Stability of an abstract-wave equation with delay and a Kelvin-Voigt damping
aa r X i v : . [ m a t h . A P ] M a y Stability of an abstract–wave equation withdelay and a Kelvin–Voigt damping
Ka¨ıs AMMARI ∗ , Serge NICAISE † and Cristina PIGNOTTI ‡ Abstract.
In this paper we consider a stabilization problem for an abstract wave equationwith delay and a Kelvin–Voigt damping. We prove an exponential stability result for appropri-ate damping coefficients. The proof of the main result is based on a frequency–domain approach. : 35B35, 35B40, 93D15, 93D20.
Keywords : Internal stabilization, Kelvin-Voigt damping, abstract wave equation withdelay.
Our main goal is to study the internal stabilization of a delayed abstract wave equationwith a Kelvin–Voigt damping. More precisely, given a constant time delay τ > , weconsider the system given by: u ′′ ( t ) + a BB ∗ u ′ ( t ) + BB ∗ u ( t − τ ) = 0 , in (0 , + ∞ ) , (1.1) u (0) = u , u ′ (0) = u , (1.2) B ∗ u ( t − τ ) = f ( t − τ ) , in (0 , τ ) , (1.3)where a > B : D ( B ) ⊂ H → H is a linear unbounded operator froma Hilbert space H into another Hilbert space H equipped with the respective norms || · || H , || · || H and inner products ( · , · ) H , ( · , · ) H , and B ∗ : D ( B ∗ ) ⊂ H → H is theadjoint of B . The initial datum ( u , u , f ) belongs to a suitable space.We suppose that the operator B ∗ satisfies the following coercivity assumption: thereexists C > k B ∗ v k H ≥ C k v k H , ∀ v ∈ D ( B ∗ ) . (1.4)For shortness we set V = D ( B ∗ ) and we assume that it is closed with the norm k v k V := k B ∗ v k H and that it is compactly embedded into H . ∗ UR Analysis and Control of Pde, UR 13ES64, Department of Mathematics, Faculty of Sciences ofMonastir, University of Monastir, 5019 Monastir, Tunisia, e-mail : [email protected] † Universit´e de Valenciennes et du Hainaut Cambr´esis, LAMAV, FR CNRS 2956, 59313 ValenciennesCedex 9, France, e-mail: [email protected] ‡ Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit`a di L’Aquila, ViaVetoio, Loc. Coppito, 67010 L’Aquila, Italy, e-mail : [email protected] a >
0. On the other hand if a = 0 and − BB ∗ corresponds to the Laplace operator withDirichlet boundary conditions in a bounded domain of R n , problem (1.1)–(1.3) is notwell–posed, see [16, 12, 19, 20]. Therefore in this paper in order to restitute the well-posedness character and its stability we propose to add the Kelvin–Voigt damping term a BB ∗ u ′ . Hence the stabilization of problem (1.1)–(1.3) is performed using a frequencydomain approach combined with a precise spectral analysis.The paper is organized as follows. The second section deals with the well–posednessof the problem while, in the third section, we perform the spectral analysis of theassociated operator. In section 4, we prove the exponential stability of the system(1.1)–(1.3) if τ ≤ a . In the last section we give an example of an application. In this section we will give a well–posedness result for problem (1.1)–(1.3) by usingsemigroup theory.Inspired from [17], we introduce the auxiliary variable z ( ρ, t ) = B ∗ u ( t − τ ρ ) , ρ ∈ (0 , , t > . (2.1)Then, problem (1.1)–(1.3) is equivalent to u ′′ ( t ) + a BB ∗ u ′ ( t ) + Bz (1 , t ) = 0 , in (0 , + ∞ ) , (2.2) τ z t ( ρ, t ) + z ρ ( ρ, t ) = 0 in (0 , × (0 , + ∞ ) , (2.3) u (0) = u , u ′ (0) = u , (2.4) z ( ρ,
