto Y n and positive constants m > 0, M > 0 independent of n such that

(3.63) m x n ≤ L n x n ≤ M x n , ∀x n ∈ X n , ∀n ≥ 1.
Assume furthermore that for each n ≥ 1, there exists a Riesz basis {f n,i } 1≤l≤I n (I n ≤ +∞) in X n and positive constants c > 0, C > 0 independent of n such that (3.64) c

I n i=1 |α n,i | 2 ≤ x n 2 ≤ C I n i=1 |α n,i | 2 , ∀x n = I n i=1 α n,i f n,i .
Then the sequence

(3.65) g n,i = L n f n,i , ∀n ≥ 1, 1 ≤ i ≤ I n ,
forms a Riesz basis in X.

Proof. Since {X n } n≥1 is a Riesz basis of subspaces in X, then for any x ∈ X there exists a unique sequence {x n } n≥1 with x n ∈ X n such that

(3.66) x = ∞ n=1 x n , c ∞ n=1 x n 2 ≤ x 2 ≤ C ∞ n=1 x n 2 .
This combining with (3.64) imply that the sequence {f n,i } n≥1,1≤i≤I n forms a Riesz basis in X. Now define the application L in X as following

(3.67) Lx = ∞ n=1 L n x n , x = ∞ n=1
x n .

It is obvious that L is a linear application in X. Moreover, since {Y n } n≥1 is a Riesz sequence of subspaces in X, we get

(3.68) c ∞ n=1 L n x n 2 ≤ Lx 2 ≤ C ∞ n=1 L n x n 2 .
Combining (3.63), (3.66) and (3.68) we get

(3.69) mc C x 2 ≤ Lx 2 ≤ M C c x 2 , ∀x ∈ X.
It follows that L is an isomorphism in X. Thus the sequence {g n,i } n≥1,1≤i≤I n , being the image of the Riesz basis {f n,i } n≥1,1≤i≤I n by the isomorphism L, forms a Riesz basis in X. This achieves the proof.

Proposition 3.3. Let γ ≤ 0 and a be a real number small enough. Then the system of eigenvectors {e + 1,n , e + 2,n , e - 2,n , e - 2,n } n≥1 of A forms a Riesz basis in H. In particular, all eigenvectors of A are of the form (3.7).

Proof. For all n ≥ 1 let

(3.70) W n = Sp{E + 1,n , E + 2,n , E - 2,n , E - 1,n }.
It is clear that {W n } n≥1 forms a Hilbert basis of subspaces and {V n } n≥1 is a Hilbert sequence of subspaces in H. The condition (3.64) is trivial since

E + 1,n , E + 2,n , E - 2,n , E - 2,
n is a Hilbert basis in the subspace W n . Let L n be defined as in (3.46). Following Proposition 3.2, L n has a constant leading term which is invertible. This together with the fact that L n is invertible for all n ≥ 1 imply the condition (3.63). Then applying Lemma 3.1, we get that the system of eigenvectors {e + 1,n , e + 2,n , e - 2,n , e - 2,n } n≥1 forms a Riesz basis in H. In particular, all eigenvectors of A are of the form (3.7). The proof is thus completed. Remark 3.2. There were many papers based on Riesz approach for which the essential part is to show that the sequence of the root vectors is quadratically close to known Riesz basis. This approach requires a very long calculation and is limited to one-dimensional problems (see [START_REF] Guo | Riesz basis approach to the stabilization of a flexible beam with a tip mass[END_REF] and the successions).

The idea of Proposition 3.3 lies in constructing the pairwise orthogonal subspaces V n without the eigenvectors e + 1,n , e + 2,n , e - 1,n , e - 2,n being orthogonal. By this way, we reduced the quadratical convergence to the Parseval equality. In order to get the condition (3.64), we need only some asymptotic expansions such as (3.47)-(3.49) which do not give any quadratical convergence.

To fix the idea, we consider the case where A := -∆ is the Laplacian on a bounded open set Ω ⊂ R N with the homogeneous Dirichlet boundary condition as in Example 4.1. Following a classical result of Agmon (Theorem 14.6 in [START_REF] Agmon | Lectures on elliptic boundary values problems[END_REF]), we know that µ n ∼ n 1/N . Let 2γ + 1 > 0, then the series of general term

(3.71) e ± k,n -E ± k,n = O 1 n (2γ+1)/N
converges in l 2 if and only if N < 2(2γ+1). This is never true if -1/2 < γ ≤ -1/4 even in one-dimension. In the best case γ = 0 which corresponds to the usual damping y t , the series of general term (3.71) converges in l 2 if and only if N < 2. This confirms that the quadratical convergence could be expected only in one-dimension. Theorem 3.1. Let γ ≤ 0 and a be a real number small enough. Then for all y 0 , z 0 ∈ D(A) and y 1 , u 1 ∈ D(A 1 2 ) the energy of the system (3.2)-(3.3) has the following polynomial decay rate

