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Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization...
Transcript of Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization...
![Page 1: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/1.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Stability of some string-beam systems
Farhat Shel
Faculte des Sciences de Monastir
ContrOpt 2017
15-19 Mai 2017, Monastir, Tunisie
Farhat Shel Stability of some string-beam systems
![Page 2: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/2.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Farhat Shel Stability of some string-beam systems
![Page 3: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/3.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Outline
1 Introduction
2 Feedback stabilizationAbstract settingAsymptotic behavior
3 Thermoelastic caseAbstract settingAsymptotic behavior
Farhat Shel Stability of some string-beam systems
![Page 4: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/4.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Outline
1 Introduction
2 Feedback stabilizationAbstract settingAsymptotic behavior
3 Thermoelastic caseAbstract settingAsymptotic behavior
Farhat Shel Stability of some string-beam systems
![Page 5: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/5.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Outline
1 Introduction
2 Feedback stabilizationAbstract settingAsymptotic behavior
3 Thermoelastic caseAbstract settingAsymptotic behavior
Farhat Shel Stability of some string-beam systems
![Page 6: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/6.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
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(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
Farhat Shel Stability of some string-beam systems
![Page 7: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/7.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
Energy
E (t) =
∫ `1
0|u1,t |2 dx+
∫ `1
0|u1,x |2 dx+
∫ `2
0|u2,t |2 dx+
∫ `2
0|u2,xx |2 dx
d
dtE (t) = 0.
The system is conservative.
Farhat Shel Stability of some string-beam systems
![Page 8: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/8.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βvtx − κθxx = 0
, u2,tt + u2,xxxx
+ βθx
= 0
θt + βvtx − κθxx = 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(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 9: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/9.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βvtx − κθxx = 0
, u2,tt + u2,xxxx
+ βθx
= 0
θt + βvtx − κθxx = 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(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 10: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/10.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βvtx − κθxx = 0
, u2,tt + u2,xxxx
+ βθx
= 0
θt + βvtx − κθxx = 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(`1, t) = −u1,t(`1, t),
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 11: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/11.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βvtx − κθxx = 0
, u2,tt + u2,xxxx
+ βθx
= 0
θt + βvtx − κθxx = 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(`1, t) = −u1,t(`1, t),
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is exponentially stable.Farhat Shel Stability of some string-beam systems
![Page 12: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/12.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βvtx − κθxx = 0
, u2,tt + u2,xxxx
+ βθx
= 0
θt + βvtx − κθxx = 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(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 13: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/13.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βvtx − κθxx = 0
, u2,tt + u2,xxxx
+ βθx
= 0
θt + βvtx − κθxx = 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(`1, t) = 0,
u2,x(`2, t) = 0, u2,xxx(`2, t) = u2,t(`2, t).
The system is
Farhat Shel Stability of some string-beam systems
![Page 14: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/14.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βvtx − κθxx = 0
, u2,tt + u2,xxxx
+ βθx
= 0
θt + βvtx − κθxx = 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(`1, t) = 0,
u2,x(`2, t) = 0, u2,xxx(`2, t) = u2,t(`2, t).
The system is polynomially stable.Farhat Shel Stability of some string-beam systems
![Page 15: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/15.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βutx − κθxx = 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) = 0.
θx(0, t)
.
Boundary conditions
u1(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 16: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/16.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
TE. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx + βθx = 0θt + βutx − κθxx = 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) = 0.
θx(0, t)
.
Boundary conditions
u1(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 17: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/17.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
TE. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx + βθx = 0θt + βutx − κθxx = 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) = 0.
θx(0, t)
.
Boundary conditions
u1(`1, t) = 0, θ(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 18: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/18.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
TE. String E. Beamℓ2ℓ1 0
u1,tt − u1,xx + βθx = 0θt + βutx − κθxx = 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) = 0.
θx(0, t)
.
Boundary conditions
u1(`1, t) = 0, θ(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is exponentially stable.Farhat Shel Stability of some string-beam systems
![Page 19: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/19.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String TE. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βutx − κθxx = 0
, u2,tt + u2,xxxx + βθx = 0θt + 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(`1, t) = 0, θ(`2, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is
Farhat Shel Stability of some string-beam systems
![Page 20: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/20.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Models
E. String TE. Beamℓ2ℓ1 0
u1,tt − u1,xx
+ βθx
= 0
θt + βutx − κθxx = 0
, u2,tt + u2,xxxx + βθx = 0θt + 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(`1, t) = 0, θ(`2, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
The system is polynomially stable.Farhat Shel Stability of some string-beam systems
![Page 21: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/21.jpg)
IntroductionFeedback stabilization
Thermoelastic case
References
Coupled string-beam systemAmmari, Jellouli, Mehrenberger, 2009.
Chain of beams and stringsAmmari et al, 2012.
String beams networkAmmari, Mehrenberger 2012
Farhat Shel Stability of some string-beam systems
![Page 22: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/22.jpg)
IntroductionFeedback stabilization
Thermoelastic case
References
Coupled string-beam systemAmmari, Jellouli, Mehrenberger, 2009.
Chain of beams and stringsAmmari et al, 2012.
String beams networkAmmari, Mehrenberger 2012
Farhat Shel Stability of some string-beam systems
![Page 23: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/23.jpg)
IntroductionFeedback stabilization
Thermoelastic case
References
Coupled string-beam systemAmmari, Jellouli, Mehrenberger, 2009.
Chain of beams and stringsAmmari et al, 2012.
String beams networkAmmari, Mehrenberger 2012
Farhat Shel Stability of some string-beam systems
![Page 24: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/24.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Energy space
E. String E. Beamℓ2ℓ1 0
L2(G) = L2(0, `1)× L2(0, `2).
V ={
f = (f1, f2) ∈ H1(0, `1)× H2(0, `2) | f satisfies (1)},
f2(`2) = 0,
δ ∈ {0, 1}
f1(0) = f2(0),∂x f2(0) = 0.
(1)
Energy space:H = V × L2(G),
〈y1, y2〉H :=⟨∂x f 1
1 , ∂x f 21
⟩+⟨∂2x f 1
2 , ∂2x f 2
2
⟩+⟨g 1
1 , g21
⟩+⟨g 1
2 , g22
⟩Hilbert space.