0) = f ( − ρτ ) , in (0 , , (2.5) z (0 , t ) = B ∗ u ( t ) , t > . (2.6)If we denote U := (cid:0) u, u ′ , z (cid:1) ⊤ , then U ′ := (cid:0) u ′ , u ′′ , z t (cid:1) ⊤ = (cid:0) u ′ , − aBB ∗ u ′ − Bz (1 , t ) , − τ − z ρ (cid:1) ⊤ . Therefore, problem (2.2)–(2.6) can be rewritten as (cid:26) U ′ = A U,U (0) = ( u , u , f ( − · τ )) ⊤ , (2.7)2here the operator A is defined by A uvz := v − aBB ∗ v − Bz ( · , − τ − z ρ , with domain D ( A ) := n ( u, v, z ) ⊤ ∈ D ( B ∗ ) × D ( B ∗ ) × H (0 , H ) : aB ∗ v + z (1) ∈ D ( B ) ,B ∗ u = z (0) o , (2.8)in the Hilbert space H := D ( B ∗ ) × H × L (0 , H ) , (2.9)equipped with the standard inner product(( u, v, z ) , ( u , v , z )) H = ( B ∗ u, B ∗ u ) H + ( v, v ) H + ξ Z ( z, z ) H dρ, where ξ > A generates a C semigroup on H by proving that A − cId ismaximal dissipative for an appropriate choice of c in function of ξ, τ and a . Namely weprove the next result. Lemma 2.1. If ξ > τa , then there exists a ∗ > such that A − a − ∗ Id is maximaldissipative in H .Proof. Take U = ( u, v, z ) T ∈ D ( A ) . Then we have( A ( u, v, z ) , ( u, v, z )) H = ( B ∗ v, B ∗ u ) H − (( B ( aB ∗ v + z (1)) , v ) H − ξτ − Z ( z ρ , z ) H dρ. Hence, we get( A ( u, v, z ) , ( u, v, z )) H = ( B ∗ v, B ∗ u ) H − ( aB ∗ v + z (1) , B ∗ v ) H − ξ τ k z (1) k H + ξ τ k z (0) k H . Hence reminding that z (0) = B ∗ u and using Young’s inequality we find that ℜ ( A ( u, v, z ) , ( u, v, z )) H ≤ ( ε − a ) k B ∗ v k H + ( ε − ξ τ ) k z (1) k H + ( ε + ξ τ ) k B ∗ u k H . Chosing ε = a , we find that ℜ ( A ( u, v, z ) , ( u, v, z )) H ≤ − a k B ∗ v k H + ( 1 a − ξ τ ) k z (1) k H + ( 1 a + ξ τ ) k B ∗ u k H . ξ is equivalent to a − ξ τ <
0, and therefore for a ∗ = (cid:16) a + ξ τ (cid:17) − , ℜ ( A ( u, v, z ) , ( u, v, z )) H ≤ − a k B ∗ v k H + ( 1 a − ξ τ ) k z (1) k H + a − ∗ k B ∗ u k H . (2.10)As k B ∗ u k H ≤ k ( u, v, z ) k H , we get ℜ (( A − a − ∗ Id )( u, v, z ) , ( u, v, z )) H ≤ − a k B ∗ v k H + ( 1 a − ξ τ ) k z (1) k H ≤ , which directly leads to the dissipativeness of A − a − ∗ Id .Let us go on with the maximality, namely let us show that λI − A is surjective fora fixed λ > . Given ( f, g, h ) T ∈ H , we look for a solution U = ( u, v, z ) T ∈ D ( A ) of( λI − A ) uvz = fgh , (2.11)that is, verifying λu − v = f,λv + B ( aB ∗ v + z (1)) = g,λz + τ − z ρ = h. (2.12)Suppose that we have found u with the appropriate regularity. Then, v = λu − f (2.13)and we can determine z. Indeed, by (2.8), z (0) = B ∗ u, (2.14)and, from (2.12), λz ( ρ ) + τ − z ρ ( ρ ) = h ( ρ ) for ρ ∈ (0 , . (2.15)Then, by (2.14) and (2.15), we obtain z ( ρ ) = B ∗ ue − λρτ + τ e − λρτ Z ρ h ( σ ) e λστ dσ. (2.16)In particular, we have z (1) = B ∗ ue − λτ + z , (2.17)with z ∈ H defined by z = τ e − λτ Z h ( σ ) e λστ dσ. (2.18)This expression in (2.12) shows that the function u verifies formally λ u + B ( aB ∗ ( λu − f ) + B ∗ ue − λτ + z ) = g + λf, that is, λ u + ( λa + e − λτ ) BB ∗ u = g + λf + B ( aB ∗ f ) − Bz . (2.19)4roblem (2.19) can be reformulated as( λ u + ( λa + e − λτ ) BB ∗ u, w ) H = ( g + λf + B ( aB ∗ f ) − Bz , w ) H , ∀ w ∈ V. (2.20)Using the definition of the adjoint of B , we get λ ( u, w ) H + ( λa + e − λτ )( B ∗ u, B ∗ w ) H = ( g + λf, w ) H + ( aB ∗ f − z , B ∗ w ) H , ∀ w ∈ V. (2.21)As the left-hand side of (2.21) is coercive on D ( B ∗ ), the Lax–Milgram lemma guaranteesthe existence and uniqueness of a solution u ∈ V of (2.21). Once u is obtained we define v by (2.13) that belongs to V and z by (2.16) that belongs to H (0 , H ). Hence wecan set r = aB ∗ v + z (1), it belongs to H but owing to (2.21), it fulfils λ ( v, w ) H + ( r, B ∗ w ) H = ( g, w ) H , ∀ w ∈ D ( B ∗ ) , or equivalently ( r, B ∗ w ) H = ( g − λv, w ) H , ∀ w ∈ D ( B ∗ ) . As g − λv ∈ H , this implies that r belongs to D ( B ) with Br = g − λv. This shows that the triple U = ( u, v, z ) belongs to D ( A ) and satisfies (2.11), hence λI − A is surjective for every λ > . We have then the following result.