(3.72) E(t) ≤ C( Ay 0 2 H + Au 0 2 H + A 1 2 y 1 2 H + A 1 2 u 1 2 H ) 1 t δ(γ) , ∀t > 0 where (3.73) δ(γ) =    1 γ+1 , 2γ + 1 > 0, -1 γ , 2γ + 1 < 0, 2, 2γ + 1 = 0.
Proof. Following Proposition 3.1, the eigenvalues of A satisfy the asymptotic expansions (2.1) with

(3.74)    α 1 = -2γ, α 2 = 2(γ + 1), δ = 1 γ+1 , 2γ + 1 > 0, α 1 = α 2 = -2γ, δ = 1 -γ , 2γ + 1 < 0, α 1 = α 2 = 1, δ = 2, 2γ + 1 = 0.
On the other hand, following Proposition 3.3 the system of eigenvectors of A forms a Riesz basis in H. Then applying Theorem 2.1, we get the polynomial energy decay rate (3.72)-(3.73). The proof is thus achieved.

The following theorem gives the repartition of energy within the two equations of the system (3.2) -(3.3). Theorem 3.2. Assume the same conditions as in Theorem 3.1. Let w 0 ∈ D(A) be an initial data and

E 1 (t) = 1 2 ( A 1 2 y(t) 2 H + z(t) 2 H ), (3.75) E 2 (t) = 1 2 ( A 1 2 u(t)| 2 H + v(t) 2 H ). (3.76)
be the corresponding energy of the first equation, respectively the second equation. Then the following estimates hold

E 1 (t) ≤ C Aw 0 2 1 t 2 , E 2 (t) ≤ C Aw 0 2 1 t 1 γ+1 , 2γ + 1 > 0, (3.77) E 1 (t) ≤ C Aw 0 2 1 t 1 -γ , E 2 (t) ≤ C Aw 0 2 1 t 1 -γ , 2γ + 1 ≤ 0. (3.78) Proof. Developing e ± 1,n , e ± 2,n on the basis E ± 1,n , E ± 2,n , we get (3.79) e ± 1,n = α ± 1,n E ± 1,n + α ± 2,n E ± 2,n , e ± 2,n = β ± 1,n E ± 1,n + β ± 2,n E ± 2,n . Writing (3.80) w 0 = ∞ n=1 a ± 1,n e ± 1,n + ∞ n=1 a ± 2,n e ± 2,n ,
then we have

w(t) = ∞ n=1 a ± 1,n e λ ± 1,n t e ± 1,n + ∞ n=1 a ± 2,n e λ ± 2,n t e ± 2,n (3.81)
= ∞ n=1 (a ± 1,n α ± 1,n e λ ± 1,n t + a ± 2,n β ± 1,n e λ ± 2,n t )E ± 1,n + ∞ n=1 (a ± 1,n α ± 2,n e λ ± 1,n t + a ± 2,n β ± 2,n e λ ± 2,n t )E ± 2,n . The orthogonality of E ± 1,n , E ± 2,n implies E 1 (t) = 1 2 ∞ n=1 |a ± 1,n α ± 1,n e λ ± 1,n t + a ± 2,n β ± 1,n e λ ± 2,n t | 2 , (3.82) E 2 (t) = 1 2 ∞ n=1 |a ± 1,n α ± 2,n e λ ± 1,n t + a ± 2,n β ± 2,n e λ ± 2,n t | 2 . (3.83) If 2γ + 1 > 0, then from (3.47) we get (3.84) α ± 1,n = O(1), α ± 2,n = O 1 µ 2γ+1 n , β ± 1,n = O 1 µ 2γ+1 n , β ± 2,n = O(1).
Inserting (3.84) into (3.82)-(3.83), we get

E 1 (t) ≤ C ∞ n=1 |µ n a ± 1,n | 2 µ 2 n e µ 2γ n t + |µ n a ± 2,n | 2 µ 4γ+4 n e a 2 t µ 2γ+2 n (3.85) E 2 (t) ≤ C ∞ n=1 |µ n a ± 1,n | 2 µ 4γ+4 n e µ 2γ n t + |µ n a ± 2,n | 2
E 1 (t) ≤ C Aw 0 2 1 t 1 -γ + 1 t 2 ≤ C Aw 0 2 1 t 2 , (3.88) E 2 (t) ≤ C Aw 0 2 1 t 2(γ+1) -γ + 1 t 1 γ+1 ≤ C Aw 0 2 1 t 1 γ+1 . ( 3
(3.91) E 1,2 (t) ≤ C ∞ n=1 (|µ n a ± 1,n | 2 + |µ n a ± 2,n | 2 ) 1 µ 2 n e µ 2γ n t
.

