Polynomial stability of the wave equation with distributed delay term on the dynamical control

: Using the frequency domain approach, we prove the rational stability for a wave equation with distributed delay on the dynamical control, after establishing the strong stability and the lack of uniform stability.


Introduction
In the literature, there are full practical processes that might be modelled by distributed delay systems, which present a wide range of applications in various elds such as micro-organism growth [25], hematopoiesis [1,2], logistics [4] and tra c ow [21]. In the recent past (last four decades), many researchers have fruitfully investigated on that subject, and successfully applied them in more widespread other areas. They have developed mathematical tools in order to establish polynomial or exponential decays of these systems. We refer readers to [20] for a list of early works, and to [8-10, 12-15, 22, 26, 27] and the references therein, for some other relevant results. In this paper, we consider the following wave equation with a distributed delay term on the dynamical control : where η denotes the dynamical control, while τ and τ are two real numbers verifying ≤ τ < τ ; moreover β is a positive constant, β : [τ , τ ] → R is a positive L ∞ function and the initial data (u , u , f ) belong to a suitable space. The damping of the system is made via the indirect damping mechanism. Throughout all paper, we assume that β : [τ , τ ] → R is a positive L ∞ function verifying : It is well known that if β = (that is no delay occurs in the system), the energy of problem (1) is polynomially decaying to zero with the rate t − ; see for instance Wehbe [28] for one dimensional case and Toufayli [24] for higher dimension. But in the presence of a delay, namely for such a below system with constants β and β verifying the following assumption that there exists a positive constant ζ verifying it has been proved in Gilbert and . al [7] that the energy of problem (3) decays polynomially with the same rate as in [24,28].
In the case of the wave equations, Nicaise and Pignotti [16] investigated exponential stability results with delay concentrated at τ for the system under the condition µ < µ , by combining inequalities due to Carleman estimates and compactnessuniqueness arguments. Later, they also obtain in [17] the exponential stability with distributed delay of the under the assumption τ τ µ (s)ds < µ .
In this paper, staying on the one dimensional space, we purpose a dynamical boundary moment control η with a distributed delay term, and we look for the possible ways to stabilize the system (1). To our knowledge polynomial stability with distributed delay term has not yet been done, even if the system (3), that is time delay concentrated at τ, decays polynomially.
The paper is organized as follows: section 2 is devoted to the well posedness of problem (1), while the section 3 deals with the strong stability of problem (1) ; furthermore, section 4 establishes the lack of uniform stability, and nally in section 5 stands on the polynomial stability of problem (1).

Well posedness
In this section, we will establish the well posedness of the problem (1), using the semigroup theory. Let us set The problem (1) is now equivalent to Let us set U = u, u t , η, z T .
Then we have Therefore problem ( ) can be rewritten in an abstract framework: where the operator A is de ned by Let us now introduce the Hilbert space So, the natural associated inner product is Proof. We see that Using Green formula, Cauchy Schwarz's inequality and the de nition of D (A) we obtain Now, the relation (2) allows to conclude that A is dissipative.
Let us prove that the operator λI − A is surjective for at least one λ > .
Suppose that we have found u with the appropriate regularity. It means that we have also found η. Then v = λu − f , and we can determine z by solving the system We obtain In particular The function u veri es now By using Lax-Milgram's Lemma, the problem (13) admits a unique weak solution. Indeed, multiplying the rst equation by v ∈ V and by integrating formally by parts, we get where the bilinear and continuous form a is given by Since a is clearly strongly coercive on V and F is continuous on V, by Lax-Milgram's Lemma, problem (13) admits a unique solution u ∈ V. By taking test functions v ∈ D( ; ), we recover the rst identity of (13). This guarantees that u belongs to H ( , ). Using now Green's formula, we see that u satis es the third identity of (13).
Finally, we de ne η and v by setting This shows that the operator A is m-dissipative on H and it generates a C semigroup of contractions in H, under Lumer-Phillips theorem. So, we have found (u, v, η, z) T ∈ D(A) which veri es (13).
We can now state on the following existence results.

Theorem 2.2.
If U = (u , u , η , f ) T belongs to H, then problem (1) has one and only one weak solution U = (u, u t , η, z) T verifying: (1) has one and only one strong solu- Proof. This result is easy to check following the Hille-Yosida theory.

