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

Introduction Feedback stabilization Thermoelastic case

Stability of some string-beam systems

Farhat Shel

Facult´ e des Sciences de Monastir

ContrOpt 2017 15-19 Mai 2017, Monastir, Tunisie

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Outline

1

Introduction

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Outline

1

Introduction

2

Feedback stabilization Abstract setting Asymptotic behavior

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Outline

1

Introduction

2

Feedback stabilization Abstract setting Asymptotic behavior

3

Thermoelastic case Abstract setting Asymptotic behavior

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt − u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

Energy E(t) = ℓ1 |u1,t|2 dx+ ℓ1 |u1,x|2 dx+ ℓ2 |u2,t|2 dx+ ℓ2 |u2,xx|2 dx d dt E(t) = 0. The system is conservative.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1,x(ℓ1, t) = −u1,t(ℓ1, t), u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1,x(ℓ1, t) = −u1,t(ℓ1, t), u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0. The system is exponentially stable.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1(ℓ1, t) = 0, u2,x(ℓ2, t) = 0, u2,xxx(ℓ2, t) = u2,t(ℓ2, t).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. Boundary conditions u1(ℓ1, t) = 0, u2,x(ℓ2, t) = 0, u2,xxx(ℓ2, t) = u2,t(ℓ2, t). The system is polynomially stable.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. . Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• TE. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx + βθx = 0 θt + βutx − κθxx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. . Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• TE. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx + βθx = 0 θt + βutx − κθxx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, θ(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. . Boundary conditions u1(ℓ1, t) = 0, θ(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• TE. String
• E. Beam

ℓ2 ℓ1

u1,tt − u1,xx + βθx = 0 θt + βutx − κθxx = 0 , u2,tt + u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, θ(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = 0. . Boundary conditions u1(ℓ1, t) = 0, θ(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0. The system is exponentially stable.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• TE. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx + βθx = 0 θt + βutxx − κθxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, θ(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = θx(0, t). Boundary conditions u1(ℓ1, t) = 0, θ(ℓ2, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

Models

• E. String
• TE. Beam

ℓ2 ℓ1

u1,tt − u1,xx = 0 , u2,tt + u2,xxxx + βθx = 0 θt + βutxx − κθxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, θ(0, t) = 0, u2,xxx(0, t) − u1,x(0, t) = θx(0, t). Boundary conditions u1(ℓ1, t) = 0, θ(ℓ2, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0. The system is polynomially stable.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

References

Coupled string-beam system Ammari, Jellouli, Mehrenberger, 2009.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

References

Coupled string-beam system Ammari, Jellouli, Mehrenberger, 2009. Chain of beams and strings Ammari et al, 2012.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case

References

Coupled string-beam system Ammari, Jellouli, Mehrenberger, 2009. Chain of beams and strings Ammari et al, 2012. String beams network Ammari, Mehrenberger 2012

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Energy space

• E. String
• E. Beam

ℓ2 ℓ1

L2(G) = L2(0, ℓ1) × L2(0, ℓ2). V =

• f = (f1, f2) ∈ H1(0, ℓ1) × H2(0, ℓ2) | f satisfies (1)
• ,

   f2(ℓ2) = 0, f1(0) = f2(0), ∂xf2(0) = 0. (1)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Energy space

• E. String
• E. Beam

ℓ2 ℓ1

L2(G) = L2(0, ℓ1) × L2(0, ℓ2). V =

• f = (f1, f2) ∈ H1(0, ℓ1) × H2(0, ℓ2) | f satisfies (1)
• ,

   f1(ℓ1) = 0, f1(0) = f2(0), ∂xf2(0) = 0. (1)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Energy space

• E. String
• E. Beam

ℓ2 ℓ1

L2(G) = L2(0, ℓ1) × L2(0, ℓ2). V =

• f = (f1, f2) ∈ H1(0, ℓ1) × H2(0, ℓ2) | f satisfies (1)
• ,

   δf2(ℓ2) = 0, (1 − δ)f1(ℓ1) = 0, f1(0) = f2(0), ∂xf2(0) = 0. (1)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Energy space

• E. String
• E. Beam

ℓ2 ℓ1

L2(G) = L2(0, ℓ1) × L2(0, ℓ2). V =

• f = (f1, f2) ∈ H1(0, ℓ1) × H2(0, ℓ2) | f satisfies (1)
• ,

   δf2(ℓ2) = 0, (1 − δ)f1(ℓ1) = 0, δ ∈ {0, 1} f1(0) = f2(0), ∂xf2(0) = 0. (1)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Energy space

• E. String
• E. Beam

ℓ2 ℓ1

L2(G) = L2(0, ℓ1) × L2(0, ℓ2). V =

• f = (f1, f2) ∈ H1(0, ℓ1) × H2(0, ℓ2) | f satisfies (1)
• ,

   δf2(ℓ2) = 0, (1 − δ)f1(ℓ1) = 0, δ ∈ {0, 1} f1(0) = f2(0), ∂xf2(0) = 0. (1) Energy space: H = V × L2(G), y1, y2H :=