Then using (3.87) into (3.91) gives we get (3.78). The proof is thus complete.

Comments on theorems 3.1 and 3.2. From (3.73) we see that the decay rate δ(γ) is an increasing function for -∞ < γ ≤ -1/2, decreasing for -1/2 ≤ γ ≤ 0 and achieves the maximum 2 at γ = -1/2. The best decay rate δ = 2 occurs for γ = -1/2.

If 2γ + 1 > 0, the system (3.2)-(3.3) is over-damped in the sense that the eigenvectors e ± 1,n are asymptotically decoupled from the eigenvectors e ± 2,n . In that case, the interferences between the two equations are very weak and the wave propagates almost independently in the two equations. The energy of the first equation E 1 (t) decays more quickly than the energy E 2 (t) of the second equation.

If 2γ + 1 ≤ 0, the system (3.2)-(3.3) is right-damped in the sense that the eigenvectors e ± 1,n are involved with the eigenvectors e ± 2,n . In that case the damping applied to the first equation is well transmitted to the second equation and the system is really coupled. Consequently, the energies E 1 (t), E 2 (t) of the two equations decay at the same rate. This balanced repartition of energies within the two equations is due to the compensation of real parts of the two branches of the eigenvalues.

A stronger damping A γ y t does not necessarily give a better total decay rate of energy. A good damping should provide a compensation of the real parts of the two branches of eigenvalues and carry the transmission of the damping from one equation to another. This can be done by means of suitably weaker damping.

We believe that the spectrum compensation is a natural phenomenon for partially damped distributed systems. It seems interesting to consider coupled systems with different operators A 1 , A 2 as in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF]. Indeed the same results could obtained without essential difficulty in the case A 2 = A 2 1 or other similar situations(see [START_REF] Afilal | Stability of coupled second order equations[END_REF], [START_REF] Khodja | Sufficient conditions for uniform stabilization of second order equations by dynamical controllers[END_REF]

0,Ω , • 1,Ω , • 2,Ω • • • the norms of the Sobolev's spaces L 2 (Ω), H 1 (Ω), H 2 (Ω) • • • respectively.
   y tt -∆y + (-∆) -1/2 y t + au = 0 in Ω, u tt -∆u + ay = 0 in Ω, y = u = 0 on Γ
where a ∈ R is small enough, and where (-∆) -1/2 is a linear continuous operator from L 2 (Ω) onto

H 1 0 (Ω). Defining the operator A in L 2 (Ω) by (4.2) A = -∆ with D(A) = H 2 (Ω) ∩ H 1 0 (Ω),
we easily check that A is a densely defined closed self-adjoint operator with compact resolvent in L 2 (Ω). Then applying Theorem 3.1 we get the following optimal polynomial energy decay rate

y(t) 2 1,Ω + y t (t) 2 0,Ω + u(t) 2 1,Ω + u t (t) 2 0,Ω (4.3)
≤ c t 2 y 0 2 2,Ω + z 0 2 1,Ω + u 0 2 2,Ω + v 0 2 1,Ω for all smooth initial data (4.4) y(x, 0) = y 0 (x) ∈ H 2 (Ω) ∩ H 1 0 (Ω), y t (x, 0) = z 0 (x) ∈ H 1 0 (Ω), u(x, 0) = u 0 (x) ∈ H 2 (Ω) ∩ H 1 0 (Ω), u t (x, 0) = v 0 (x) ∈ H 1 0 (Ω),
If we have taken a stronger damping y t as it was done in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF], instead of 1/t 2 we could only get 1/t as the energy decay rate for the same initial data.

Example 4.2. Now we consider the system of weakly coupled plate equations (4.5)

   y tt + ∆ 2 y + (∆ 2 ) -1/2 y t + au = 0 in Ω, u tt + ∆ 2 u + ay = 0 in Ω, y = ∂y ∂n = u = ∂u ∂n = 0 on Γ
where a ∈ R is small enough, and where (∆ 2 ) -1/2 is a linear continuous operator from L 2 (Ω) onto H 2 0 (Ω). Defining the operator A in L 2 (Ω) by

(4.6) A = ∆ 2 with D(A) = H 4 (Ω) ∩ H 2 0 (Ω),
we check easily all the conditions of Theorem 3.1. Then for any smooth initial data

(4.7) y(x, 0) = y 0 (x) ∈ H 4 (Ω) ∩ H 2 0 (Ω), y t (x, 0) = z 0 (x) ∈ H 2 0 (Ω), u(x, 0) = u 0 (x) ∈ H 4 (Ω) ∩ H 2 0 (Ω), u t (x, 0) = v 0 (x) ∈ H 2 0 (Ω),
we have

y(t) 2 2,Ω + y t (t) 2 0,Ω + u(t) 2 2,Ω + u t (t) 2 0,Ω (4.8)
≤ c t 2 y 0 2 4,Ω + z 0 2 2,Ω + u 0 2 4,Ω + v 0 2 2,Ω .
Similarly if we have taken a stronger feedback control y t as it was done in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF], we could only get 1/t as the energy decay rate for the same initial data.