Farhat Shel Stability of some string-beam systems
![Page 25: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/25.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Energy space
E. String E. Beamℓ2ℓ1 0
L2(G) = L2(0, `1)× L2(0, `2).
V ={
f = (f1, f2) ∈ H1(0, `1)× H2(0, `2) | f satisfies (1)},
f1(`1) = 0,
δ ∈ {0, 1}
f1(0) = f2(0),∂x f2(0) = 0.
(1)
Energy space:H = V × L2(G),
〈y1, y2〉H :=⟨∂x f 1
1 , ∂x f 21
⟩+⟨∂2x f 1
2 , ∂2x f 2
2
⟩+⟨g 1
1 , g21
⟩+⟨g 1
2 , g22
⟩Hilbert space.
Farhat Shel Stability of some string-beam systems
![Page 26: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/26.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Energy space
E. String E. Beamℓ2ℓ1 0
L2(G) = L2(0, `1)× L2(0, `2).
V ={
f = (f1, f2) ∈ H1(0, `1)× H2(0, `2) | f satisfies (1)},
δf2(`2) = 0, (1− δ)f1(`1) = 0,
δ ∈ {0, 1}
f1(0) = f2(0),∂x f2(0) = 0.
(1)
Energy space:H = V × L2(G),
〈y1, y2〉H :=⟨∂x f 1
1 , ∂x f 21
⟩+⟨∂2x f 1
2 , ∂2x f 2
2
⟩+⟨g 1
1 , g21
⟩+⟨g 1
2 , g22
⟩Hilbert space.
Farhat Shel Stability of some string-beam systems
![Page 27: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/27.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Energy space
E. String E. Beamℓ2ℓ1 0
L2(G) = L2(0, `1)× L2(0, `2).
V ={
f = (f1, f2) ∈ H1(0, `1)× H2(0, `2) | f satisfies (1)},
δf2(`2) = 0, (1− δ)f1(`1) = 0, δ ∈ {0, 1}f1(0) = f2(0),∂x f2(0) = 0.
(1)
Energy space:H = V × L2(G),
〈y1, y2〉H :=⟨∂x f 1
1 , ∂x f 21
⟩+⟨∂2x f 1
2 , ∂2x f 2
2
⟩+⟨g 1
1 , g21
⟩+⟨g 1
2 , g22
⟩Hilbert space.
Farhat Shel Stability of some string-beam systems
![Page 28: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/28.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Energy space
E. String E. Beamℓ2ℓ1 0
L2(G) = L2(0, `1)× L2(0, `2).
V ={
f = (f1, f2) ∈ H1(0, `1)× H2(0, `2) | f satisfies (1)},
δf2(`2) = 0, (1− δ)f1(`1) = 0, δ ∈ {0, 1}f1(0) = f2(0),∂x f2(0) = 0.
(1)
Energy space:H = V × L2(G),
〈y1, y2〉H :=⟨∂x f 1
1 , ∂x f 21
⟩+⟨∂2x f 1
2 , ∂2x f 2
2
⟩+⟨g 1
1 , g21
⟩+⟨g 1
2 , g22
⟩
Hilbert space.
Farhat Shel Stability of some string-beam systems
![Page 29: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/29.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Energy space
E. String E. Beamℓ2ℓ1 0
L2(G) = L2(0, `1)× L2(0, `2).
V ={
f = (f1, f2) ∈ H1(0, `1)× H2(0, `2) | f satisfies (1)},
δf2(`2) = 0, (1− δ)f1(`1) = 0, δ ∈ {0, 1}f1(0) = f2(0),∂x f2(0) = 0.
(1)
Energy space:H = V × L2(G),
〈y1, y2〉H :=⟨∂x f 1
1 , ∂x f 21
⟩+⟨∂2x f 1
2 , ∂2x f 2
2
⟩+⟨g 1
1 , g21
⟩+⟨g 1
2 , g22
⟩Hilbert space.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Evolution equation
Then the system (S) may be rewritten as the first order evolutionequation 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
∂2xu1
−∂4xu2
.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Evolution equation
Then the system (S) may be rewritten as the first order evolutionequation 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
∂2xu1
−∂4xu2
.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Evolution equation
D(A) ={
y = (u, v) ∈ V 2 | u1 ∈ H2(0, `1),u2 ∈ H4(0, `2),y satisfies (3)}
(1− δ)∂xu1(`1) = −(1− δ)v1(`1),
(1− δ)∂2xu2(`2) = 0,
δ∂3xu2(`2) = δv2(`2), δ∂xu2(`2) = 0,
∂xu1(0)− ∂3xu2(0) = 0.
(3)
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Theorem
The operator A generates a C0-semigroup S(t) = eAt ofcontraction on H.
For an initial datum y0 ∈ H there exists a unique solution
y ∈ C ([0,+∞),H)
of 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
![Page 34: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/34.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Theorem
The operator A generates a C0-semigroup S(t) = eAt ofcontraction on H.
For an initial datum y0 ∈ H there exists a unique solution
y ∈ C ([0,+∞),H)
of 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
![Page 35: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/35.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Theorem
The operator A generates a C0-semigroup S(t) = eAt ofcontraction on H.
For an initial datum y0 ∈ H there exists a unique solution
y ∈ C ([0,+∞),H)
of 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
![Page 36: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/36.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Theorem
The operator A generates a C0-semigroup S(t) = eAt ofcontraction on H.
For an initial datum y0 ∈ H there exists a unique solution
y ∈ C ([0,+∞),H)
of 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
![Page 37: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/37.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Theorem
The operator A generates a C0-semigroup S(t) = eAt ofcontraction on H.
For an initial datum y0 ∈ H there exists a unique solution
y ∈ C ([0,+∞),H)
of 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
![Page 38: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/38.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Theorem
The operator A generates a C0-semigroup S(t) = eAt ofcontraction on H.
For an initial datum y0 ∈ H there exists a unique solution
y ∈ C ([0,+∞),H)
of 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
![Page 39: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/39.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Theorem
The operator A generates a C0-semigroup S(t) = eAt ofcontraction on H.