• ∂xf 1

1 , ∂xf 2 1

• +
• ∂2

xf 1 2 , ∂2 xf 2 2

• +
• g1

1 , g2 1

• +
• g1

2 , g2 2

• Farhat Shel

Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Energy space

• E. String
• E. Beam

ℓ2 ℓ1

L2(G) = L2(0, ℓ1) × L2(0, ℓ2). V =

• f = (f1, f2) ∈ H1(0, ℓ1) × H2(0, ℓ2) | f satisfies (1)
• ,

   δf2(ℓ2) = 0, (1 − δ)f1(ℓ1) = 0, δ ∈ {0, 1} f1(0) = f2(0), ∂xf2(0) = 0. (1) Energy space: H = V × L2(G), y1, y2H :=

• ∂xf 1

1 , ∂xf 2 1

• +
• ∂2

xf 1 2 , ∂2 xf 2 2

• +
• g1

1 , g2 1

• +
• g1

2 , g2 2

• Hilbert space.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Evolution equation

Then the system (S) may be rewritten as the first order evolution equation on H, y′(t) = Ay(t), t > 0, y(0) = y0 (2) where y = (u, ut), y0 = (u0, u1).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Evolution equation

Then the system (S) may be rewritten as the first order evolution equation on H, y′(t) = Ay(t), t > 0, y(0) = y0 (2) where y = (u, ut), y0 = (u0, u1). A     u1 u2 v1 v2     =     v1 v2 ∂2

xu1

−∂4

xu2

    .

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Evolution equation

D(A) =

• y = (u, v) ∈ V 2 | u1 ∈ H2(0, ℓ1),u2 ∈ H4(0, ℓ2),y satisfies (3)
• 

                  (1 − δ)∂xu1(ℓ1) = −(1 − δ)v1(ℓ1), (1 − δ)∂2

xu2(ℓ2) = 0,

δ∂3

xu2(ℓ2) = δv2(ℓ2),

δ∂xu2(ℓ2) = 0, ∂xu1(0) − ∂3

xu2(0) = 0.

(3)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Theorem The operator A generates a C0-semigroup S(t) = eAt of contraction on H.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Theorem The operator A generates a C0-semigroup S(t) = eAt of contraction on H. For an initial datum y0 ∈ H there exists a unique solution y ∈ C([0, +∞), H)

• f the Cauchy problem (2).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Theorem The operator A generates a C0-semigroup S(t) = eAt of contraction on H. For an initial datum y0 ∈ H there exists a unique solution y ∈ C([0, +∞), H)

• f the Cauchy problem (2).

Moreover if y0 ∈ D(A), then y ∈ C([0, +∞), D(A)) ∩ C 1([0, +∞), H).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Theorem The operator A generates a C0-semigroup S(t) = eAt of contraction on H. For an initial datum y0 ∈ H there exists a unique solution y ∈ C([0, +∞), H)

• f the Cauchy problem (2).

Moreover if y0 ∈ 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 Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Theorem The operator A generates a C0-semigroup S(t) = eAt of contraction on H. For an initial datum y0 ∈ H there exists a unique solution y ∈ C([0, +∞), H)

• f the Cauchy problem (2).

Moreover if y0 ∈ 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 Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Theorem The operator A generates a C0-semigroup S(t) = eAt of contraction on H. For an initial datum y0 ∈ H there exists a unique solution y ∈ C([0, +∞), H)

• f the Cauchy problem (2).

Moreover if y0 ∈ 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 Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Theorem The operator A generates a C0-semigroup S(t) = eAt of contraction on H. For an initial datum y0 ∈ H there exists a unique solution y ∈ C([0, +∞), H)

• f the Cauchy problem (2).

Moreover if y0 ∈ 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 Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

exponential stability ⇐ ⇒ S(t) = eAt is exponentially stable: S(t)y0 ≤ Ce−wt y0 ∀t > 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

exponential stability ⇐ ⇒ S(t) = eAt is exponentially stable: S(t)y0 ≤ Ce−wt y0 ∀t > 0. Lemma [Gearhard-Pr¨ uss-Huang] A C0-semigroup of contraction etB is exponentially stable if, and

• nly if,

iR = {iβ | β ∈ R} ⊆ ρ(B) (4) and lim sup

|β|→∞

• (iβ − B)−1

< ∞. (5)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

polynomial stability ⇐ ⇒ S(t) = eAt is polynomially stable: S(t)y0 ≤ C tα y0D(A) ∀t > 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

polynomial stability ⇐ ⇒ S(t) = eAt is polynomially stable: S(t)y0 ≤ C tα y0D(A) ∀t > 0. Lemma [Borichev-Tomilov] A C0-semigroup of contraction etB on a Hilbert space H satisfies

