Stability of some string-beam systems Farhat Shel Facult e des - PowerPoint PPT Presentation

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Energy space ℓ 1 ℓ 2 0 E. String E. Beam L 2 ( G ) = L 2 (0 , ℓ 1 ) × L 2 (0 , ℓ 2 ) . f = ( f 1 , f 2 ) ∈ H 1 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) | f satisfies (1) � � V = ,  δ f 2 ( ℓ 2 ) = 0 , (1 − δ ) f 1 ( ℓ 1 ) = 0 ,  f 1 (0) = f 2 (0) , (1) ∂ x f 2 (0) = 0 .  Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Energy space ℓ 1 ℓ 2 0 E. String E. Beam L 2 ( G ) = L 2 (0 , ℓ 1 ) × L 2 (0 , ℓ 2 ) . f = ( f 1 , f 2 ) ∈ H 1 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) | f satisfies (1) � � V = ,  δ f 2 ( ℓ 2 ) = 0 , (1 − δ ) f 1 ( ℓ 1 ) = 0 , δ ∈ { 0 , 1 }  f 1 (0) = f 2 (0) , (1) ∂ x f 2 (0) = 0 .  Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Energy space ℓ 1 ℓ 2 0 E. String E. Beam L 2 ( G ) = L 2 (0 , ℓ 1 ) × L 2 (0 , ℓ 2 ) . f = ( f 1 , f 2 ) ∈ H 1 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) | f satisfies (1) � � V = ,  δ f 2 ( ℓ 2 ) = 0 , (1 − δ ) f 1 ( ℓ 1 ) = 0 , δ ∈ { 0 , 1 }  f 1 (0) = f 2 (0) , (1) ∂ x f 2 (0) = 0 .  Energy space: H = V × L 2 ( G ) , ∂ x f 1 1 , ∂ x f 2 ∂ 2 x f 1 2 , ∂ 2 x f 2 g 1 1 , g 2 g 1 2 , g 2 � � � � � � � � � y 1 , y 2 � H := + + + 1 2 1 2 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Energy space ℓ 1 ℓ 2 0 E. String E. Beam L 2 ( G ) = L 2 (0 , ℓ 1 ) × L 2 (0 , ℓ 2 ) . f = ( f 1 , f 2 ) ∈ H 1 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) | f satisfies (1) � � V = ,  δ f 2 ( ℓ 2 ) = 0 , (1 − δ ) f 1 ( ℓ 1 ) = 0 , δ ∈ { 0 , 1 }  f 1 (0) = f 2 (0) , (1) ∂ x f 2 (0) = 0 .  Energy space: H = V × L 2 ( G ) , ∂ x f 1 1 , ∂ x f 2 ∂ 2 x f 1 2 , ∂ 2 x f 2 g 1 1 , g 2 g 1 2 , g 2 � � � � � � � � � y 1 , y 2 � H := + + + 1 2 1 2 Hilbert space. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Evolution equation Then the system ( S ) may be rewritten as the first order evolution equation on H , � y ′ ( t ) = A y ( t ) , t > 0 , (2) y (0) = y 0 where y = ( u , u t ) , y 0 = ( u 0 , u 1 ) . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Evolution equation Then the system ( S ) may be rewritten as the first order evolution equation on H , � y ′ ( t ) = A y ( t ) , t > 0 , (2) y (0) = y 0 where y = ( u , u t ) , y 0 = ( u 0 , u 1 ) .     u 1 v 1 u 2 v 2     A  =  .    ∂ 2  v 1 x u 1   − ∂ 4 v 2 x u 2 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Evolution equation y = ( u , v ) ∈ V 2 | u 1 ∈ H 2 (0 , ℓ 1 ), u 2 ∈ H 4 (0 , ℓ 2 ), y satisfies (3) � � D ( A ) =  (1 − δ ) ∂ x u 1 ( ℓ 1 ) = − (1 − δ ) v 1 ( ℓ 1 ) ,       (1 − δ ) ∂ 2 x u 2 ( ℓ 2 ) = 0,    (3) δ∂ 3 x u 2 ( ℓ 2 ) = δ v 2 ( ℓ 2 ) , δ∂ x u 2 ( ℓ 2 ) = 0 ,         ∂ x u 1 (0) − ∂ 3  x u 2 (0) = 0. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Theorem The operator A generates a C 0 -semigroup S ( t ) = e A t of contraction on H . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Theorem The operator A generates a C 0 -semigroup S ( t ) = e A t of contraction on H . For an initial datum y 0 ∈ H there exists a unique solution y ∈ C ([0 , + ∞ ) , H ) of the Cauchy problem (2). Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Theorem The operator A generates a C 0 -semigroup S ( t ) = e A t of contraction on H . For an initial datum y 0 ∈ H there exists a unique solution y ∈ C ([0 , + ∞ ) , H ) of the Cauchy problem (2). Moreover if y 0 ∈ D ( A ) , then y ∈ C ([0 , + ∞ ) , D ( A )) ∩ C 1 ([0 , + ∞ ) , H ) . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Theorem The operator A generates a C 0 -semigroup S ( t ) = e A t of contraction on H . For an initial datum y 0 ∈ H there exists a unique solution y ∈ C ([0 , + ∞ ) , H ) of the Cauchy problem (2). Moreover if y 0 ∈ D ( A ) , then y ∈ C ([0 , + ∞ ) , D ( A )) ∩ C 1 ([0 , + ∞ ) , H ) . Proof (of the theorem). Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Theorem The operator A generates a C 0 -semigroup S ( t ) = e A t of contraction on H . For an initial datum y 0 ∈ H there exists a unique solution y ∈ C ([0 , + ∞ ) , H ) of the Cauchy problem (2). Moreover if y 0 ∈ D ( A ) , then y ∈ C ([0 , + ∞ ) , D ( A )) ∩ C 1 ([0 , + ∞ ) , H ) . Proof (of the theorem). A is a dissipative operator on H . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Theorem The operator A generates a C 0 -semigroup S ( t ) = e A t of contraction on H . For an initial datum y 0 ∈ H there exists a unique solution y ∈ C ([0 , + ∞ ) , H ) of the Cauchy problem (2). Moreover if y 0 ∈ D ( A ) , then y ∈ C ([0 , + ∞ ) , D ( A )) ∩ C 1 ([0 , + ∞ ) , H ) . Proof (of the theorem). A is a dissipative operator on H . ]0 , + ∞ ) ⊂ ρ ( A ): the resolvent set of A . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Theorem The operator A generates a C 0 -semigroup S ( t ) = e A t of contraction on H . For an initial datum y 0 ∈ H there exists a unique solution y ∈ C ([0 , + ∞ ) , H ) of the Cauchy problem (2). Moreover if y 0 ∈ D ( A ) , then y ∈ C ([0 , + ∞ ) , D ( A )) ∩ C 1 ([0 , + ∞ ) , H ) . Proof (of the theorem). A is a dissipative operator on H . ]0 , + ∞ ) ⊂ ρ ( A ): the resolvent set of A . Conclusion: by Lumer phillips theorem. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case ⇒ S ( t ) = e A t is exponentially stable: exponential stability ⇐ � S ( t ) y 0 � ≤ Ce − wt � y 0 � ∀ t > 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case ⇒ S ( t ) = e A t is exponentially stable: exponential stability ⇐ � S ( t ) y 0 � ≤ Ce − wt � y 0 � ∀ t > 0 . Lemma [Gearhard-Pr¨ uss-Huang] A C 0 -semigroup of contraction e t B is exponentially stable if, and only if, i R = { i β | β ∈ R } ⊆ ρ ( B ) (4) and � ( i β − B ) − 1 � � < ∞ . � lim sup (5) | β |→∞ Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case ⇒ S ( t ) = e A t is polynomially stable: polynomial stability ⇐ � S ( t ) y 0 � ≤ C t α � y 0 � D ( A ) ∀ t > 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case ⇒ S ( t ) = e A t is polynomially stable: polynomial stability ⇐ � S ( t ) y 0 � ≤ C t α � y 0 � D ( A ) ∀ t > 0 . Lemma [Borichev-Tomilov] A C 0 -semigroup of contraction e t B on a Hilbert space H satisfies � ≤ C � � e t B y 0 � � y 0 � D ( B ) 1 t α for some constant C > 0 and for α > 0 if, and only if, (4) holds and | β |→∞ sup 1 � ( i β − B ) − 1 � � < ∞ � lim (6) β α Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability Theorem If the feedback is applied at the exterior end of the string then, the system ( S ) is exponentially stable. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof The operator A satisfies condition (4). It suffices to prove that (5) holds. Suppose the conclusion is false. Then there exists a sequense ( β n ) of real numbers, without loss of generality, with β n − → + ∞ , and a sequence of vectors ( y n ) = ( u n , v n ) in D ( A ) with � y n � H = 1, such that � ( i β n I − A ) y n � H − → 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof The operator A satisfies condition (4). It suffices to prove that (5) holds. Suppose the conclusion is false. Then there exists a sequense ( β n ) of real numbers, without loss of generality, with β n − → + ∞ , and a sequence of vectors ( y n ) = ( u n , v n ) in D ( A ) with � y n � H = 1, such that � ( i β n I − A ) y n � H − → 0 . We prove that this condition yields the contradiction � y n � H − → 0 as n − → 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability in H 1 (0 , ℓ 1 ) , i β n u 1 , n − v 1 , n = f 1 , n − → 0 , in H 2 (0 , ℓ 2 ) , i β n u 2 , n − v 2 , n = f 2 , n − → 0 , i β n v 2 , n − ∂ 2 in L 2 (0 , ℓ 1 ) , x u 2 , n = g 2 , n − → 0 , i β n v 2 , n + ∂ 4 in L 2 (0 , ℓ 2 ) . x u 2 , n = g 2 , n − → 0 , Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability in H 1 (0 , ℓ 1 ) , i β n u 1 , n − v 1 , n = f 1 , n − → 0 , in H 2 (0 , ℓ 2 ) , i β n u 