For an initial datum y0 ∈ H there exists a unique solution
y ∈ C ([0,+∞),H)
of 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
![Page 40: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/40.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
exponential stability ⇐⇒ S(t) = eAt is exponentially stable:
‖S(t)y0‖ ≤ Ce−wt ‖y0‖ ∀t > 0.
Lemma [Gearhard-Pruss-Huang]
A C0-semigroup of contraction etB is exponentially stable if, andonly if,
iR = {iβ | β ∈ R} ⊆ ρ(B) (4)
andlim sup|β|→∞
∥∥(iβ − B)−1∥∥ <∞. (5)
Farhat Shel Stability of some string-beam systems
![Page 41: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/41.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
exponential stability ⇐⇒ S(t) = eAt is exponentially stable:
‖S(t)y0‖ ≤ Ce−wt ‖y0‖ ∀t > 0.
Lemma [Gearhard-Pruss-Huang]
A C0-semigroup of contraction etB is exponentially stable if, andonly if,
iR = {iβ | β ∈ R} ⊆ ρ(B) (4)
andlim sup|β|→∞
∥∥(iβ − B)−1∥∥ <∞. (5)
Farhat Shel Stability of some string-beam systems
![Page 42: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/42.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
polynomial stability ⇐⇒ S(t) = eAt is polynomially stable:
‖S(t)y0‖ ≤C
tα‖y0‖D(A) ∀t > 0.
Lemma [Borichev-Tomilov]
A C0-semigroup of contraction etB on a Hilbert space H satisfies∥∥etBy0
∥∥ ≤ C
t1α
‖y0‖D(B)
for some constant C > 0 and for α > 0 if, and only if, (4) holdsand
lim|β|→∞
sup1
βα∥∥(iβ − B)−1
∥∥ <∞ (6)
Farhat Shel Stability of some string-beam systems
![Page 43: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/43.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
polynomial stability ⇐⇒ S(t) = eAt is polynomially stable:
‖S(t)y0‖ ≤C
tα‖y0‖D(A) ∀t > 0.
Lemma [Borichev-Tomilov]
A C0-semigroup of contraction etB on a Hilbert space H satisfies∥∥etBy0
∥∥ ≤ C
t1α
‖y0‖D(B)
for some constant C > 0 and for α > 0 if, and only if, (4) holdsand
lim|β|→∞
sup1
βα∥∥(iβ − B)−1
∥∥ <∞ (6)
Farhat Shel Stability of some string-beam systems
![Page 44: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/44.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
Theorem
If the feedback is applied at the exterior end of the string then, thesystem (S) is exponentially stable.
Farhat Shel Stability of some string-beam systems
![Page 45: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/45.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
The operator A satisfies condition (4). It suffices to prove that (5)holds. Suppose the conclusion is false. Then there exists asequense (βn) of real numbers, without loss of generality, withβn −→ +∞, and a sequence of vectors (yn) = (un, vn) in D(A)with ‖yn‖H = 1, such that
‖(iβnI −A)yn‖H −→ 0.
We prove that this condition yields the contradiction ‖yn‖H −→ 0as n −→ 0.
Farhat Shel Stability of some string-beam systems
![Page 46: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/46.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
The operator A satisfies condition (4). It suffices to prove that (5)holds. Suppose the conclusion is false. Then there exists asequense (βn) of real numbers, without loss of generality, withβn −→ +∞, and a sequence of vectors (yn) = (un, vn) in D(A)with ‖yn‖H = 1, such that
‖(iβnI −A)yn‖H −→ 0.
We prove that this condition yields the contradiction ‖yn‖H −→ 0as n −→ 0.
Farhat Shel Stability of some string-beam systems
![Page 47: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/47.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
iβnu1,n − v1,n = f1,n −→ 0, in H1(0, `1),
iβnu2,n − v2,n = f2,n −→ 0, in H2(0, `2),
iβnv2,n − ∂2xu2,n = g2,n −→ 0, in L2(0, `1),
iβnv2,n + ∂4xu2,n = g2,n −→ 0, in L2(0, `2).
Then
−β2nu1,n − ∂2
xu1,n = g1,n + iβnf1,n, (7)
−β2nu2,n + ∂4
xu2,n = g2,n + iβnf2,n (8)
and‖vj ,n‖2 − β2
n ‖uj ,n‖2 −→ 0, j = 1, 2.
Farhat Shel Stability of some string-beam systems
![Page 48: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/48.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
iβnu1,n − v1,n = f1,n −→ 0, in H1(0, `1),
iβnu2,n − v2,n = f2,n −→ 0, in H2(0, `2),
iβnv2,n − ∂2xu2,n = g2,n −→ 0, in L2(0, `1),
iβnv2,n + ∂4xu2,n = g2,n −→ 0, in L2(0, `2).
Then
−β2nu1,n − ∂2
xu1,n = g1,n + iβnf1,n, (7)
−β2nu2,n + ∂4
xu2,n = g2,n + iβnf2,n (8)
and‖vj ,n‖2 − β2
n ‖uj ,n‖2 −→ 0, j = 1, 2.
Farhat Shel Stability of some string-beam systems
![Page 49: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/49.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
I βnu1,n(`1) −→ 0, ∂xu1,n(`1) −→ 0.
I (7) ∗ q∂xu1,n : ‖∂xu1,n‖2 + β2n ‖u1,n‖2 −→ 0,
I βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −→ 0,
I (8) ∗ q∂xu2,n :
−12
∣∣∂2xu2,n(`2)
∣∣2 + 12β
2n ‖u2,n‖2 + 3
2
∥∥∂2xu2,n
∥∥2 → 0,
I (8) ∗ 1
β1/2n
e−β1/2n x : ∂2
xu2,n(`2)→ 0,
I1
2β2n ‖u2,n‖2 +
3
2
∥∥∂2xu2,n
∥∥2 → 0.
In conclusion ‖yn‖ converge to 0, which contradict the hypothesisthat ‖yn‖ = 1.
Farhat Shel Stability of some string-beam systems
![Page 50: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/50.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
I βnu1,n(`1) −→ 0, ∂xu1,n(`1) −→ 0.