Proposition 2.2.
The system (1.1) – (1.3) is well–posed. More precisely, for every ( u , u , f ) ∈ H , there exists a unique solution ( u, v, z ) ∈ C (0 , + ∞ , H ) of (2.7) . More-over, if ( u , u , f ) ∈ D ( A ) then ( u, v, z ) ∈ C (0 , + ∞ , D ( A )) ∩ C (0 , + ∞ , H ) with v = u ′ and u is indeed a solution of (1.1) – (1.3) . As D ( B ∗ ) is compactly embedded into H , the operator BB ∗ : D ( BB ∗ ) ⊂ H → H has a compact resolvent. Hence let ( λ k ) k ∈ N ∗ be the set of eigenvalues of BB ∗ repeatedaccording to their multiplicity (that are positive real numbers and are such that λ k → + ∞ as k → + ∞ ) and denote by ( ϕ k ) k ∈ N ∗ the corresponding eigenvectors that form anorthonormal basis of H (in particular for all k ∈ N ∗ , BB ∗ ϕ k = λ k ϕ k ). We have the following lemma.
Lemma 3.1. If τ ≤ a , then any eigenvalue λ of A satisfies ℜ λ < . roof. Let λ ∈ C and U = ( u, v, z ) ⊤ ∈ D ( A ) be such that( λI − A ) uvz = 0 , or equivalently v = λu, − B ( aB ∗ v + z ( · , λv, − τ − z ρ = λz. (3.1)By (2.14), we find that z ( ρ ) = λ − B ∗ ve − λρτ . (3.2)Using this property in (3.1), we find that u ∈ D ( B ∗ ) is solution of λ u + ( aλ + e − λτ ) BB ∗ u = 0 . Hence a non trivial solution exists if and only if there exists k ∈ N ∗ such that λ aλ + e − λτ = − λ k . (3.3)This condition implies that λ does not belong toΣ := { λ ∈ C : aλ + e − λτ = 0 } , (3.4)and that e − λτ + λ λ k + aλ = 0 . (3.5)Writing λ = x + iy , with x, y ∈ R , we see that this identity is equivalent to e − τx cos( τ y ) + x − y λ k + ax = 0 , (3.6) − e − τx sin( τ y ) + 2 xyλ k + ay = 0 . (3.7)The second equation is equivalent to e τx (cid:16) xλ k + a (cid:17) y = sin( τ y ) . Hence if y = 0, we will get e τx τ (cid:16) xλ k + a (cid:17) = sin( τ y ) τ y . As the modulus of the right-hand side is ≤
1, we obtain (cid:12)(cid:12)(cid:12) e τx τ (cid:16) xλ k + a (cid:17)(cid:12)(cid:12)(cid:12) ≤ ,
6r equivalently (cid:12)(cid:12)(cid:12) xλ k + a (cid:12)(cid:12)(cid:12) ≤ τ e − τx . Therefore if x ≥
0, we find that2 xλ k + a ≤ τ e − τx ≤ τ, which implies that 2 xλ k ≤ τ − a. For τ < a , we arrive to a contradiction. For τ = a , the sole possibility is x = 0 and by(3.7), we find that sin( τ y ) = τ y, which yields y = 0 and again we obtain a contradiction.If y = 0, we see that (3.7) always holds and (3.6) is equivalent to e − τx = − x ( xλ k + a ) . This equation has no non–negative solutions x since for x ≥
0, the left hand side ispositive while the right–hand side is non positive, hence again if a solution x exists, ithas to be negative.The proof of the lemma is complete.If a < τ , we now show that there exist some pairs of ( a, τ ) for which the system(1.1)–(1.3) becomes unstable. Hence the condition τ ≤ a is optimal for the stability ofthis system. Lemma 3.2.