• etBy0
• ≤ C

t

1 α

y0D(B) for some constant C > 0 and for α > 0 if, and only if, (4) holds and lim

|β|→∞ sup 1

βα

• (iβ − B)−1

< ∞ (6)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

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 Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

The operator A satisfies condition (4). It suffices to prove that (5)

• holds. Suppose the conclusion is false. Then there exists a

sequense (βn) of real numbers, without loss of generality, with βn − → +∞, and a sequence of vectors (yn) = (un, vn) in D(A) with ynH = 1, such that (iβnI − A)ynH − → 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

The operator A satisfies condition (4). It suffices to prove that (5)

• holds. Suppose the conclusion is false. Then there exists a

sequense (βn) of real numbers, without loss of generality, with βn − → +∞, and a sequence of vectors (yn) = (un, vn) in D(A) with ynH = 1, such that (iβnI − A)ynH − → 0. We prove that this condition yields the contradiction ynH − → 0 as n − → 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

iβnu1,n − v1,n = f1,n − → 0, in H1(0, ℓ1), iβnu2,n − v2,n = f2,n − → 0, in H2(0, ℓ2), iβnv2,n − ∂2

xu2,n

= g2,n − → 0, in L2(0, ℓ1), iβnv2,n + ∂4

xu2,n

= g2,n − → 0, in L2(0, ℓ2).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

iβnu1,n − v1,n = f1,n − → 0, in H1(0, ℓ1), iβnu2,n − v2,n = f2,n − → 0, in H2(0, ℓ2), iβnv2,n − ∂2

xu2,n

= g2,n − → 0, in L2(0, ℓ1), iβnv2,n + ∂4

xu2,n

= g2,n − → 0, in L2(0, ℓ2). Then −β2

nu1,n − ∂2 xu1,n

= g1,n + iβnf1,n, (7) −β2

nu2,n + ∂4 xu2,n

= g2,n + iβnf2,n (8) and vj,n2 − β2

n uj,n2 −

→ 0, j = 1, 2.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

◮ βnu1,n(ℓ1) −

→ 0, ∂xu1,n(ℓ1) − → 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

◮ βnu1,n(ℓ1) −

→ 0, ∂xu1,n(ℓ1) − → 0.

◮ (7) ∗ q∂xu1,n :

∂xu1,n2 + β2

n u1,n2 −

→ 0,

◮ βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −

→ 0,

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

◮ βnu1,n(ℓ1) −

→ 0, ∂xu1,n(ℓ1) − → 0.

◮ (7) ∗ q∂xu1,n :

∂xu1,n2 + β2

n u1,n2 −

→ 0,

◮ βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −

→ 0,

◮ (8) ∗ q∂xu2,n :

− 1

2

• ∂2

xu2,n(ℓ2)

• 2 + 1

2β2 n u2,n2 + 3 2

• ∂2

xu2,n

• 2 → 0,

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

◮ βnu1,n(ℓ1) −

→ 0, ∂xu1,n(ℓ1) − → 0.

◮ (7) ∗ q∂xu1,n :

∂xu1,n2 + β2

n u1,n2 −

→ 0,

◮ βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −

→ 0,

◮ (8) ∗ q∂xu2,n :

− 1

2

• ∂2

xu2,n(ℓ2)

• 2 + 1

2β2 n u2,n2 + 3 2

• ∂2

xu2,n

• 2 → 0,

◮ (8) ∗

1 β1/2

n

e−β1/2

n

x :

∂2

xu2,n(ℓ2) → 0,

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

◮ βnu1,n(ℓ1) −

→ 0, ∂xu1,n(ℓ1) − → 0.

◮ (7) ∗ q∂xu1,n :

∂xu1,n2 + β2

n u1,n2 −

→ 0,

◮ βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −

→ 0,

◮ (8) ∗ q∂xu2,n :

− 1

2

• ∂2

xu2,n(ℓ2)

• 2 + 1

2β2 n u2,n2 + 3 2

• ∂2

xu2,n

• 2 → 0,

◮ (8) ∗

1 β1/2

n

e−β1/2

n

x :

∂2

xu2,n(ℓ2) → 0,

◮

1 2β2

n u2,n2 + 3

2

• ∂2

xu2,n

• 2 → 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Exponential stability

◮ βnu1,n(ℓ1) −

→ 0, ∂xu1,n(ℓ1) − → 0.

◮ (7) ∗ q∂xu1,n :

∂xu1,n2 + β2

n u1,n2 −

→ 0,

◮ βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −

→ 0,

◮ (8) ∗ q∂xu2,n :

− 1

2

• ∂2

xu2,n(ℓ2)

• 2 + 1

2β2 n u2,n2 + 3 2

• ∂2

xu2,n

• 2 → 0,

◮ (8) ∗

1 β1/2

n

e−β1/2

n

x :

∂2

xu2,n(ℓ2) → 0,

◮

1 2β2

n u2,n2 + 3

2

• ∂2

xu2,n

• 2 → 0.