In [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF] more general systems with different operator A 1 , A 2 were considered under an artificial condition D(A j/2

2 ) ⊂ D(A 1 ), j ≥ 2 which restricts the applications to the distributed systems of the same kind boundary conditions (see examples 6.1-6.5 in [START_REF] Alabau | Indirect internal stabilization of weakly coupled evolution equations[END_REF]). The following example shows that the above mentioned condition is not necessary and the polynomial energy decay rate should be true for more general coupled systems. It is easy to prove that the system (4.8) generates a C 0 -semigroup of contractions on the space H for b > 0 and a small enough. Moreover, setting the energy Using the same procedure as in Proposition 3.3, we can prove that the system {e ± 1,n , e ± 2,n } n≥1 form a Riesz basis in H. But in the one-dimensional case, the Riesz basis property is evident because the eigenvectors e ± 1,n , e ± 2,n are quadratically closed to a Hilbert basis due to the expansions (4.17)-(4.18), therefore form a Riesz basis in H (Theorem 4.2.3 in [START_REF] Gohberg | Introduction to the theory of linear non selfadjoint operators[END_REF] and Theorem 4.1 in [START_REF] Rao | Optimal energy decay rate in the Rayleigh beam equation, Optimization Methods in Partial Differential Equations[END_REF] for the original idea on the eigenvalues of lower frequence). Moreover, (4.15)- (4.16) show that the eigenvalues of the system (4.11) satisfy the condition (2.1) with µ 1,n = µ 2,n = n 2 and α 1 = 0, α 2 = 1. Then applying Theorem 2.1, we get the following polynomial energy decay rate for the solution of the system (4.8) with smooth initial data y 0 ∈ H 4 (0, π) ∩ V, u 0 ∈ H 4 (0, π) ∩ W and z 0 ∈ V, v 0 ∈ W .

Remark 4.1. The coupling terms ay x , au x do not appear in the expression (4.10). Nevertheless the energy decays at the rate 1/t 2 instead of 1/t for the system coupled through the displacements ay, au.

  1 = 0, using (3.14)-(3.15), we have

  the estimate for any θ > 0, α > 0 (3.87) µ 2θ n e t µ α n ≥ Ct 2θ α in (3.85)-(3.86), it follows that

  .89) If 2γ + 1 ≤ 0, then from (3.48)-(3.49) we get (3.90) α ± 1,n = O(1), α ± 2,n = O(1), β ± 1,n = O(1), β ± 2,n = O(1). Inserting (3.90) into (3.82)-(3.83) we get

Example 4 . 1 .

 41 Let us consider the system of weakly coupled wave equations(4.1)

Example 4 . 3 .

 43 Consider the following system of compactly coupled and partially damped Euler-Bernoulli beam equations (y tt + y xxxx + by t + au x + y = 0, 0 < x < π u tt + u xxxx -ay x = 0, 0 < x < π, y x (0) = y xxx (0) = y x (π) = y xxx (π) = 0, u(0) = u xx (0) = u(π) = u xx (π) = 0. Putting V = {y ∈ H 2 (0, π), y (0) = y (π) = 0}, W = {u ∈ H 2 (0, π), u(0) = u(π) = 0}we define the energy space H = V × L 2 (0, π) × W × L 2 (0, π).

0 (|y| 2 +λ 2 y

 022 |y xx | 2 + |y t | 2 + |u xx | 2 + |u t | 2 -ay x u -auy x )dx, + y xxxx + bλy + au x + y = 0 0 < x < π λ 2 u + u xxxx -ay x = 0 0 < x < π, y x (0) = y xxx (0) = y x (π) = y xxx (π) = 0, u(0) = u xx (0) = u(π) = u xx (π) = 0.Let (4.12) y = α n cos nx, u = β n sin nx be an eigenfunction. Then λ must satisfy the system (4.13) α n (λ 2 + bλ + n 4 + 1) -aβ n n = 0, β n (λ 2 + n 4 ) + α n an = 0 which has non trivial solution if and only if (4.14) (λ 2 + n 4 )(λ 2 + bλ + n 4 + 1) + a 2 n 2 = 0. Proceeding as in Proposition 3.1, we find easily that λ ± 1,n = ±in 2 -

  for example). But in general we can no long calculate explicitly the eigenvalues as in Proposition 3.1. §4. Examples of application. Let Ω ⊂ R n a bounded open set with smooth boundary Γ. We denote by •