There exist pairs of ( a, τ ) such that < a < τ and for which the associatedoperator A has a pure imaginary eigenvalue.Proof. We look for a purely imaginary eigenvalue iy of A , hence system (3.6)–(3.7)reduces to cos( τ y ) = y λ k , (3.8)sin( τ y ) = ay. (3.9)Such a solution exists if y λ k + a y = 1 . (3.10)One solution of this equation is y k = − a λ k + q a λ k + 4 λ k .
7e now take any τ ∈ (0 , π y k ) and a = sin( τy k ) y k . Then (3.9) automatically holds, while(3.8) is valid owing to (3.10) (as cos( τ y k ) > a < τ because aτ = sin( τ y k ) τ y k < . Therefore with such a choice of a and τ , the operator A has a purely imaginary eigen-value equal to iy k . Inspired from section 3 of [1], by using a Fredholm alternative technique, we performthe spectral analysis of the operator A .Recall that an operator T from a Hilbert space X into itself is called singular if thereexists a sequence u n ∈ D ( T ) with no convergent subsequence such that k u n k X = 1 and T u n → X , see [21]. According to Theorem 1.14 of [21] T is singular if and only ifits kernel is infinite dimensional or its range is not closed. Let Σ be the set defined in(3.4). The following results hold: Theorem 3.3.
1. If λ ∈ Σ , then λI − A is singular.2. If λ Σ , then λI − A is a Fredholm operator of index zero.Proof. For the proof of point 1, let us fix λ ∈ Σ and for all k ∈ N ∗ set U k = ( u k , λu k , B ∗ u k e − λτ · ) ⊤ , with u k = √ λ k ϕ k . Then U k belongs to D ( A ) and easy calculations yield (due to theassumption λ ∈ Σ) ( λI − A ) U k = λ (0 , u k , ⊤ . Therefore we deduce that k ( λI − A ) U k k H → , as k → ∞ . Moreover due to the property k B ∗ u k k H = 1, there exist positive constants c, C, suchthat c ≤ k U k k H ≤ C, ∀ k ∈ N ∗ . This shows that λI − A is singular.For all λ ∈ C , introduce the (linear and continuous) mapping A λ from V into itsdual by h A λ v, w i V ′ − V = λ ( v, w ) H + ( aλ + e − λτ )( B ∗ v, B ∗ w ) H , ∀ v, w ∈ D ( B ∗ ) . Then from the proof of Lemma 2.1, we know that for λ > A λ is an isomorphism.Now for λ ∈ C \ Σ, we can introduce the operator B λ = ( aλ + e − λτ ) − A λ . λ ∈ C \ Σ, A λ is a Fredholm operator of index 0 if and only if B λ is a Fredholmoperator of index 0. Furthermore for λ, µ ∈ C \ Σ as B λ − B µ is a multiple of the identityoperator, due to the compact embedding of V into V ′ , and as B µ is an isomorphism for µ >
0, we finally deduce that A λ is a Fredholm operator of index 0 for all λ ∈ C \ Σ.Now we readily check that, for any λ ∈ C \ Σ, we have the equivalence u ∈ ker A λ ⇐⇒ ( u, λu, B ∗ ue − λτ · ) ⊤ ∈ ker( λI − A ) . (3.11)This equivalence implies thatdim ker( λI − A ) = dim ker A λ , ∀ λ ∈ C \ Σ . (3.12)For the range property for all λ ∈ C \ Σ introduce the inner product( u, z ) λ,V := (cid:16) ( u, λu, B ∗ ue − λτ · ) ⊤ , ( z, λz, B ∗ ze − λτ · ) ⊤ (cid:1) H , on V whose associated norm is equivalent to the standard one.Denote by { y ( i ) } Ni =1 an orthonormal basis of ker A λ for this new inner product (forshortness the dependence of λ is dropped), i.e.