Strong stability
In this section, we establish strong stability result. The main result of this subsection is the following.

= .
Proof. We use the spectral decomposition theory of Sz-Nagy-Foias and Foguel [3,6,23]. Following this theory, since the resolvent of A is compact, it su ces to establish that A has no eigenvalue on the imaginary axis. For our purpose, it is easy to prove that the resolvent of the operator A de ned in (10) is compact. We are ready now to achieve the proof of theorem 3.1 with the following lemma.

Lemma 3.2.
There is no eigenvalue of A on the imaginary axis, that is Proof. By contradiction argument, we assume that there exists at least one iλ ∈ σ(A), λ ∈ R, λ ≠ on the imaginary axis. Let U = (u, v, η, z) T ∈ D(A) be the corresponding normalized eigenvector, that is verifying which is equivalent to Recalling the dissipativity of A , it follows that that is η = .
Owing to the de nition of z in § , we deduce that η = z = .
From the rst equation of (20), we deduce that u( ) = Setting v = iλu, it remains to nd u ∈ V which veri es Therefore, from the general theory of ordinary di erential equations, we deduce that u = , on ( , ).
Now it follows that (u, v, η, z) T = ( , , , ) T which contradicts the fact that U = . We conclude that A has no eigenvalue on the imaginary axis.
As the conditions of the spectral decomposition theory of Sz-Nagy-Foias and Foguel are full satis ed, the proof of theorem 3.1 is thus completed.

Lack of exponential stability
In this section, we will show that the system (1) is lack of exponential decay rate. Our future computations are based on frequency domain approach for exponential stability (see Huang [11] and Prüss [19]), more precisely on the below result.
for some positive constants C and ω, if and only if and sup where ρ(A) denotes the resolvent set of the operator A.
The main result of the current section is the following.
The relation (26) is equivalent to We look for a particular solution, de ned for f n = f n = f n = , and f n will be chosen later. Then (29) becomes The fourth equation of (30) combining with the condition z( ) = η gives z(ρ, s) = ηe −iλn sρ , that is Combining the rst and the second equation of (30), and using the fact that (u, v, η, z) T ∈ D (A), it follows that The homogeneous equation associated to (32) can be solved as As W (u , u ) ≠ , the family (u , u ) forms a fundamental system of solutions. Consequently we can search the particular solution of (32) in the form where k and k are functions which verify The equation ( Now the general solution of (32) can be written as On the one hand we have On the other hand we compute Consequently the general solution of (32) can be rewritten as Let us set Before computing η, let us demonstrate that Πn ≠ with the choice λn = nπ + √ n .
We have  Consequently we have where we set Let Recalling the choice of λn, we have that sin(λn) ≈ √ n , cos(λn) ≈ and λn ≈ nπ. So we get Then it follow that that is Furthermore we have with C (positive) and C are generic constants.
On the other hand, according to the choice of Fn we have Finally we have found some sequences λn, Un and Fn which veri es (26)-(28). Consequently system (1) is not uniformly stable.

Rational stabilization result
In this section, we shall prove that problem (1) is polynomially stable under assumption (2 ). To obtain this, we use method based on the following result due to Borichev and Tomilov [5]: The main result of this section is the next theorem.
Theorem 5.2. The semigroup of system ( ) decays polynomially as Proof. It su ces to show following the results in [18,28] and the above theorem, that for any U = (u, v, η, z) T ∈ D(A) and F = (f , g, h, k) T ∈ H, the solution of where λ > and C a positive constant.
Problem (1) without delay is the following one The generator of its semigroup is A de ned by Thanks to [28], the operator A generates a polynomial stable semigroup with optimal decay rate t − . Therefore the solution u * , v * , η * T of where C is a positive constant.
On the other hand the system (59) can be rewritten as Let ε be a positive constant, the choice of which will be made later. With the help of integrations by parts and using (61), we have Recalling (53) and using (56), we deduce from the above relation that So using (64) and (63) in (62), we get where C and C de ned below are constants not dependent on λ Setting ε = C λ in the above relation such that C λε = , we have where C = C + C is a constant not dependent on λ.
Using the triangular inequality, it follows from (67) that On the one hand, by Cauchy-Schwarz's inequality we obtain On the other hand, Young's inequality guarantees that Combining (68), (69) and (70)