In conclusion yn converge to 0, which contradict the hypothesis that yn = 1.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Polynomial stability

Theorem If no control is applied on the string then, the C0-semigroup is polynomially stable. More precisely, there is C > 0 such that

• etAy0
• ≤ C

t y0D(A) for every y0 ∈ D(A). Proof It suffices to prove that (6) holds for α = 1. Suppose the conclusion is false. There exists a sequence (βn) of real numbers, without loss of generality, with βn − → +∞, and a sequence of vectors (yn) = (un, vn) in D(A) with ynH = 1, such that βα

n (iβnI − A)ynH −

→ 0

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Polynomial stability

Lemma [Gagliardo-Nirenberg] (1) There are two positive constants C1 and C2 such that for any w in H1(0, ℓj), w∞ ≤ C1 ∂xw1/2 w1/2 + C2 w . (9) (2) There are two positive constants C3 and C4 such that for any w in H2(0, ℓj), ∂xw ≤ C3

• ∂2

xw

• 1/2 w1/2 + C4 w .

(10)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Non exponential stability

                   u1,tt − u1,xx = 0 in (0, π) × (0, ∞), u2,tt + u2,xxxx = 0 in (0, π) × (0, ∞), u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) = u1,x(0, t), u1(π, t) = 0, u2,xxx(π, t) = u2,t(π, t), u2,x(π, t) = 0, uj(x, 0) = u0

j (x), uj,t(x, 0) = u1 j (x), j = 1, 2.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Non exponential stability

                   u1,tt − u1,xx = 0 in (0, π) × (0, ∞), u2,tt + u2,xxxx = 0 in (0, π) × (0, ∞), u1(0, t) = u2(0, t), u2,x(0, t) = 0, u2,xxx(0, t) = u1,x(0, t), u1(π, t) = 0, u2,xxx(π, t) = u2,t(π, t), u2,x(π, t) = 0, uj(x, 0) = u0

j (x), uj,t(x, 0) = u1 j (x), j = 1, 2.

The system is not exponentially stable.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

We prove that the corresponding semigroup etA is not exponentially stable. Let

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

We prove that the corresponding semigroup etA is not exponentially stable. Let

◮ βn = n2 + 2√n + 1

n, βn → +∞

◮ fn = (0, 0, − sin βnx, 0), fn is in H and is bounded. ◮ yn = (u1,n, u2,n, v1,n, v2,n) ∈ D(A) such that (A − iβn)yn = fn.

We will prove that yn → +∞.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

We prove that the corresponding semigroup etA is not exponentially stable. Let

◮ βn = n2 + 2√n + 1

n, βn → +∞

◮ fn = (0, 0, − sin βnx, 0), fn is in H and is bounded. ◮ yn = (u1,n, u2,n, v1,n, v2,n) ∈ D(A) such that (A − iβn)yn = fn.

We will prove that yn → +∞.

◮

u1,n = c1 sin(βnx) + (− x 2βn + c2) cos(βnx), u2,n = d1 sin(

• βnx) + d2 cos(
• βnx)

+d3 sinh(

• βnx) + d4 cosh(
• βnx).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

◮

2β3/2

n

d1 ∼+∞ π2 2 √n.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

◮

2β3/2

n

d1 ∼+∞ π2 2 √n.

◮

−π 2

• − 1

2βn + βnc1

• 2

− π 2 |βnc2|2 = −1 2(β2

n

• u1

n

• 2 +
• ∂xu1

n

• 2) + Re(

π sin(βnx)(π − x)∂xu1

ndx).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

◮

2β3/2

n

d1 ∼+∞ π2 2 √n.

◮

−π 2

• − 1

2βn + βnc1

• 2

− π 2 |βnc2|2 = −1 2(β2

n

• u1

n

• 2 +
• ∂xu1

n

• 2) + Re(

π sin(βnx)(π − x)∂xu1

ndx).

β2

n

• u1

n

• 2 +
• ∂xu1

n

• 2 must be not bounded. In conclusion yn is

not bounded.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Remarks

Let ε > 0. By taking βn = n2 + 2n1−α +

1 n2α with 0 < α < ε

and such that n1−α is integer and even and yn is such that fn = (β

1 2 −ε

n

(A − iβn))yn, then we can prove that yn is not bounded and then the polynomial stability of (S) can’t be butter than 1

t2 .

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Remarks

Let ε > 0. By taking βn = n2 + 2n1−α +

1 n2α with 0 < α < ε

and such that n1−α is integer and even and yn is such that fn = (β

1 2 −ε

n

(A − iβn))yn, then we can prove that yn is not bounded and then the polynomial stability of (S) can’t be butter than 1

t2 .

If we replace the boundary conditions by the followings δu1(ℓ1, t) = 0, (1 − δ)u1

xx(ℓ1, t) = 0,

(1 − δ)u1,x(ℓ1, t) = −(1 − δ)u1,t(ℓ1, t), δu2,xx(ℓ2, t) = −δu2,tx(ℓ2, t), u2(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Remarks

Let ε > 0. By taking βn = n2 + 2n1−α +

1 n2α with 0 < α < ε

and such that n1−α is integer and even and yn is such that fn = (β

1 2 −ε

n

(A − iβn))yn, then we can prove that yn is not bounded and then the polynomial stability of (S) can’t be butter than 1

t2 .