2 , n − v 2 , n = f 2 , n − → 0 , i β n v 2 , n − ∂ 2 in L 2 (0 , ℓ 1 ) , x u 2 , n = g 2 , n − → 0 , i β n v 2 , n + ∂ 4 in L 2 (0 , ℓ 2 ) . x u 2 , n = g 2 , n − → 0 , Then − β 2 n u 1 , n − ∂ 2 x u 1 , n = g 1 , n + i β n f 1 , n , (7) − β 2 n u 2 , n + ∂ 4 x u 2 , n = g 2 , n + i β n f 2 , n (8) and � v j , n � 2 − β 2 n � u j , n � 2 − → 0 , j = 1 , 2 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability ◮ β n u 1 , n ( ℓ 1 ) − → 0 , ∂ x u 1 , n ( ℓ 1 ) − → 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability ◮ β n u 1 , n ( ℓ 1 ) − → 0 , ∂ x u 1 , n ( ℓ 1 ) − → 0 . � ∂ x u 1 , n � 2 + β 2 n � u 1 , n � 2 − ◮ (7) ∗ q ∂ x u 1 , n : → 0 , ◮ β n u 1 , n (0) , ∂ x u 1 , n (0) , Re ( i β n f 1 , n (0) u 1 , n (0)) − → 0 , Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability ◮ β n u 1 , n ( ℓ 1 ) − → 0 , ∂ x u 1 , n ( ℓ 1 ) − → 0 . � ∂ x u 1 , n � 2 + β 2 n � u 1 , n � 2 − ◮ (7) ∗ q ∂ x u 1 , n : → 0 , ◮ β n u 1 , n (0) , ∂ x u 1 , n (0) , Re ( i β n f 1 , n (0) u 1 , n (0)) − → 0 , ◮ (8) ∗ q ∂ x u 2 , n : � 2 + 1 � 2 → 0 , n � u 2 , n � 2 + 3 − 1 � ∂ 2 � � 2 β 2 � � ∂ 2 � x u 2 , n ( ℓ 2 ) x u 2 , n 2 2 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability ◮ β n u 1 , n ( ℓ 1 ) − → 0 , ∂ x u 1 , n ( ℓ 1 ) − → 0 . � ∂ x u 1 , n � 2 + β 2 n � u 1 , n � 2 − ◮ (7) ∗ q ∂ x u 1 , n : → 0 , ◮ β n u 1 , n (0) , ∂ x u 1 , n (0) , Re ( i β n f 1 , n (0) u 1 , n (0)) − → 0 , ◮ (8) ∗ q ∂ x u 2 , n : � 2 + 1 � 2 → 0 , n � u 2 , n � 2 + 3 − 1 � � ∂ 2 � 2 β 2 � ∂ 2 � � x u 2 , n ( ℓ 2 ) x u 2 , n 2 2 e − β 1 / 2 x : 1 ∂ 2 ◮ (8) ∗ x u 2 , n ( ℓ 2 ) → 0 , n β 1 / 2 n Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability ◮ β n u 1 , n ( ℓ 1 ) − → 0 , ∂ x u 1 , n ( ℓ 1 ) − → 0 . � ∂ x u 1 , n � 2 + β 2 n � u 1 , n � 2 − ◮ (7) ∗ q ∂ x u 1 , n : → 0 , ◮ β n u 1 , n (0) , ∂ x u 1 , n (0) , Re ( i β n f 1 , n (0) u 1 , n (0)) − → 0 , ◮ (8) ∗ q ∂ x u 2 , n : � 2 + 1 � 2 → 0 , n � u 2 , n � 2 + 3 − 1 � � ∂ 2 � 2 β 2 � � ∂ 2 � x u 2 , n ( ℓ 2 ) x u 2 , n 2 2 e − β 1 / 2 x : 1 ∂ 2 ◮ (8) ∗ x u 2 , n ( ℓ 2 ) → 0 , n β 1 / 2 n ◮ 1 n � u 2 , n � 2 + 3 � 2 → 0 . 2 β 2 � ∂ 2 � � x u 2 , n 2 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability ◮ β n u 1 , n ( ℓ 1 ) − → 0 , ∂ x u 1 , n ( ℓ 1 ) − → 0 . � ∂ x u 1 , n � 2 + β 2 n � u 1 , n � 2 − ◮ (7) ∗ q ∂ x u 1 , n : → 0 , ◮ β n u 1 , n (0) , ∂ x u 1 , n (0) , Re ( i β n f 1 , n (0) u 1 , n (0)) − → 0 , ◮ (8) ∗ q ∂ x u 2 , n : � 2 + 1 � 2 → 0 , n � u 2 , n � 2 + 3 − 1 � � ∂ 2 � 2 β 2 � ∂ 2 � � x u 2 , n ( ℓ 2 ) x u 2 , n 2 2 e − β 1 / 2 x : 1 ∂ 2 ◮ (8) ∗ x u 2 , n ( ℓ 2 ) → 0 , n β 1 / 2 n ◮ 1 n � u 2 , n � 2 + 3 � 2 → 0 . 2 β 2 � ∂ 2 � � x u 2 , n 2 In conclusion � y n � converge to 0 , which contradict the hypothesis that � y n � = 1 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Polynomial stability Theorem If no control is applied on the string then, the C 0 -semigroup is polynomially stable. More precisely, there is C > 0 such that � ≤ C � � e t A y 0 � t � y 0 � D ( A ) for every y 0 ∈ D ( A ) . Proof It suffices to prove that (6) holds for α = 1. Suppose the conclusion is false. There exists a sequence ( β n ) of real numbers, without loss of generality, with β n − → + ∞ , and a sequence of vectors ( y n ) = ( u n , v n ) in D ( A ) with � y n � H = 1, such that � β α n ( i β n I − A ) y n � H − → 0 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Polynomial stability Lemma [Gagliardo-Nirenberg] (1) There are two positive constants C 1 and C 2 such that for any w in H 1 (0 , ℓ j ) , � w � ∞ ≤ C 1 � ∂ x w � 1 / 2 � w � 1 / 2 + C 2 � w � . (9) (2) There are two positive constants C 3 and C 4 such that for any w in H 2 (0 , ℓ j ) , � 1 / 2 � w � 1 / 2 + C 4 � w � . � � ∂ 2 � � ∂ x w � ≤ C 3 x w (10) Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Non exponential stability  u 1 , tt − u 1 , xx = 0 in (0 , π ) × (0 , ∞ ) ,   u 2 , tt + u 2 , xxxx = 0 in (0 , π ) × (0 , ∞ ),        u 1 (0 , t ) = u 2 (0 , t ) , u 2 , x (0 , t ) = 0 , u 2 , xxx (0 , t ) = u 1 , x (0 , t ) , u 1 ( π, t ) = 0 , u 2 , xxx ( π, t ) = u 2 , t ( π, t ) , u 2 , x ( π, t ) = 0 ,         u j ( x , 0) = u 0 j ( x ) , u j , t ( x , 0) = u 1 j ( x ) , j = 1 , 2 .  Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Non exponential stability  u 1 , tt − u 1 , xx = 0 in (0 , π ) × (0 , ∞ ) ,   u 2 , tt + u 2 , xxxx = 0 in (0 , π ) × (0 , ∞ ),        u 1 (0 , t ) = u 2 (0 , t ) , u 2 , x (0 , t ) = 0 , u 2 , xxx (0 , t ) = u 1 , x (0 , t ) , u 1 ( π, t ) = 0 , u 2 , xxx ( π, t ) = u 2 , t ( π, t ) , u 2 , x ( π, t ) = 0 ,         u j ( x , 0) = u 0 j ( x ) , u j , t ( x , 0) = u 1 j ( x ) , j = 1 , 2 .  The system is not exponentially stable. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof We prove that the corresponding semigroup e t A is not exponentially stable. Let Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof We prove that the corresponding semigroup e t A is not exponentially stable. Let ◮ β n = n 2 + 2 √ n + 1 n , β n → + ∞ ◮ f n = (0 , 0 , − sin β n x , 0) , f n is in H and is bounded. ◮ y n = ( u 1 , n , u 2 , n , v 1 , n , v 2 , n ) ∈ D ( A ) such that ( A − i β n ) y n = f n . We will prove that y n → + ∞ . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof We prove that the corresponding semigroup e t A is not exponentially stable. Let ◮ β n = n 2 + 2 √ n + 1 n , β n → + ∞ ◮ f n = (0 , 0 , − sin β n x , 0) , f n is in H and is bounded. ◮ y n = ( u 1 , n , u 2 , n , v 1 , n , v 2 , n ) ∈ D ( A ) such that ( A − i β n ) y n = f n . We will prove that y n → + ∞ . ◮ u 1 , n = c 1 sin( β n x ) + ( − x + c 2 ) cos( β n x ) , 2 β n � � u 2 , n = d 1 sin( β n x ) + d 2 cos( β n x ) � � + d 3 sinh( β n x ) + d 4 cosh( β n x ) . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof ◮ π 2 √ n . 2 β 3 / 2 d 1 ∼ + ∞ n 2 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof ◮ π 2 √ n . 2 β 3 / 2 d 1 ∼ + ∞ n 2 ◮ 2 � � − 1 � − π − π 2 | β n c 2 | 2 � � + β n c 1 � � 2 2 β n � � π = − 1 � 2 + � 2 ) + Re ( 2( β 2 � u 1 � ∂ x u 1 � � � � sin( β n x )( π − x ) ∂ x u 1 n dx ) . n n n 0 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof ◮ π 2 √ n . 2 β 3 / 2 d 1 ∼ + ∞ n 2 ◮ 2 � − 1 � � − π − π 2 | β n c 2 | 2 � � + β n c 1 � � 2 2 β n � � π = − 1 � 2 + � 2 ) + Re ( 2( β 2 � u 1 � ∂ x u 1 � � � � sin( β n x )( π − x ) ∂ x u 1 n dx ) . n n n 0 � 2 + � 2 must be not bounded. In conclusion y n is β 2 � � u 1 � � � ∂ x u 1 � n n n not bounded. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Remarks Let ε > 0 . By taking β n = n 2 + 2 n 1 − α + 1 n 2 α with 0 < α < ε and such that n 1 − α is integer and even and y n is such that 1 2 − ε f n = ( β ( A − i β n )) y n , then we can prove that y n is not n bounded and then the polynomial stability of ( S ) can’t be butter than 1 t 2 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Remarks Let ε > 0 . By taking β n = n 2 + 2 n 1 − α + 1 n 2 α with 0 < α < ε and such that n 1 − α is integer and even and y n is such that 1 2 − ε f n = ( β ( A − i β n )) y n , then we can prove that y n is not n bounded and then the polynomial stability of ( S ) can’t be butter than 1 t 2 . If we replace the boundary conditions by the followings 0 , (1 − δ ) u 1 δ u 1 ( ℓ 1 , t ) = xx ( ℓ 1 , t ) = 0 , (1 − δ ) u 1 , x ( ℓ 1 , t ) = − (1 − δ ) u 1 , t ( ℓ 1 , t ) , δ u 2 , xx ( ℓ 2 , t ) = − δ u 2 , tx ( ℓ 2 , t ) , u 2 ( ℓ 2 , t ) = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Remarks Let ε > 0 . By taking β n = n 2 + 2 n 1 − α + 1 n 2 α with 0 < α < ε and such that n 1 − α is integer and even and y n is such that 1 2 − ε f n = ( β ( A − i β n )) y n , then we can prove that y n is not n bounded and then the polynomial stability of ( S ) can’t be butter than 1 t 2 . If we replace the boundary conditions by the followings 0 , (1 − δ ) u 1 δ u 1 ( ℓ 1 , t ) = xx ( ℓ 1 , t ) = 0 , (1 − δ ) u 1 , x ( ℓ 1 , t ) = − (1 − δ ) u 1 , t ( ℓ 1 , t ) , δ u 2 , xx ( ℓ 2 , t ) = − δ u 2 , tx ( ℓ 2 , t ) , u 2 ( ℓ 2 , t ) = 0 . then we obtain the same results. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case System ℓ 1 ℓ 2 0 E. String E. Beam u 1 , tt − α 1 u 1 , xx = 0 , u 2 , tt + α 2 u 2 , xxxx = 0 Transmission conditions u 1 (0 , t ) = u 2 (0 , t ) , u 2 , x (0 , t ) = 0 , α 2 u 2 , xxx (0 , t ) = α 1 u 1 , x (0 , t ) , . Boundary conditions u 1 ( ℓ 1 , t ) = 0 , u 2 ( ℓ 2 , t ) = 0 , u 2 , xx ( ℓ 2 , t ) = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case System ℓ 1 ℓ 2 0 TE. String TE. Beam u 1 , tt − α 1 u 1 , xx = 0 , u 2 , tt + α 2 u 2 , xxxx = 0 Transmission conditions u 1 (0 , t ) = u 2 (0 , t ) , u 2 , x (0 , t ) = 0 , α 2 u 2 , xxx (0 , t ) = α 1 u 1 , x (0 , t ) , . Boundary conditions u 1 ( ℓ 1 , t ) = 0 , u 2 ( ℓ 2 , t ) = 0 , u 2 , xx ( ℓ 2 , t ) = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case System ℓ 1 ℓ 2 0 TE. String TE. Beam u 1 , tt − α 1 u 1 , xx + β 1 θ 1 , x = 0 , u 2 , tt + α 2 u 2 , xxxx + β 2 θ 2 , x = 0 θ 1 , t + β 1 u 1 , tx − κ 1 θ 1 , xx = 0 θ 2 , t + β 2 u 2 , txx − κ 2 θ 2 , xx = 0 Transmission conditions u 1 (0 , t ) = u 2 (0 , t ) , u 2 , x (0 , t ) = 0 , α 2 u 2 , xxx (0 , t ) = α 1 u 1 , x (0 , t ) , . Boundary conditions u 1 ( ℓ 1 , t ) = 0 , u 2 ( ℓ 2 , t ) = 0 , u 2 , xx ( ℓ 2 , t ) = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case System ℓ 1 ℓ 2 0 TE. String TE. Beam u 1 , tt − α 1 u 1 , xx + β 1 θ 1 , x = 0 , u 2 , tt + α 2 u 2 , xxxx + β 2 θ 2 , x = 0 θ 1 , t + β 1 u 1 , tx − κ 1 θ 1 , xx = 0 θ 2 , t + β 2 u 2 , txx − κ 2 θ 2 , xx = 0 Transmission conditions u 1 (0 , t ) = u 2 (0 , t ) , u 2 , x (0 , t ) = 0 , θ 1 (0 , t ) = θ 2 (0 , t ) , α 2 u 2 , xxx (0 , t ) − β 2 θ 2 , x (0 , t ) = α 1 u 1 , x (0 , t ) − β 1 θ 1 (0 , t ) , κ 1 θ 1 , x (0 , t ) + κ 2 θ 2 , x (0 , t ) = 0 . Boundary conditions u 1 ( ℓ 1 , t ) = 0 , θ ( ℓ 1 , t ) = 0 , θ ( ℓ 2 , t ) = 0 , u 2 ( ℓ 2 , t ) = 0 , u 2 , xx ( ℓ 2 , t ) = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case System ℓ 1 ℓ 2 0 TE. String TE. Beam u 1 , tt − α 1 u 1 , xx + β 1 θ 1 , x = 0 , u 2 , tt + α 2 u 2 , xxxx + β 2 θ 2 , x = 0 θ 1 , t + β 1 u 1 , tx − κ 1 θ 1 , xx = 0 θ 2 , t + β 2 u 2 , txx − κ 2 θ 2 , xx = 0 Transmission conditions u 1 (0 , t ) = u 2 (0 , t ) , u 2 , x (0 , t ) = 0 , θ 1 (0 , t ) = θ 2 (0 , t ) , α 2 u 2 , xxx (0 , t ) − β 2 θ 2 , x (0 , t ) = α 1 u 1 , x (0 , t ) − β 1 θ 1 (0 , t ) , κ 1 κ 2 ( κ 1 θ 1 , x (0 , t ) + κ 2 θ 2 , x (0 , t )) = 0 . Boundary conditions u 1 ( ℓ 1 , t ) = 0 , θ ( ℓ 1 , t ) = 0 , θ ( ℓ 2 , t ) = 0 , u 2 ( ℓ 2 , t ) = 0 , u 2 , xx ( ℓ 2 , t ) = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case System For a solution ( u , v , θ ) of ( S ) the energy is defined as � ℓ 1 E ( t ) = 1 � | u 1 , t | 2 + α 1 | u 1 , x | 2 + | θ 1 | 2 � dx 2 0 � ℓ 2 +1 � | u 2 , t | 2 + α 2 | u 2 , xx | 2 + | θ 2 | 2 � dx . 2 0 Differentiate formally the energy function with respect to time t , we get d dt E ( t ) = − κ 1 � ∂ x θ 1 � 2 − κ 2 � ∂ x θ 2 � 2 and the system is dissipative. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Let us consider f = ( f 1 , f 2 ) ∈ H 1 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) | f satisfies (11) � � V = where f 1 ( ℓ 1 ) = 0 , f 2 ( ℓ 2 ) = 0 , f 1 (0) = f 2 (0) and ∂ x f 2 (0) = 0 . (11) Define the Hilbert space H L 2 (0 , ℓ 1 ) × L 2 (0 , ℓ 2 ) � � H = V × × W with W = L 2 (0 , ℓ 1 ) × L 2 (0 , ℓ 2 ) if e 1 and e 2 are thermoelastic, W = L 2 (0 , ℓ 1 ) × { 0 } if only e 1 is thermoelastic and W = { 0 } × L 2 (0 , ℓ 2 ) if only e 1 is purely elastic, and norm given by 2 � 2 + � z � H := α 1 � ∂ x f 1 � 2 + α 2 � g j � 2 + � h j � 2 � � � ∂ 2 � � � x f 2 j =1 where z = ( f = ( f 1 , f 2 ) , g = ( g 1 , g 2 ) , h = ( h 1 , h 2 )) . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case � y = ( u , v , θ ) ∈ V ∩ ( H 2 (0 , ℓ 1 ) × H 4 (0 , ℓ 2 )) × V × W 2 | � D ( A ) = and y satisfies (12) with W 2 = H 2 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) if e 1 and e 2 are T.... and where ∂ 2  x u 2 ( ℓ 2 ) = 0 , θ 1 ( ℓ 1 ) = θ 2 ( ℓ 2 ) = 0 ,   θ 1 (0) = θ 2 (0) ,  (12) α 2 ∂ 3 x u 2 (0) − β 2 ∂ x θ 2 (0) = α 1 ∂ x u 1 (0) − β 1 θ 1 (0) ,   κ 1 κ 2 ( κ 1 ∂ x θ 1 (0) + κ 2 ∂ x θ 2 (0)) = 0 .  Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case � y = ( u , v , θ ) ∈ V ∩ ( H 2 (0 , ℓ 1 ) × H 4 (0 , ℓ 2 )) × V × W 2 | � D ( A ) = and y satisfies (12) with W 2 = H 2 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) if e 1 and e 2 are T.... and where ∂ 2  x u 2 ( ℓ 2 ) = 0 , θ 1 ( ℓ 1 ) = θ 2 ( ℓ 2 ) = 0 ,   θ 1 (0) = θ 2 (0) ,  (12) α 2 ∂ 3 x u 2 (0) − β 2 ∂ x θ 2 (0) = α 1 ∂ x u 1 (0) − β 1 θ 1 (0) ,   κ 1 κ 2 ( κ 1 ∂ x θ 1 (0) + κ 2 ∂ x θ 2 (0)) = 0 .  with β j = 0 and κ j = 0 if e j is purely elastic, Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case � y = ( u , v , θ ) ∈ V ∩ ( H 2 (0 , ℓ 1 ) × H 4 (0 , ℓ 2 )) × V × W 2 | � D ( A ) = and y satisfies (12) with W 2 = H 2 (0 , ℓ 1 ) × H 2 (0 , ℓ 2 ) if e 1 and e 2 are T.... and where ∂ 2  x u 2 ( ℓ 2 ) = 0 , θ 1 ( ℓ 1 ) = θ 2 ( ℓ 2 ) = 0 ,   θ 1 (0) = θ 2 (0) ,  (12) α 2 ∂ 3 x u 2 (0) − β 2 ∂ x θ 2 (0) = α 1 ∂ x u 1 (0) − β 1 θ 1 (0) ,   κ 1 κ 2 ( κ 1 ∂ x θ 1 (0) + κ 2 ∂ x θ 2 (0)) = 0 .  with β j = 0 and κ j = 0 if e j is purely elastic, and  u 1   v 1  u 2 v 2     α 1 ∂ 2     v 1 x u 1 − β 1 ∂ x θ 1     A = .    − α 2 ∂ 4 x u 2 + β 2 ∂ 2  v 2 x θ 2        − β 1 ∂ x v 1 + κ 1 ∂ 2  θ 1 x θ 1     − β 2 ∂ xx v 2 + κ 2 ∂ 2 θ 2 x θ 2 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Then, putting y = ( u , u t , θ ) , we write the system ( S ) in the three cases, into the following first order evolution equation d � dt y = A y (13) y (0) = y 0 on the energy space H , where y 0 = ( u 0 , v 0 , θ 0 ) . We have the following result, Lemma The operator A is the infinitesimal generator of a C 0 -semigroup of contraction S ( t ) . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Exponential stability Lemma The semigroup S ( t ) , generated by the operator A is asymptotically stable. Theorem If the string is thermoelastic, then the system ( S ) is exponentially stable. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof It suffices to prove that (5) holds. Suppose the conclusion is false. Then there exists a sequence ( w n ) of real numbers, with w n − → + ∞ and a sequence of vectors ( y n ) = ( u n , v n , θ n ) in D ( A ) with � y n � H = 1, such that � ( i w n I − A ) y n � H F − → 0 which is equivalent to in H 1 (0 , ℓ 1 ) , i w n u 1 , n − v 1 , n = f 1 , n − → 0 , i w n v 1 , n − α 1 ∂ 2 in L 2 (0 , ℓ 1 ) , x u 1 , n + β 1 ∂ x θ 1 , n = g 1 , n − → 0 , i w n θ 1 , n + β 1 ∂ x v 1 , n − κ 1 ∂ 2 in L 2 (0 , ℓ 1 ) , x θ 1 , n = h 1 , n − → 0 , and in H 2 (0 , ℓ 2 ) , i w 2 , n u 2 , n − v 2 , n = f 2 , n − → 0 , i w n v 2 , n + α 2 ∂ 4 in L 2 (0 , ℓ 2 ) , x u 2 , n = g 2 , n − → 0 , Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case We get w 2 n u 1 , n + α 1 ∂ 2 x u 1 , n − β 1 ∂ x θ 1 , n = − g 1 , n − i w n f 1 , n , (14) − w 2 n u 2 , n + α 2 ∂ 4 x u 2 , n = g 2 , n + i w n f 2 , n , (15) and � v j , n � 2 − w 2 n � u j , n � 2 − → 0 , j = 1 , 2 . First, since Re ( � ( i w n − A ) y n , y n � H ) = − κ 1 � ∂ x θ 1 � 2 we obtain that ∂ x θ 1 , n converges to 0 in L 2 (0 , ℓ 2 ). As in [ ? ] one can get � w n u 1 , n � , � ∂ x u 1 , n � , � θ 1 , n � − → 0 . Moreover w n u 1 , n (0) , ∂ x u 1 , n (0) , θ 1 , n (0) − → 0 . (16) Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof Taking the inner product of (15) with p = ( ℓ 2 − x ) ∂ x u 2 , n ( x ) , � ℓ 2 � ℓ 2 − 1 � 2 ℓ 2 +1 n | u 2 , n | 2 dx +3 � 2 dx → 0 � ∂ 2 � � w 2 � � ∂ 2 � 2 α 2 x u 2 , n (0) 2 α 2 x u 2 , n 2 0 0 e − aw 1 / 2 1 x Now the inner product of the first member of (15) by n w 1 / 2 n 1 gives, with a = , α 1 / 4 2 α 2 ∂ 3 x u 2 , n (0) + α 2 a ∂ 2 x u 2 , n (0) = o (1) w 1 / 2 n then ∂ 2 x u 2 , n (0) = o (1) Return back to(4), � ℓ 2 � ℓ 2 � 2 dx , converge to zero n | u 2 , n | 2 dx , w 2 � ∂ 2 � � x u 2 , n 0 0 Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Lack of exponential stability In this part the string is purely elastic . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Lack of exponential stability In this part the string is purely elastic . We take ℓ 1 = ℓ 2 = π , κ 2 << α 2 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Lack of exponential stability In this part the string is purely elastic . We take ℓ 1 = ℓ 2 = π , κ 2 << α 2 . Theorem If the string is purely elastic then the system ( S ) is not exponential stable in the energy space H . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof We prove that the corresponding semigroup ( S ( t )) t ≥ 0 is not exponentially stable. For n ∈ N , let f n = (0 , 0 , − α 1 sin β n x , 0 , 0) , with β n → + ∞ and f n is in H and is bounded. Let y n = ( u 1 , n , u 2 , n , v 1 , n , v 2 , n , θ 2 , n ) ∈ D ( A ) such that ( A − id n ) y n = f n . We will prove that y n → + ∞ . We have w 2 n u 1 , n + α 1 ∂ 2 x u 1 , n = α 1 sin β n x with w n = √ α 1 β n , and in H 2 (0 , π ) , (17) i w 2 , n u 2 , n − v 2 , n = 0 , − w 2 n u 2 , n + α 2 ∂ 4 x u 2 , n − β 2 ∂ 2 in L 2 (0 , π ) , = 0 , (18) x θ 2 , n i w n θ 2 , n + i w n β 2 ∂ 2 x u 2 , n − κ 2 ∂ 2 in L 2 (0 , π ) . x θ 2 , n = 0 , (19) Notations: α 2 = α , β 2 = β , κ 2 = κ . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof The function u 1 , n is of the form u 1 , n = c 1 sin( w n x ) + ( − x + c 2 ) cos( w n x ) , 2 w n Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof The function u 1 , n is of the form u 1 , n = c 1 sin( w n x ) + ( − x + c 2 ) cos( w n x ) , 2 w n Using (18) and (19) we obtain that ακ∂ 6 x u 2 , n − i w n ( α + β 2 ) ∂ 4 x u 2 , n − κ w 2 n ∂ 2 x u 2 , n + i w 3 n u 2 , n = 0 , (20) By taking A = 3 ακ 2 + ( α + β 2 ) 2 , B = 9 ακ 2 ( α + β 2 ) + 2( α + β 2 ) 3 − 27 α 2 κ 2 , � √ � √ � 1 / 3 � 1 / 3 B 2 + 4 A 3 + B B 2 + 4 A 3 − B 1 1 a 1 = , b 1 = 2 1 / 3 2 1 / 3 and r = α + β 2 , the squares of the solutions of (20) are Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof � √ �� w n 3 � r + 1 x 1 = 2 ( a 1 − a 2 ) + i 2 ( a 1 − a 2 ) 3 ακ √ � �� w n 3 � r + 1 x 2 = − 2 a 1 + i 2 a 1 + a 2 , 3 ακ � √ �� � w n 3 r − a 1 − 1 x 3 = 2 a 2 + i 2 a 2 3 ακ Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof Let x 2 , x ′ 2 and x ′′ 2 the squares of the real parts of solutions of (20). √ � 1 / 2 � 3 4( a 1 − a 2 ) 2 + ( r + 1 3 2 x 2 2( a 1 − a 2 )) 2 = + 2 ( a 1 − a 2 ) , √ � 1 / 2 � 3 1 + ( r + 1 3 2 x ′ 2 4( a 2 2 a 1 + a 2 ) 2 = − 2 a 1 , √ � 1 / 2 � 3 2 + ( r − a 1 − 1 3 2 x ′′ 2 4( a 2 2 a 2 ) 2 = + 2 a 2 . 2 x 2 > 2 x ′′ 2 > 2 x ′ 2 . The equation (20) admits six simple solutions ±√ w n R 1 , ±√ w n R 2 , ±√ w n R 3 , with 0 < Re ( R 3 ) < Re ( R 2 ) < Re ( R 1 ) . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof 3 √ w n R k x + b k e −√ w n R k x ) . � u 2 , n = ( d k e k =1 Return back to (18), 3 √ w n R k x + b k e −√ w n R k x ) � β∂ 2 x θ 2 , n = w 2 ( − 1 + α R 4 k )( d k e n k =1 Then there exist two constants a ′ and b ′ such that 3 ( − 1 √ w n R k x + b k e −√ w n R k x ) + a ′ x + b ′ . � + α R 2 βθ 2 , n = w n k )( d k e R 2 k k =1 Moreover, the equation (19) is verified if and only if a ′ = b ′ = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof The transmission and boundary conditions are expressed as follow 3 3 � � ( d k + b k ) = c 2 , R k ( d k − b k ) = 0 , (21) k =1 k =1 3 1 1 w 3 / 2 � α ( d k − b k ) = − + w n c 1 , (22) n R k 2 w n k =1 3 3 1 √ wnRk π + b k e −√ wnRk π ) = 0 , � + α R 2 � ( − k )( d k + b k ) = 0 , ( d k e (23) R 2 k =1 k =1 k 3 3 1 √ wnRk π + b k e −√ wnRk π ) √ wnRk π + b k e −√ wnRk π ) = 0 , R 2 � � k ( d k e = 0 , ( d k e (24) R 2 k =1 k =1 k π c 1 sin( β n π ) + ( − + c 2 ) cos( β n π ) = 0 . (25) 2 β n Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof After some calculus 1 π √ wn ( R 1+2 R 2) π + ... √ wn ( R 1+2 R 2) π + ... � � + β n c 1 ) = w 3 / 2 � � 2 a 4 e ( − ( − c 1 tan( β n π )) a 3 e . n 2 β n 2 β n and then w 3 / 2 2 a 4 ( − 1 c 1 tan( β n π ) ∼ π + β n c 1 ) + a 3 w 3 / 2 n . n 2 β n 2 β n Hence, with β n = 2 n + 1 n , tan( β n π ) = π n + ... = π √ α 1 w 3 / 2 ( − 1 √ w n . + β n c 1 ) ∼ π n 2 β n 4 a 4 4 a 4 β n The real part of the inner product of (6) with ( π − x ) ∂ x u 1 , n gives � π π 1 2 π 1 � � | w n c 2 | 2 = − � 2 + ( w 2 � 2 ) + Re ( � � � � � � − � − + w n c 1 − � u 1 , n � ∂ x u 1 , n sin( w n x )( π − x ) ∂ x u 1 , n dx ) . � � n 2 2 w n 2 2 � 0 In conclusion y n is not bounded. Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Polynomial stability Theorem If the string is purely elastic, then the system ( S ) is polynomially stable. More precisely, (for every γ < 2 ) there exists c > 0 such that � S ( t ) y 0 � ≤ 1 t γ � y 0 � D ( A ) . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof Let 1 > α > 1 2 . It suffices to prove that (5) holds. Suppose the conclusion is false. Then there exists a sequence ( w n ) of real numbers, with w n − → + ∞ and a sequence of vectors ( y n ) = ( u n , v n , θ n ) in D ( A ) with � y n � H = 1, such that � w α n ( i w n I − A ) y n � H − → 0 which is equivalent to w α in H 1 , n ( i w n u 1 , n − v 1 , n ) = f 1 , n − → 0 , (26) w α i w n v 1 , n − α 1 ∂ 2 in L 2 , � � x u 1 , n = g 1 , n − → 0 (27) n and w α in H 2 , n ( i w n u 2 , n − v 2 , n ) = f 2 , n − → 0 , (28) w α i w n v 2 , n + α 2 ∂ 4 x u 2 , n − β 2 ∂ 2 in L 2 , � � x θ 2 , n = g 2 , n − → 0 , (29) n w α i w n θ 2 , n + β 2 ∂ 2 x v 2 , n − κ 2 ∂ 2 in L 2 . � � = h 2 , n − → 0 , (30) x θ 2 , n n Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Substituting (26) into (27) and (28) into (30) respectively to get w α w 2 n u 1 , n + α 1 ∂ 2 � � x u 1 , n = − g 1 , n − i w n f 1 , n , (31) n � θ 2 , n − 1 � 1 w α κ 2 ∂ 2 x θ 2 , n + β 2 ∂ 2 ( h 2 , n + ∂ 2 x u 2 , n = x f 2 , n ) (32) n i w n i w n First, w α/ 2 ∂ x θ 2 , n converge to 0 in L 2 (0 , ℓ 2 ). Then w α/ 2 θ 2 , n n n converge to 0 in L 2 (0 , ℓ 2 ) since θ 2 , n (0) = 0 . Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Substituting (26) into (27) and (28) into (30) respectively to get w α w 2 n u 1 , n + α 1 ∂ 2 � � x u 1 , n = − g 1 , n − i w n f 1 , n , (31) n � θ 2 , n − 1 � 1 w α κ 2 ∂ 2 x θ 2 , n + β 2 ∂ 2 ( h 2 , n + ∂ 2 x u 2 , n = x f 2 , n ) (32) n i w n i w n First, w α/ 2 ∂ x θ 2 , n converge to 0 in L 2 (0 , ℓ 2 ). Then w α/ 2 θ 2 , n n n converge to 0 in L 2 (0 , ℓ 2 ) since θ 2 , n (0) = 0 . 1 ∂ 2 Multiplying (32) by x u 2 , n w α/ 2 n � 2 + w α/ 2 β 2 w α/ 2 � ∂ 2 θ 2 , n , ∂ 2 � � � � x u 2 , n x u 2 , n (33) n n x u 2 , n (0) − i κ 2 w α/ 2 − 1 − i κ 2 w α/ 2 − 1 ∂ x θ 2 , n (0) ∂ 2 ∂ x θ 2 , n , ∂ 3 � � x u 2 , n = 0 . n Then we prove that � 2 − β 2 w α/ 2 � ∂ 2 � � x u 2 , n → 0 . n Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof � v 2 , n � 2 → 0 . Using (29) we prove that w α/ 8 n Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof � v 2 , n � 2 → 0 . Using (29) we prove that w α/ 8 n We built two sequences of positive numbers r m and s m such that w r m / 2 � → 0 , w r m / 2 � θ 2 , n � → 0 , w s m / 2 � � ∂ 2 � x u 2 , n � v 2 , n � → 0 n n n Farhat Shel Stability of some string-beam systems

Introduction Abstract setting Feedback stabilization Asymptotic behavior Thermoelastic case Proof � v 2 , n � 2 → 0 . Using (29) we prove that w α/ 8 n We built two sequences of positive numbers r m and s m such that w r m / 2 � → 0 , w r m / 2 � θ 2 , n � → 0 , w s m / 2 � � ∂ 2 � x u 2 , n � v 2 , n � → 0 n n n and r m and s m converge to 1 + α . Farhat Shel Stability of some string-beam systems