I (7) ∗ q∂xu1,n : ‖∂xu1,n‖2 + β2n ‖u1,n‖2 −→ 0,
I βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −→ 0,
I (8) ∗ q∂xu2,n :
−12
∣∣∂2xu2,n(`2)
∣∣2 + 12β
2n ‖u2,n‖2 + 3
2
∥∥∂2xu2,n
∥∥2 → 0,
I (8) ∗ 1
β1/2n
e−β1/2n x : ∂2
xu2,n(`2)→ 0,
I1
2β2n ‖u2,n‖2 +
3
2
∥∥∂2xu2,n
∥∥2 → 0.
In conclusion ‖yn‖ converge to 0, which contradict the hypothesisthat ‖yn‖ = 1.
Farhat Shel Stability of some string-beam systems
![Page 51: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/51.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
I βnu1,n(`1) −→ 0, ∂xu1,n(`1) −→ 0.
I (7) ∗ q∂xu1,n : ‖∂xu1,n‖2 + β2n ‖u1,n‖2 −→ 0,
I βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −→ 0,
I (8) ∗ q∂xu2,n :
−12
∣∣∂2xu2,n(`2)
∣∣2 + 12β
2n ‖u2,n‖2 + 3
2
∥∥∂2xu2,n
∥∥2 → 0,
I (8) ∗ 1
β1/2n
e−β1/2n x : ∂2
xu2,n(`2)→ 0,
I1
2β2n ‖u2,n‖2 +
3
2
∥∥∂2xu2,n
∥∥2 → 0.
In conclusion ‖yn‖ converge to 0, which contradict the hypothesisthat ‖yn‖ = 1.
Farhat Shel Stability of some string-beam systems
![Page 52: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/52.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
I βnu1,n(`1) −→ 0, ∂xu1,n(`1) −→ 0.
I (7) ∗ q∂xu1,n : ‖∂xu1,n‖2 + β2n ‖u1,n‖2 −→ 0,
I βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −→ 0,
I (8) ∗ q∂xu2,n :
−12
∣∣∂2xu2,n(`2)
∣∣2 + 12β
2n ‖u2,n‖2 + 3
2
∥∥∂2xu2,n
∥∥2 → 0,
I (8) ∗ 1
β1/2n
e−β1/2n x : ∂2
xu2,n(`2)→ 0,
I1
2β2n ‖u2,n‖2 +
3
2
∥∥∂2xu2,n
∥∥2 → 0.
In conclusion ‖yn‖ converge to 0, which contradict the hypothesisthat ‖yn‖ = 1.
Farhat Shel Stability of some string-beam systems
![Page 53: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/53.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
I βnu1,n(`1) −→ 0, ∂xu1,n(`1) −→ 0.
I (7) ∗ q∂xu1,n : ‖∂xu1,n‖2 + β2n ‖u1,n‖2 −→ 0,
I βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −→ 0,
I (8) ∗ q∂xu2,n :
−12
∣∣∂2xu2,n(`2)
∣∣2 + 12β
2n ‖u2,n‖2 + 3
2
∥∥∂2xu2,n
∥∥2 → 0,
I (8) ∗ 1
β1/2n
e−β1/2n x : ∂2
xu2,n(`2)→ 0,
I1
2β2n ‖u2,n‖2 +
3
2
∥∥∂2xu2,n
∥∥2 → 0.
In conclusion ‖yn‖ converge to 0, which contradict the hypothesisthat ‖yn‖ = 1.
Farhat Shel Stability of some string-beam systems
![Page 54: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/54.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
I βnu1,n(`1) −→ 0, ∂xu1,n(`1) −→ 0.
I (7) ∗ q∂xu1,n : ‖∂xu1,n‖2 + β2n ‖u1,n‖2 −→ 0,
I βnu1,n(0), ∂xu1,n(0), Re (iβnf1,n(0)u1,n(0)) −→ 0,
I (8) ∗ q∂xu2,n :
−12
∣∣∂2xu2,n(`2)
∣∣2 + 12β
2n ‖u2,n‖2 + 3
2
∥∥∂2xu2,n
∥∥2 → 0,
I (8) ∗ 1
β1/2n
e−β1/2n x : ∂2
xu2,n(`2)→ 0,
I1
2β2n ‖u2,n‖2 +
3
2
∥∥∂2xu2,n
∥∥2 → 0.
In conclusion ‖yn‖ converge to 0, which contradict the hypothesisthat ‖yn‖ = 1.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Polynomial stability
Theorem
If no control is applied on the string then, the C0-semigroup ispolynomially stable. More precisely, there is C > 0 such that∥∥etAy0
∥∥ ≤ C
t‖y0‖D(A)
for every y0 ∈ D(A).
ProofIt suffices to prove that (6) holds for α = 1. Suppose theconclusion is false. There exists a sequence (βn) of real numbers,without loss of generality, with βn −→ +∞, and a sequence ofvectors (yn) = (un, vn) in D(A) with ‖yn‖H = 1, such that
‖βαn (iβnI −A)yn‖H −→ 0
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Polynomial stability
Lemma [Gagliardo-Nirenberg]
(1) There are two positive constants C1 and C2 such that for anyw in H1(0, `j),
‖w‖∞ ≤ C1 ‖∂xw‖1/2 ‖w‖1/2 + C2 ‖w‖ . (9)
(2) There are two positive constants C3 and C4 such that for anyw in H2(0, `j),
‖∂xw‖ ≤ C3
∥∥∂2xw∥∥1/2 ‖w‖1/2 + C4 ‖w‖ . (10)
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic 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) = u0j (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
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic 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) = u0j (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
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
We prove that the corresponding semigroup etA is notexponentially stable. Let
I βn = n2 + 2√
n + 1n , βn → +∞
I fn = (0, 0,− sinβnx , 0), fn is in H and is bounded.
I yn = (u1,n, u2,n, v1,n, v2,n) ∈ D(A) such that (A− iβn)yn = fn.We will prove that yn → +∞.
Iu1,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
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
We prove that the corresponding semigroup etA is notexponentially stable. Let
I βn = n2 + 2√
n + 1n , βn → +∞
I fn = (0, 0,− sinβnx , 0), fn is in H and is bounded.