If we replace the boundary conditions by the followings δu1(ℓ1, t) = 0, (1 − δ)u1

xx(ℓ1, t) = 0,

(1 − δ)u1,x(ℓ1, t) = −(1 − δ)u1,t(ℓ1, t), δu2,xx(ℓ2, t) = −δu2,tx(ℓ2, t), u2(ℓ2, t) = 0. then we obtain the same results.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

System

• E. String
• E. Beam

ℓ2 ℓ1

u1,tt − α1u1,xx = 0 , u2,tt + α2u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, α2u2,xxx(0, t) = α1u1,x(0, t) , . Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

System

• TE. String
• TE. Beam

ℓ2 ℓ1

u1,tt − α1u1,xx = 0 , u2,tt + α2u2,xxxx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, α2u2,xxx(0, t) = α1u1,x(0, t) , . Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

System

• TE. String
• TE. Beam

ℓ2 ℓ1

u1,tt − α1u1,xx + β1θ1,x = 0 θ1,t + β1u1,tx − κ1θ1,xx = 0 , u2,tt + α2u2,xxxx + β2θ2,x = 0 θ2,t + β2u2,txx − κ2θ2,xx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, α2u2,xxx(0, t) = α1u1,x(0, t) , . Boundary conditions u1(ℓ1, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

System

• TE. String
• TE. Beam

ℓ2 ℓ1

u1,tt − α1u1,xx + β1θ1,x = 0 θ1,t + β1u1,tx − κ1θ1,xx = 0 , u2,tt + α2u2,xxxx + β2θ2,x = 0 θ2,t + β2u2,txx − κ2θ2,xx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, θ1(0, t) = θ2(0, t), α2u2,xxx(0, t) − β2θ2,x(0, t) = α1u1,x(0, t) − β1θ1(0, t), κ1θ1,x(0, t) + κ2θ2,x(0, t) = 0. Boundary conditions u1(ℓ1, t) = 0, θ(ℓ1, t) = 0, θ(ℓ2, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

System

• TE. String
• TE. Beam

ℓ2 ℓ1

u1,tt − α1u1,xx + β1θ1,x = 0 θ1,t + β1u1,tx − κ1θ1,xx = 0 , u2,tt + α2u2,xxxx + β2θ2,x = 0 θ2,t + β2u2,txx − κ2θ2,xx = 0 Transmission conditions u1(0, t) = u2(0, t), u2,x(0, t) = 0, θ1(0, t) = θ2(0, t), α2u2,xxx(0, t) − β2θ2,x(0, t) = α1u1,x(0, t) − β1θ1(0, t), κ1κ2(κ1θ1,x(0, t) + κ2θ2,x(0, t)) = 0. Boundary conditions u1(ℓ1, t) = 0, θ(ℓ1, t) = 0, θ(ℓ2, t) = 0, u2(ℓ2, t) = 0, u2,xx(ℓ2, t) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

System

For a solution (u, v, θ) of (S) the energy is defined as E(t) = 1 2 ℓ1

• |u1,t|2 + α1 |u1,x|2 + |θ1|2

dx +1 2 ℓ2

• |u2,t|2 + α2 |u2,xx|2 + |θ2|2

dx. Differentiate formally the energy function with respect to time t,we get d dt E(t) = −κ1 ∂xθ12 − κ2 ∂xθ22 and the system is dissipative.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Let us consider V =

• f = (f1, f2) ∈ H1(0, ℓ1) × H2(0, ℓ2) | f satisfies (11)
• where

f1(ℓ1) = 0, f2(ℓ2) = 0, f1(0) = f2(0) and ∂xf2(0) = 0. (11) Define the Hilbert space H H = V ×

• L2(0, ℓ1) × L2(0, ℓ2)
• × W

with W = L2(0, ℓ1) × L2(0, ℓ2) if e1 and e2 are thermoelastic, W = L2(0, ℓ1) × {0} if only e1 is thermoelastic and W = {0} × L2(0, ℓ2) if only e1 is purely elastic, and norm given by zH := α1 ∂xf12 + α2

• ∂2

xf2

• 2 +

2

• j=1
• gj2 + hj2

where z = (f = (f1, f2), g = (g1, g2), h = (h1, h2)) .