( y ( i ) , y ( j ) ) λ,V = δ ij , ∀ i, j = 1 , . . . , N. Finally, for all i = 1 , . . . , N , we set Z ( i ) = ( y ( i ) , λy ( i ) , B ∗ y ( i ) e − λτ · ) ⊤ , the element of ker( λI − A ) associated with y ( i ) that are orthonormal with respect tothe inner product of H .Let us now show that for all λ ∈ C \ Σ, the range R ( λI − A ) of λI − A is closed.Indeed, let us consider a sequence U n = ( u n , v n , z n ) ⊤ ∈ D ( A ) such that( λI − A ) U n = F n = ( f n , g n , h n ) ⊤ → F = ( f, g, h ) ⊤ in H . (3.13)Without loss of generality we can assume that( U n , Z ( i ) ) H = − α n,i , ∀ i = 1 , . . . , N, (3.14)where α n,i := ((0 , f n , − τ e − λτ · Z · h n ( σ ) e λστ dσ ) ⊤ , Z ( i ) ) H . Indeed, if this is not the case, we can consider˜ U n = U n − N X i =1 β i Z ( i ) that still belongs to D ( A ) and satisfies( λI − A ) ˜ U n = F n ,
9s well as ( ˜ U n , Z ( i ) ) H = − α n,i , ∀ i = 1 , . . . , N, by setting β i = ( U n , Z ( i ) ) H + α n,i , ∀ i = 1 , . . . , N. Note that the condition (3.14) is equivalent to( u n , y ( i ) ) λ,V = 0 , ∀ i = 1 , . . . , N. In other words, u n ∈ (ker A λ ) ⊥ λ,V , (3.15)where ⊥ λ,V means that the orthogonality is taken with respect to the inner product( · , · ) λ,V .Returning to (3.13), the arguments of the proof of Lemma 2.1 imply that A λ u n = L F n in V ′ , where L F is defined by L F ( w ) := ( g, w ) H − τ e − λτ ( Z h ( σ ) e λστ dσ, B ∗ w ) H + ( λf + aB ∗ f, w ) H , when F = ( f, g, h ) ⊤ . But it is easy to check that L F n → L F in V ′ . Moreover, as λ ∈ C \ Σ, A λ is an isomorphism from (ker A λ ) ⊥ λ,V into R ( A λ ), hence by(3.15) we deduce that there exists a positive constant C ( λ ) such that k u n − u m k V ≤ C ( λ ) k L F n − L F m k V ′ , ∀ n, m ∈ N . Hence, ( u n ) n is a Cauchy sequence in V , and therefore there exists u ∈ V such that u n → u in V, as well as A λ u = L F in V ′ . Then defining v by (2.13) and z by (2.16), we deduce that U := ( u, v, z ) ⊤ belongs to D ( A ) and ( λI − A ) U = F. In other words, F belongs to R ( λI − A ). The closedness of R ( λI − A ) is thus proved.At this stage, for any λ ∈ C \ Σ, we show thatcodim R ( A λ ) = codim R ( λI − A ) , (3.16)where for W ⊂ H , codim W is the dimension of the orthogonal in H of W , while for W ′ ⊂ V ′ , codim W ′ is the dimension of the annihilator A := { v ∈ V : h v, w i V − V ′ = 0 , ∀ w ∈ W ′ } , W ′ in V .Indeed, let us set N = codim R ( A λ ), then there exist N elements ϕ i ∈ V, i =1 , . . . , N, such that f ∈ R ( A λ ) ⇐⇒ f ∈ V ′ and h f, ϕ i i V ′ − V = 0 , ∀ i = 1 , . . . , N. Consequently, for F ∈ H , if L F (that belongs to V ′ ) satisfies L F ( ϕ i ) = 0 , ∀ i = 1 , . . . , N, (3.17)then there exists