I yn = (u1,n, u2,n, v1,n, v2,n) ∈ D(A) such that (A− iβn)yn = fn.We will prove that yn → +∞.
Iu1,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
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
We prove that the corresponding semigroup etA is notexponentially stable. Let
I βn = n2 + 2√
n + 1n , βn → +∞
I fn = (0, 0,− sinβnx , 0), fn is in H and is bounded.
I yn = (u1,n, u2,n, v1,n, v2,n) ∈ D(A) such that (A− iβn)yn = fn.We will prove that yn → +∞.
Iu1,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
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
I
2β3/2n d1 ∼+∞
π2
2
√n.
I
−π2
∣∣∣∣− 1
2βn+ βnc1
∣∣∣∣2 − π
2|βnc2|2
= −1
2(β2
n
∥∥u1n
∥∥2+∥∥∂xu1
n
∥∥2) + Re(
∫ π
0sin(βnx)(π − x)∂xu1
ndx).
β2n
∥∥u1n
∥∥2+∥∥∂xu1
n
∥∥2must be not bounded. In conclusion yn is
not bounded.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
I
2β3/2n d1 ∼+∞
π2
2
√n.
I
−π2
∣∣∣∣− 1
2βn+ βnc1
∣∣∣∣2 − π
2|βnc2|2
= −1
2(β2
n
∥∥u1n
∥∥2+∥∥∂xu1
n
∥∥2) + Re(
∫ π
0sin(βnx)(π − x)∂xu1
ndx).
β2n
∥∥u1n
∥∥2+∥∥∂xu1
n
∥∥2must be not bounded. In conclusion yn is
not bounded.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
I
2β3/2n d1 ∼+∞
π2
2
√n.
I
−π2
∣∣∣∣− 1
2βn+ βnc1
∣∣∣∣2 − π
2|βnc2|2
= −1
2(β2
n
∥∥u1n
∥∥2+∥∥∂xu1
n
∥∥2) + Re(
∫ π
0sin(βnx)(π − x)∂xu1
ndx).
β2n
∥∥u1n
∥∥2+∥∥∂xu1
n
∥∥2must be not bounded. In conclusion yn is
not bounded.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Remarks
Let ε > 0. By taking βn = n2 + 2n1−α + 1n2α with 0 < α < ε
and such that n1−α is integer and even and yn is such that
fn = (β12−ε
n (A− iβn))yn, then we can prove that yn is notbounded and then the polynomial stability of (S) can’t bebutter than 1
t2 .
If we replace the boundary conditions by the followings
δu1(`1, t) = 0, (1− δ)u1xx(`1, t) = 0,
(1− δ)u1,x(`1, t) = −(1− δ)u1,t(`1, t),
δu2,xx(`2, t) = −δu2,tx(`2, t), u2(`2, t) = 0.
then we obtain the same results.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Remarks
Let ε > 0. By taking βn = n2 + 2n1−α + 1n2α with 0 < α < ε
and such that n1−α is integer and even and yn is such that
fn = (β12−ε
n (A− iβn))yn, then we can prove that yn is notbounded and then the polynomial stability of (S) can’t bebutter than 1
t2 .
If we replace the boundary conditions by the followings
δu1(`1, t) = 0, (1− δ)u1xx(`1, t) = 0,
(1− δ)u1,x(`1, t) = −(1− δ)u1,t(`1, t),
δu2,xx(`2, t) = −δu2,tx(`2, t), u2(`2, t) = 0.
then we obtain the same results.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Remarks
Let ε > 0. By taking βn = n2 + 2n1−α + 1n2α with 0 < α < ε
and such that n1−α is integer and even and yn is such that
fn = (β12−ε
n (A− iβn))yn, then we can prove that yn is notbounded and then the polynomial stability of (S) can’t bebutter than 1
t2 .
If we replace the boundary conditions by the followings
δu1(`1, t) = 0, (1− δ)u1xx(`1, t) = 0,
(1− δ)u1,x(`1, t) = −(1− δ)u1,t(`1, t),
δu2,xx(`2, t) = −δu2,tx(`2, t), u2(`2, t) = 0.
then we obtain the same results.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
System
E. String E. Beamℓ2ℓ1 0
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(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
System
TE. String TE. Beamℓ2ℓ1 0
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(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
System
TE. String TE. Beamℓ2ℓ1 0
u1,tt − α1u1,xx + β1θ1,x = 0θ1,t + β1u1,tx − κ1θ1,xx = 0
, u2,tt + α2u2,xxxx + β2θ2,x = 0θ2,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(`1, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
System
TE. String TE. Beamℓ2ℓ1 0
u1,tt − α1u1,xx + β1θ1,x = 0θ1,t + β1u1,tx − κ1θ1,xx = 0
, u2,tt + α2u2,xxxx + β2θ2,x = 0θ2,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(`1, t) = 0, θ(`1, t) = 0, θ(`2, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
System
TE. String TE. Beamℓ2ℓ1 0
u1,tt − α1u1,xx + β1θ1,x = 0θ1,t + β1u1,tx − κ1θ1,xx = 0
, u2,tt + α2u2,xxxx + β2θ2,x = 0θ2,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(`1, t) = 0, θ(`1, t) = 0, θ(`2, t) = 0,
u2(`2, t) = 0, u2,xx(`2, t) = 0.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
System
For a solution (u, v , θ) of (S) the energy is defined as
E (t) =1
2
∫ `1
0
(|u1,t |2 + α1 |u1,x |2 + |θ1|2
)dx
+1
2
∫ `2
0
(|u2,t |2 + α2 |u2,xx |2 + |θ2|2
)dx .
Differentiate formally the energy function with respect to time t,weget
d
dtE (t) = −κ1 ‖∂xθ1‖2 − κ2 ‖∂xθ2‖2
and the system is dissipative.