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

D(A) = y = (u, v, θ) ∈ V ∩ (H2(0, ℓ1) × H4(0, ℓ2)) × V × W2 | and y satisfies (12)

• with W2 = H2(0, ℓ1) × H2(0, ℓ2) if e1 and e2 are T.... and where

       ∂2

xu2(ℓ2) = 0, θ1(ℓ1) = θ2(ℓ2) = 0,

θ1(0) = θ2(0), α2∂3

xu2(0) − β2∂xθ2(0) = α1∂xu1(0) − β1θ1(0),

κ1κ2 (κ1∂xθ1(0) + κ2∂xθ2(0)) = 0. (12)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

D(A) = y = (u, v, θ) ∈ V ∩ (H2(0, ℓ1) × H4(0, ℓ2)) × V × W2 | and y satisfies (12)

• with W2 = H2(0, ℓ1) × H2(0, ℓ2) if e1 and e2 are T.... and where

       ∂2

xu2(ℓ2) = 0, θ1(ℓ1) = θ2(ℓ2) = 0,

θ1(0) = θ2(0), α2∂3

xu2(0) − β2∂xθ2(0) = α1∂xu1(0) − β1θ1(0),

κ1κ2 (κ1∂xθ1(0) + κ2∂xθ2(0)) = 0. (12) with βj = 0 and κj = 0 if ej is purely elastic,

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

D(A) = y = (u, v, θ) ∈ V ∩ (H2(0, ℓ1) × H4(0, ℓ2)) × V × W2 | and y satisfies (12)

• with W2 = H2(0, ℓ1) × H2(0, ℓ2) if e1 and e2 are T.... and where

       ∂2

xu2(ℓ2) = 0, θ1(ℓ1) = θ2(ℓ2) = 0,

θ1(0) = θ2(0), α2∂3

xu2(0) − β2∂xθ2(0) = α1∂xu1(0) − β1θ1(0),

κ1κ2 (κ1∂xθ1(0) + κ2∂xθ2(0)) = 0. (12) with βj = 0 and κj = 0 if ej is purely elastic, and A         u1 u2 v1 v2 θ1 θ2         =         v1 v2 α1∂2

xu1 − β1∂xθ1

−α2∂4

xu2 + β2∂2 xθ2

−β1∂xv1 + κ1∂2

xθ1

−β2∂xxv2 + κ2∂2

xθ2

        .

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Then, putting y = (u, ut, θ), we write the system (S) in the three cases, into the following first order evolution equation

• d

dt y = Ay

y(0) = y0 (13)

• n the energy space H, where y0 = (u0, v0, θ0).

We have the following result, Lemma The operator A is the infinitesimal generator of a C0-semigroup of contraction S(t).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

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 Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

It suffices to prove that (5) holds. Suppose the conclusion is false. Then there exists a sequence (wn) of real numbers, with wn − → +∞ and a sequence of vectors (yn) = (un, vn, θn) in D(A) with ynH = 1, such that (iwnI − A)ynHF − → 0 which is equivalent to iwnu1,n − v1,n = f1,n − → 0, in H1(0, ℓ1), iwnv1,n − α1∂2

xu1,n + β1∂xθ1,n

= g1,n − → 0, in L2(0, ℓ1), iwnθ1,n + β1∂xv1,n − κ1∂2

xθ1,n

= h1,n − → 0, in L2(0, ℓ1), and iw2,nu2,n − v2,n = f2,n − → 0, in H2(0, ℓ2), iwnv2,n + α2∂4

xu2,n

= g2,n − → 0, in L2(0, ℓ2),

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

We get w2

nu1,n + α1∂2 xu1,n − β1∂xθ1,n

= −g1,n − iwnf1,n, (14) −w2

nu2,n + α2∂4 xu2,n

= g2,n + iwnf2,n, (15) and vj,n2 − w2

n uj,n2 −

→ 0, j = 1, 2. First, since Re((iwn − A)yn, ynH) = −κ1 ∂xθ12 we obtain that ∂xθ1,n converges to 0 in L2(0, ℓ2). As in [?] one can get wnu1,n , ∂xu1,n , θ1,n − → 0. Moreover wnu1,n(0), ∂xu1,n(0), θ1,n(0) − → 0. (16)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Taking the inner product of (15) with p = (ℓ2 − x)∂xu2,n(x), −1 2α2

• ∂2

xu2,n(0)

• 2 ℓ2+1

2 ℓ2 w2

n |u2,n|2 dx+3

2α2 ℓ2

• ∂2

xu2,n

• 2 dx → 0

Now the inner product of the first member of (15) by

1 w1/2

n

e−aw1/2

n

x

gives, with a =

1 α1/4

2

, α2 w1/2

n

∂3

xu2,n(0) + α2a∂2 xu2,n(0) = o(1)

then ∂2

xu2,n(0) = o(1)

Return back to(4), ℓ2 w2

n |u2,n|2 dx,

ℓ2

• ∂2

xu2,n

• 2 dx, converge to zero

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Lack of exponential stability

In this part the string is purely elastic.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Lack of exponential stability

In this part the string is purely elastic. We take ℓ1 = ℓ2 = π, κ2 << α2.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Lack of exponential stability

In this part the string is purely elastic. We take ℓ1 = ℓ2 = π, κ2 << α2. 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 Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