a solution u ∈ V of A λ u = L F in V’ , and the arguments of the proof of Lemma 2.1 imply that F is in R ( λI − A ). Hence, the N conditions on F ∈ H from (3.17) allow to show that it belongs to R ( λI − A ), andtherefore codim R ( λI − A ) ≤ N = codim R ( A λ ) . (3.18)This shows that λI − A is a Fredholm operator.Conversely, set M = codim R ( λI −A ), then there exist M elements Ψ i = ( u i , v i , z i ) ∈H , i = 1 , . . . , M, such that F ∈ R ( λI − A ) ⇐⇒ F ∈ H and ( F, Ψ i ) H = 0 , ∀ i = 1 , . . . , M. Then, for any g ∈ H , if( g, v i ) H = ((0 , g, ⊤ , Ψ i ) H = 0 , ∀ i = 1 , . . . , M, (3.19)there exists U = ( u, v, z ) ⊤ ∈ D ( A ) such that( λI − A ) U = (0 , g, , which implies that A λ u = g. This shows that R ( A λ ) ⊃ H , where H := { g ∈ H satisfying (3 . } . This inclusion implies that (here ⊥ means theannihilator of the set in V ) R ( A λ ) ⊥ ⊂ H ⊥ . Therefore R ( A λ ) ⊥ ⊂ { v ∈ V : h v, g i V − V ′ = 0 , ∀ g ∈ H } = { v ∈ V : ( v, g ) H = 0 , ∀ g ∈ H }⊂ Span { v i } Mi =1 ∩ V. Hence, codim R ( A λ ) ≤ M = codim R ( λI − A ) . (3.20)The inequalities (3.18) and (3.20) imply (3.16).11 emma 3.4. If τ ≤ a , then Σ ⊂ { λ ∈ C : ℜ λ < } . Proof.
Let λ = x + iy ∈ Σ, with x, y ∈ R we deduce that ax + e − τx cos( τ y ) = 0 ,ay − e − τx sin( τ y ) = 0 . This corresponds to the system (3.6)–(3.7) with k = ∞ , hence the arguments as in theproof of Lemma 3.1 yield the result. Corollary 3.5.
It holds σ ( A ) = σ pp ( A ) ∪ Σ , and therefore if τ ≤ a σ ( A ) ⊂ { λ ∈ C : ℜ λ < } . Proof.
By Theorem 3.3, C \ Σ ⊂ σ pp ( A ) ∪ ρ ( A ) . The first assertion directly follows.The second assertion follows from Lemmas 3.1 and 3.4.
In this section, we show that if τ ≤ a and ξ > τa , the semigroup e t A decays to the nullsteady state with an exponential decay rate. To obtain this, our technique is based ona frequency domain approach and combines a contradiction argument to carry out aspecial analysis of the resolvent. Theorem 4.1. If ξ > τa and τ ≤ a , then there exist constants C, ω > such that thesemigroup e t A satisfies the following estimate (cid:13)(cid:13) e t A (cid:13)(cid:13) L ( H ) ≤ C e − ωt , ∀ t > . (4.21) Proof of theorem 4.1.
We will employ the following frequency domain theorem for uni-form stability from [15, Thm 8.1.4] of a C semigroup on a Hilbert space: Lemma 4.2. A C semigroup e t L on a Hilbert space H satisfies || e t L || L ( H ) ≤ C e − ωt , for some constant C > and for ω > if and only if ℜ λ < , ∀ λ ∈ σ ( L ) , (4.22) and sup ℜ λ ≥ k ( λI − L ) − k L ( H ) < ∞ . (4.23) where σ ( L ) denotes the spectrum of the operator L . A is fully included into ℜ λ <
0, whichclearly implies (4.22). Then the proof of Theorem 4.1 is based on the following lemmathat shows that (4.23) holds with L = A . Lemma 4.3.
The resolvent operator of A satisfies condition sup ℜ λ ≥ k ( λI − A ) − k L ( H ) < ∞ . (4.24) Proof.