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Let us consider
V ={
f = (f1, f2) ∈ H1(0, `1)× H2(0, `2) | f satisfies (11)}
where
f1(`1) = 0, f2(`2) = 0, f1(0) = f2(0) and ∂x f2(0) = 0. (11)
Define the Hilbert space H
H = V ×(L2(0, `1)× L2(0, `2)
)×W
with W = L2(0, `1)× L2(0, `2) if e1 and e2 are thermoelastic,W = L2(0, `1)× {0} if only e1 is thermoelastic andW = {0} × L2(0, `2) if only e1 is purely elastic, and norm given by
‖z‖H := α1 ‖∂x f1‖2 + α2
∥∥∂2x f2
∥∥2+
2∑j=1
(‖gj‖2 + ‖hj‖2
)where z = (f = (f1, f2), g = (g1, g2), h = (h1, h2)) .
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
D(A) =
{y = (u, v , θ) ∈ V ∩ (H2(0, `1)× H4(0, `2))× V ×W2 |
and y satisfies (12)
}with W2 = H2(0, `1)× H2(0, `2) if e1 and e2 are T.... and where
∂2xu2(`2) = 0, θ1(`1) = θ2(`2) = 0,θ1(0) = θ2(0),α2∂
3xu2(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∂2xu1 − β1∂xθ1
−α2∂4xu2 + β2∂
2xθ2
−β1∂xv1 + κ1∂2xθ1
−β2∂xxv2 + κ2∂2xθ2
.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
D(A) =
{y = (u, v , θ) ∈ V ∩ (H2(0, `1)× H4(0, `2))× V ×W2 |
and y satisfies (12)
}with W2 = H2(0, `1)× H2(0, `2) if e1 and e2 are T.... and where
∂2xu2(`2) = 0, θ1(`1) = θ2(`2) = 0,θ1(0) = θ2(0),α2∂
3xu2(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∂2xu1 − β1∂xθ1
−α2∂4xu2 + β2∂
2xθ2
−β1∂xv1 + κ1∂2xθ1
−β2∂xxv2 + κ2∂2xθ2
.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
D(A) =
{y = (u, v , θ) ∈ V ∩ (H2(0, `1)× H4(0, `2))× V ×W2 |
and y satisfies (12)
}with W2 = H2(0, `1)× H2(0, `2) if e1 and e2 are T.... and where
∂2xu2(`2) = 0, θ1(`1) = θ2(`2) = 0,θ1(0) = θ2(0),α2∂
3xu2(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∂2xu1 − β1∂xθ1
−α2∂4xu2 + β2∂
2xθ2
−β1∂xv1 + κ1∂2xθ1
−β2∂xxv2 + κ2∂2xθ2
.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Then, putting y = (u, ut , θ), we write the system (S) in the threecases, into the following first order evolution equation{
ddt y = Ayy(0) = y0
(13)
on the energy space H, where y0 = (u0, v 0, θ0).We have the following result,
Lemma
The operator A is the infinitesimal generator of a C0-semigroup ofcontraction S(t).
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Exponential stability
Lemma
The semigroup S(t), generated by the operator A is asymptoticallystable.
Theorem
If the string is thermoelastic, then the system (S) is exponentiallystable.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
It suffices to prove that (5) holds. Suppose the conclusion is false.Then there exists a sequence (wn) of real numbers, withwn −→ +∞ and a sequence of vectors (yn) = (un, vn, θn) in D(A)with ‖yn‖H = 1, such that
‖(iwnI −A)yn‖HF−→ 0
which is equivalent to
iwnu1,n − v1,n = f1,n −→ 0, in H1(0, `1),
iwnv1,n − α1∂2xu1,n + β1∂xθ1,n = g1,n −→ 0, in L2(0, `1),
iwnθ1,n + β1∂xv1,n − κ1∂2xθ1,n = h1,n −→ 0, in L2(0, `1),
and
iw2,nu2,n − v2,n = f2,n −→ 0, in H2(0, `2),
iwnv2,n + α2∂4xu2,n = g2,n −→ 0, in L2(0, `2),
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
We get
w 2nu1,n + α1∂
2xu1,n − β1∂xθ1,n = −g1,n − iwnf1,n, (14)
−w 2nu2,n + α2∂
4xu2,n = g2,n + iwnf2,n, (15)
and‖vj ,n‖2 − w 2
n ‖uj ,n‖2 −→ 0, j = 1, 2.
First, since
Re(〈(iwn −A)yn, yn〉H) = −κ1 ‖∂xθ1‖2
we obtain that ∂xθ1,n converges to 0 in L2(0, `2).As in [?] one can get
‖wnu1,n‖ , ‖∂xu1,n‖ , ‖θ1,n‖ −→ 0.
Moreoverwnu1,n(0), ∂xu1,n(0), θ1,n(0) −→ 0. (16)
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Taking the inner product of (15) with p = (`2 − x)∂xu2,n(x),
−1
2α2
∣∣∂2xu2,n(0)
∣∣2 `2+1
2
∫ `2
0w 2n |u2,n|2 dx+
3
2α2
∫ `2
0
∣∣∂2xu2,n
∣∣2 dx → 0
Now the inner product of the first member of (15) by 1
w1/2n
e−aw1/2n x
gives, with a = 1
α1/42
,
α2
w1/2n
∂3xu2,n(0) + α2a∂2
xu2,n(0) = o(1)
then∂2xu2,n(0) = o(1)
Return back to(4),∫ `2
0w 2n |u2,n|2 dx ,
∫ `2
0
∣∣∂2xu2,n
∣∣2 dx , converge to zero
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Lack of exponential stability
In this part the string is purely elastic.
We take `1 = `2 = π, κ2 << α2.
Theorem
If the string is purely elastic then the system (S) is not exponentialstable in the energy space H.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Lack of exponential stability
In this part the string is purely elastic.
We take `1 = `2 = π, κ2 << α2.
Theorem
If the string is purely elastic then the system (S) is not exponentialstable in the energy space H.
Farhat Shel Stability of some string-beam systems
![Page 85: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/85.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Lack of exponential stability
In this part the string is purely elastic.
We take `1 = `2 = π, κ2 << α2.
Theorem
If the string is purely elastic then the system (S) is not exponentialstable in the energy space H.