We prove that the corresponding semigroup (S(t))t≥0 is not exponentially stable. For n ∈ N, let fn = (0, 0, −α1 sin βnx, 0, 0), with βn → +∞ and fn is in H and is bounded. Let yn = (u1,n, u2,n, v1,n, v2,n, θ2,n) ∈ D(A) such that (A − idn)yn = fn. We will prove that yn → +∞. We have w2

nu1,n + α1∂2 xu1,n = α1 sin βnx

with wn = √α1βn, and iw2,nu2,n − v2,n = 0, in H2(0, π), (17) −w2

nu2,n + α2∂4 xu2,n − β2∂2 xθ2,n

= 0, in L2(0, π), (18) iwnθ2,n + iwnβ2∂2

xu2,n − κ2∂2 xθ2,n

= 0, in L2(0, π). (19) Notations: α2 = α, β2 = β, κ2 = κ.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

The function u1,n is of the form u1,n = c1 sin(wnx) + (− x 2wn + c2) cos(wnx),

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

The function u1,n is of the form u1,n = c1 sin(wnx) + (− x 2wn + c2) cos(wnx), Using (18) and (19) we obtain that ακ∂6

xu2,n − iwn(α + β2)∂4 xu2,n − κw2 n∂2 xu2,n + iw3 nu2,n = 0,

(20) By taking A = 3ακ2 + (α + β2)2, B = 9ακ2(α + β2) + 2(α + β2)3 − 27α2κ2, a1 =

1 21/3

√ B2 + 4A3 + B 1/3 , b1 =

1 21/3

√ B2 + 4A3 − B 1/3 and r = α + β2, the squares of the solutions of (20) are

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

x1 = wn 3ακ √ 3 2 (a1 − a2) + i

• r + 1

2 (a1 − a2)

• x2

= wn 3ακ

• −

√ 3 2 a1 + i

• r + 1

2a1 + a2

• ,

x3 = wn 3ακ √ 3 2 a2 + i

• r − a1 − 1

2a2

• Farhat Shel

Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Let x2, x′2 and x′′2 the squares of the real parts of solutions of (20). 2x2 = 3 4(a1 − a2)2 + (r + 1 2(a1 − a2))2 1/2 + √ 3 2 (a1 − a2), 2x′2 = 3 4(a2

1 + (r + 1

2a1 + a2)2 1/2 − √ 3 2 a1, 2x′′2 = 3 4(a2

2 + (r − a1 − 1

2a2)2 1/2 + √ 3 2 a2. 2x2 > 2x′′2 > 2x′2. The equation (20) admits six simple solutions ±√wnR1, ±√wnR2, ±√wnR3, with 0 < Re(R3) < Re(R2) < Re(R1).

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

u2,n =

3

• k=1

(dke

√wnRkx + bke−√wnRkx).

Return back to (18), β∂2

xθ2,n = w2 n 3

• k=1

(−1 + αR4

k)(dke √wnRkx + bke−√wnRkx)

Then there exist two constants a′ and b′ such that βθ2,n = wn

3

• k=1

(− 1 R2

k

+ αR2

k)(dke √wnRkx + bke−√wnRkx) + a′x + b′.

Moreover, the equation (19) is verified if and only if a′ = b′ = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

The transmission and boundary conditions are expressed as follow

3

• k=1

(dk + bk ) = c2,

3

• k=1

Rk (dk − bk ) = 0, (21) w3/2

n

α

3

• k=1

1 Rk (dk − bk ) = − 1 2wn + wnc1, (22)

3

• k=1

(− 1 R2

k

+ αR2

k )(dk + bk )

= 0,

3

• k=1

(dk e

√wnRk π + bk e−√wnRk π) = 0,

(23)

3

• k=1

R2

k (dk e √wnRk π + bk e−√wnRk π)

= 0,

3

• k=1

1 R2

k

(dk e

√wnRk π + bk e−√wnRk π) = 0,

(24) c1 sin(βnπ) + (− π 2βn + c2) cos(βnπ) = 0. (25) Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

After some calculus

• 2a4e

√wn(R1+2R2)π + ...

• (−

1 2βn + βnc1) = w3/2

n

( π 2βn − c1 tan(βnπ))

• a3e

√wn(R1+2R2)π + ...

• .

and then 2a4(− 1 2βn + βnc1) + a3w3/2

n

c1 tan(βnπ) ∼ π 2 w3/2

n

βn . Hence, with βn = 2n + 1

n, tan(βnπ) = π n + ...

(− 1 2βn + βnc1) ∼ π 4a4 w3/2

n

βn = π√α1 4a4 √wn. The real part of the inner product of (6) with (π − x)∂xu1,n gives

− π 2

• −

1 2wn + wnc1

• 2

− π 2 |wnc2|2 = − 1 2 (w2

n

• u1,n
• 2 +
• ∂x u1,n
• 2) + Re(

π sin(wnx)(π − x)∂x u1,ndx).

In conclusion yn is not bounded.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

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)y0 ≤ 1 tγ y0D(A) .