Suppose that condition (4.24) is false. By the Banach-Steinhaus Theorem (see[8]), there exists a sequence of complex numbers λ n such that ℜ λ n ≥ , | λ n | → + ∞ and a sequence of vectors Z n = ( u n , v n , z n ) t ∈ D ( A ) with k Z n k H = 1 (4.25)such that || ( λ n I − A ) Z n || H → n → ∞ , (4.26)i.e., λ n u n − v n ≡ f n → D ( B ∗ ) , (4.27) λ n v n + a B ( B ∗ v n + z n (1)) ≡ g n → H, (4.28) λ n z n + τ − ∂ ρ z n ≡ h n → L ((0 , H ) . (4.29)Our goal is to derive from (4.26) that || Z n || H converges to zero, that furnishes acontradiction.We notice that from (2.10) and (4.27) we have || ( λ n I − A ) Z n || H ≥ |ℜ (( λ n I − A ) Z n , Z n ) H |≥ ℜ λ n − a − ∗ k B ∗ u n k H + (cid:18) ξ τ − a (cid:19) k z n (1) k H + a k B ∗ v n k H = ℜ λ n − a − ∗ (cid:13)(cid:13)(cid:13)(cid:13) B ∗ v n + B ∗ f n λ n (cid:13)(cid:13)(cid:13)(cid:13) H + (cid:18) ξ τ − a (cid:19) k z n (1) k H + a k B ∗ v n k H . Hence using the inequality k B ∗ v n + B ∗ f n k H ≤ k B ∗ v n k H + 2 k B ∗ f n k H , we obtain that || ( λ n I − A ) Z n || H ≥ ℜ λ n − a − ∗ | λ n | − k B ∗ f n k H + (cid:18) ξ τ − a (cid:19) k z n (1) k H +( a − a − ∗ | λ n | − ) k B ∗ v n k H . Hence for n large enough, say n ≥ n ∗ , we can suppose that a − a − ∗ | λ n | − ≥ a . n ≥ n ∗ , we get || ( λ n I − A ) Z n || H ≥ ℜ λ n − a − ∗ | λ n | − k B ∗ f n k H + (cid:18) ξ τ − a (cid:19) k z n (1) k H + a k B ∗ v n k H . By this estimate, (4.26) and (4.27), we deduce that z n (1) → , B ∗ v n → , in H , as n → ∞ , (4.30)and in particular, from the coercivity (1.4), that v n → , in H, as n → ∞ . This implies according to (4.27) that u n = 1 λ n v n + 1 λ n f n → , in D ( B ∗ ) , as n → ∞ , (4.31)as well as z n (0) = B ∗ u n → , in H , as n → ∞ . (4.32)By integration of the identity (4.29), we have z n ( ρ ) = z n (0) e − τλ n ρ + τ Z ρ e − τλ n ( ρ − γ ) h n ( γ ) dγ. (4.33)Hence recalling that ℜ λ n ≥ Z k z n ( ρ ) k H dρ ≤ k z n (0) k H + 2 τ Z Z ρ k h n ( γ ) k H dγρ dρ → , as n → ∞ . All together we have shown that k Z n k H converges to zero, that clearly contradicts k Z n k H = 1.The two hypotheses of Lemma 4.2 are proved, then (4.21) holds. The proof ofTheorem 4.1 is then finished. We study the internal stabilization of a delayed wave equation. More precisely, weconsider the system given by : u tt ( x, t ) − a ∆ u t ( x, t ) − ∆ u ( x, t − τ ) = 0 , in Ω × (0 , + ∞ ) , (5.1) u = 0 , on ∂ Ω × (0 , + ∞ ) , (5.2) u ( x,
0) = u ( x ) , u t ( x,
0) = u ( x ) , in Ω , (5.3) ∇ u ( x, t − τ ) = f ( t − τ ) , in Ω × (0 , τ ) , (5.4)14here Ω is a smooth open bounded domain of R n and a, τ > H = L (Ω) , B = − div : D ( B ) = H (Ω) n → L (Ω) , B ∗ = ∇ : D ( B ∗ ) = H (Ω) → H := L (Ω) n , the assumption (1.4)being satisfied owing to Poincar´e’s inequality. The operator A is then given by A uvz := va ∆ v + div z ( · , − τ − z ρ , with domain D ( A ) := n ( u, v, z ) ⊤ ∈ H (Ω) × H (Ω) × L (Ω; H (0 , a ∇ v + z ( · , ∈ H (Ω) , ∇ u = z ( · ,
0) in Ω o , (5.5)in the Hilbert space H := H (Ω) × L (Ω) × L (Ω × (0 , . (5.6)According to Lemma 3.5 and Theorem 4.1 we have: Corollary 5.1. If τ ≤ a , the system (5.1) – (5.4) is exponentially stable in H , namelyfor ξ > τa , the energy E ( t ) = 12 (cid:18)Z Ω ( |∇ u ( x, t ) | + | u t ( x, t ) | ) dx + ξ Z Ω Z |∇ u ( x, t − τ ρ ) | dxdρ (cid:19) , satisfies E ( t ) ≤ M e − ωt E (0) , ∀ t > , for some positive constants M and ω . Conclusion
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