Farhat Shel Stability of some string-beam systems
![Page 86: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/86.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
We prove that the corresponding semigroup (S(t))t≥0 is notexponentially stable.For n ∈ N, let fn = (0, 0,−α1 sinβnx , 0, 0), with βn → +∞ and fnis 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
w 2nu1,n + α1∂
2xu1,n = α1 sinβnx
with wn =√α1βn, and
iw2,nu2,n − v2,n = 0, in H2(0, π), (17)
−w 2nu2,n + α2∂
4xu2,n − β2∂
2xθ2,n = 0, in L2(0, π), (18)
iwnθ2,n + iwnβ2∂2xu2,n − κ2∂
2xθ2,n = 0, in L2(0, π). (19)
Notations: α2 = α, β2 = β, κ2 = κ.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic 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∂6xu2,n − iwn(α + β2)∂4
xu2,n − κw 2n∂
2xu2,n + iw 3
nu2,n = 0, (20)
By taking A = 3ακ2 + (α + β2)2,B = 9ακ2(α + β2) + 2(α + β2)3 − 27α2κ2,
a1 = 121/3
(√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
![Page 88: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/88.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic 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∂6xu2,n − iwn(α + β2)∂4
xu2,n − κw 2n∂
2xu2,n + iw 3
nu2,n = 0, (20)
By taking A = 3ακ2 + (α + β2)2,B = 9ακ2(α + β2) + 2(α + β2)3 − 27α2κ2,
a1 = 121/3
(√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
![Page 89: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/89.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
x1 =wn
3ακ
[√3
2(a1 − a2) + i
(r +
1
2(a1 − a2)
)]
x2 =wn
3ακ
[−√
3
2a1 + i
(r +
1
2a1 + a2
)],
x3 =wn
3ακ
[√3
2a2 + i
(r − a1 −
1
2a2
)]
Farhat Shel Stability of some string-beam systems
![Page 90: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/90.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic 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
2a1,
2x ′′2 =
(3
4(a2
2 + (r − a1 −1
2a2)2
)1/2
+
√3
2a2.
2x2 > 2x ′′2 > 2x ′2.The equation (20) admits six simple solutions
±√
wnR1, ±√
wnR2, ±√
wnR3,
with0 < Re(R3) < Re(R2) < Re(R1).
Farhat Shel Stability of some string-beam systems
![Page 91: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/91.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
u2,n =3∑
k=1
(dke√wnRkx + bke−
√wnRkx).
Return back to (18),
β∂2xθ2,n = w 2
n
3∑k=1
(−1 + αR4k )(dke
√wnRkx + bke−
√wnRkx)
Then there exist two constants a′ and b′ such that
βθ2,n = wn
3∑k=1
(− 1
R2k
+αR2k )(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
![Page 92: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/92.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic 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)
w3/2n α
3∑k=1
1
Rk
(dk − bk ) = −1
2wn+ wnc1, (22)
3∑k=1
(−1
R2k
+ αR2k )(dk + bk ) = 0,
3∑k=1
(dk e√
wnRkπ + bk e−√wnRkπ) = 0, (23)
3∑k=1
R2k (dk e
√wnRkπ + bk e
−√wnRkπ) = 0,3∑
k=1
1
R2k
(dk e√
wnRkπ + bk e−√wnRkπ) = 0, (24)
c1 sin(βnπ) + (−π
2βn+ c2) cos(βnπ) = 0. (25)
Farhat Shel Stability of some string-beam systems
![Page 93: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/93.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic 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) + a3w
3/2n c1 tan(βnπ) ∼ π
2
w3/2n
βn.
Hence, with βn = 2n + 1n , tan(βnπ) = π
n + ...
(− 1
2βn+ βnc1) ∼ π
4a4
w3/2n
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 +
∥∥∂xu1,n∥∥2) + Re(
∫ π0
sin(wnx)(π − x)∂xu1,ndx).
In conclusion yn is not bounded.Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Polynomial stability
Theorem
If the string is purely elastic, then the system (S) is polynomiallystable. More precisely, (for every γ < 2) there exists c > 0 suchthat
‖S(t)y0‖ ≤1
tγ‖y0‖D(A) .
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Let 1 > α > 12 . It suffices to prove that (5) holds. Suppose the
conclusion is false. Then there exists a sequence (wn) of realnumbers, with wn −→ +∞ and a sequence of vectors(yn) = (un, vn, θn) in D(A) with ‖yn‖H = 1, such that
‖wαn (iwnI −A)yn‖H −→ 0
which is equivalent to
wαn (iwnu1,n − v1,n) = f1,n −→ 0, in H1, (26)
wαn
(iwnv1,n − α1∂
2xu1,n
)= g1,n −→ 0 in L2, (27)
and
wαn (iwnu2,n − v2,n) = f2,n −→ 0, in H2,(28)
wαn
(iwnv2,n + α2∂
4xu2,n − β2∂
2xθ2,n
)= g2,n −→ 0, in L2,(29)
wαn
(iwnθ2,n + β2∂
2xv2,n − κ2∂
2xθ2,n
)= h2,n −→ 0, in L2.(30)
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Substituting (26) into (27) and (28) into (30) respectively to get
wαn
(w 2nu1,n + α1∂
2xu1,n
)= −g1,n − iwnf1,n, (31)
wαn
(θ2,n −
1
iwnκ2∂
2xθ2,n + β2∂
2xu2,n
)=
1
iwn(h2,n + ∂2
x f2,n)(32)
First, wα/2n ∂xθ2,n converge to 0 in L2(0, `2). Then w
α/2n θ2,n
converge to 0 in L2(0, `2) since θ2,n(0) = 0.
Multiplying (32) by 1
wα/2n
∂2xu2,n
β2wα/2n
∥∥∂2xu2,n
∥∥2+ w
α/2n
⟨θ2,n, ∂
2xu2,n
⟩(33)
−iκ2wα/2−1∂xθ2,n(0)∂2xu2,n(0)− iκ2w
α/2−1n
⟨∂xθ2,n, ∂
3xu2,n
⟩= 0.
Then we prove that
β2wα/2n
∥∥∂2xu2,n
∥∥2 −→ 0.