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Let 1 > α > 1

• 2. It suffices to prove that (5) holds. Suppose the

conclusion is false. Then there exists a sequence (wn) of real numbers, with wn − → +∞ and a sequence of vectors (yn) = (un, vn, θn) in D(A) with ynH = 1, such that wα

n (iwnI − A)ynH −

→ 0 which is equivalent to wα

n (iwnu1,n − v1,n)

= f1,n − → 0, in H1, (26) wα

n

• iwnv1,n − α1∂2

xu1,n

• =

g1,n − → 0 in L2, (27) and wα

n (iwnu2,n − v2,n)

= f2,n − → 0, in H2, (28) wα

n

• iwnv2,n + α2∂4

xu2,n − β2∂2 xθ2,n

• =

g2,n − → 0, in L2, (29) wα

n

• iwnθ2,n + β2∂2

xv2,n − κ2∂2 xθ2,n

• =

h2,n − → 0, in L2. (30)

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Substituting (26) into (27) and (28) into (30) respectively to get wα

n

• w2

nu1,n + α1∂2 xu1,n

• =

−g1,n − iwnf1,n, (31) wα

n

• θ2,n − 1

iwn κ2∂2

xθ2,n + β2∂2 xu2,n

• =

1 iwn (h2,n + ∂2

xf2,n)

(32) First, wα/2

n

∂xθ2,n converge to 0 in L2(0, ℓ2). Then wα/2

n

θ2,n converge to 0 in L2(0, ℓ2) since θ2,n(0) = 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Substituting (26) into (27) and (28) into (30) respectively to get wα

n

• w2

nu1,n + α1∂2 xu1,n

• =

−g1,n − iwnf1,n, (31) wα

n

• θ2,n − 1

iwn κ2∂2

xθ2,n + β2∂2 xu2,n

• =

1 iwn (h2,n + ∂2

xf2,n)

(32) First, wα/2

n

∂xθ2,n converge to 0 in L2(0, ℓ2). Then wα/2

n

θ2,n converge to 0 in L2(0, ℓ2) since θ2,n(0) = 0. Multiplying (32) by

1 wα/2

n

∂2

xu2,n

β2wα/2

n

• ∂2

xu2,n

• 2 + wα/2

n

• θ2,n, ∂2

xu2,n

• (33)

−iκ2wα/2−1∂xθ2,n(0)∂2

xu2,n(0) − iκ2wα/2−1 n

• ∂xθ2,n, ∂3

xu2,n

• = 0.

Then we prove that β2wα/2

n

• ∂2

xu2,n

• 2 −

→ 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Using (29) we prove that wα/8

n

v2,n2 → 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Using (29) we prove that wα/8

n

v2,n2 → 0. We built two sequences of positive numbers rm and sm such that wrm/2

n

• ∂2

xu2,n

• → 0, wrm/2

n

θ2,n → 0, wsm/2

n

v2,n → 0

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Using (29) we prove that wα/8

n

v2,n2 → 0. We built two sequences of positive numbers rm and sm such that wrm/2

n

• ∂2

xu2,n

• → 0, wrm/2

n

θ2,n → 0, wsm/2

n

v2,n → 0 and rm and sm converge to 1 + α.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Using (29) we prove that wα/8

n

v2,n2 → 0. We built two sequences of positive numbers rm and sm such that wrm/2

n

• ∂2

xu2,n

• → 0, wrm/2

n

θ2,n → 0, wsm/2

n

v2,n → 0 and rm and sm converge to 1 + α. α2∂3

xu2,n(0) − β2∂xθ(0) → 0.

wn u2,n(0) → 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Using (29) we prove that wα/8

n

v2,n2 → 0. We built two sequences of positive numbers rm and sm such that wrm/2

n

• ∂2

xu2,n

• → 0, wrm/2

n

θ2,n → 0, wsm/2

n

v2,n → 0 and rm and sm converge to 1 + α. α2∂3

xu2,n(0) − β2∂xθ(0) → 0.

wn u2,n(0) → 0. ℓ2

• |∂xu1,n(x)|2 + w2

n |u1,n(x)|2

dx − → 0.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

Proof

Using (29) we prove that wα/8

n

v2,n2 → 0. We built two sequences of positive numbers rm and sm such that wrm/2

n

• ∂2

xu2,n

• → 0, wrm/2

n

θ2,n → 0, wsm/2

n

v2,n → 0 and rm and sm converge to 1 + α. α2∂3

xu2,n(0) − β2∂xθ(0) → 0.

wn u2,n(0) → 0. ℓ2

• |∂xu1,n(x)|2 + w2

n |u1,n(x)|2

dx − → 0. In summary, we have ynH − → 0. This resul contradicts the hypothesis that yn has the unit norm.

Farhat Shel Stability of some string-beam systems

Introduction Feedback stabilization Thermoelastic case Abstract setting Asymptotic behavior

THANKS!

Farhat Shel Stability of some string-beam systems