Farhat Shel Stability of some string-beam systems
![Page 97: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/97.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Substituting (26) into (27) and (28) into (30) respectively to get
wαn
(w 2nu1,n + α1∂
2xu1,n
)= −g1,n − iwnf1,n, (31)
wαn
(θ2,n −
1
iwnκ2∂
2xθ2,n + β2∂
2xu2,n
)=
1
iwn(h2,n + ∂2
x f2,n)(32)
First, wα/2n ∂xθ2,n converge to 0 in L2(0, `2). Then w
α/2n θ2,n
converge to 0 in L2(0, `2) since θ2,n(0) = 0.Multiplying (32) by 1
wα/2n
∂2xu2,n
β2wα/2n
∥∥∂2xu2,n
∥∥2+ w
α/2n
⟨θ2,n, ∂
2xu2,n
⟩(33)
−iκ2wα/2−1∂xθ2,n(0)∂2xu2,n(0)− iκ2w
α/2−1n
⟨∂xθ2,n, ∂
3xu2,n
⟩= 0.
Then we prove that
β2wα/2n
∥∥∂2xu2,n
∥∥2 −→ 0.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Using (29) we prove that wα/8n ‖v2,n‖2 → 0.
We built two sequences of positive numbers rm and sm suchthat
wrm/2n
∥∥∂2xu2,n
∥∥→ 0, wrm/2n ‖θ2,n‖ → 0, w
sm/2n ‖v2,n‖ → 0
and rm and sm converge to 1 + α.
α2∂3xu2,n(0)− β2∂xθ(0)→ 0.
wn ‖u2,n(0)‖ → 0.∫ `2
0
(|∂xu1,n(x)|2 + w 2
n |u1,n(x)|2)
dx −→ 0.
In summary, we have ‖yn‖H −→ 0. This resul contradicts thehypothesis that yn has the unit norm.
Farhat Shel Stability of some string-beam systems
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IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Using (29) we prove that wα/8n ‖v2,n‖2 → 0.
We built two sequences of positive numbers rm and sm suchthat
wrm/2n
∥∥∂2xu2,n
∥∥→ 0, wrm/2n ‖θ2,n‖ → 0, w
sm/2n ‖v2,n‖ → 0
and rm and sm converge to 1 + α.
α2∂3xu2,n(0)− β2∂xθ(0)→ 0.
wn ‖u2,n(0)‖ → 0.∫ `2
0
(|∂xu1,n(x)|2 + w 2
n |u1,n(x)|2)
dx −→ 0.
In summary, we have ‖yn‖H −→ 0. This resul contradicts thehypothesis that yn has the unit norm.
Farhat Shel Stability of some string-beam systems
![Page 100: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/100.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Using (29) we prove that wα/8n ‖v2,n‖2 → 0.
We built two sequences of positive numbers rm and sm suchthat
wrm/2n
∥∥∂2xu2,n
∥∥→ 0, wrm/2n ‖θ2,n‖ → 0, w
sm/2n ‖v2,n‖ → 0
and rm and sm converge to 1 + α.
α2∂3xu2,n(0)− β2∂xθ(0)→ 0.
wn ‖u2,n(0)‖ → 0.∫ `2
0
(|∂xu1,n(x)|2 + w 2
n |u1,n(x)|2)
dx −→ 0.
In summary, we have ‖yn‖H −→ 0. This resul contradicts thehypothesis that yn has the unit norm.
Farhat Shel Stability of some string-beam systems
![Page 101: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/101.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Using (29) we prove that wα/8n ‖v2,n‖2 → 0.
We built two sequences of positive numbers rm and sm suchthat
wrm/2n
∥∥∂2xu2,n
∥∥→ 0, wrm/2n ‖θ2,n‖ → 0, w
sm/2n ‖v2,n‖ → 0
and rm and sm converge to 1 + α.
α2∂3xu2,n(0)− β2∂xθ(0)→ 0.
wn ‖u2,n(0)‖ → 0.
∫ `2
0
(|∂xu1,n(x)|2 + w 2
n |u1,n(x)|2)
dx −→ 0.
In summary, we have ‖yn‖H −→ 0. This resul contradicts thehypothesis that yn has the unit norm.
Farhat Shel Stability of some string-beam systems
![Page 102: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/102.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Using (29) we prove that wα/8n ‖v2,n‖2 → 0.
We built two sequences of positive numbers rm and sm suchthat
wrm/2n
∥∥∂2xu2,n
∥∥→ 0, wrm/2n ‖θ2,n‖ → 0, w
sm/2n ‖v2,n‖ → 0
and rm and sm converge to 1 + α.
α2∂3xu2,n(0)− β2∂xθ(0)→ 0.
wn ‖u2,n(0)‖ → 0.∫ `2
0
(|∂xu1,n(x)|2 + w 2
n |u1,n(x)|2)
dx −→ 0.
In summary, we have ‖yn‖H −→ 0. This resul contradicts thehypothesis that yn has the unit norm.
Farhat Shel Stability of some string-beam systems
![Page 103: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/103.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
Proof
Using (29) we prove that wα/8n ‖v2,n‖2 → 0.
We built two sequences of positive numbers rm and sm suchthat
wrm/2n
∥∥∂2xu2,n
∥∥→ 0, wrm/2n ‖θ2,n‖ → 0, w
sm/2n ‖v2,n‖ → 0
and rm and sm converge to 1 + α.
α2∂3xu2,n(0)− β2∂xθ(0)→ 0.
wn ‖u2,n(0)‖ → 0.∫ `2
0
(|∂xu1,n(x)|2 + w 2
n |u1,n(x)|2)
dx −→ 0.
In summary, we have ‖yn‖H −→ 0. This resul contradicts thehypothesis that yn has the unit norm.
Farhat Shel Stability of some string-beam systems
![Page 104: Stability of some string-beam systems · 2017-05-19 · Introduction Feedback stabilization Thermoelastic case Stability of some string-beam systems Farhat Shel Facult e des Sciences](https://reader033.fdocuments.in/reader033/viewer/2022042003/5e6e5aeb8fbbdf7ed300cef6/html5/thumbnails/104.jpg)
IntroductionFeedback stabilization
Thermoelastic case
Abstract settingAsymptotic behavior
THANKS!
Farhat Shel Stability of some string-beam systems