We consider the hybrid laminated Timoshenko beam model. This structure is given by two identical layers uniform on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We suppose that the beam is fastened securely on the left while on the right it’s free and has an attached container. Using the semigroup approach and a result of Borichev and Tomilov, we prove that the solution is polynomially stable.

1.
Abbas
,
B. A. H.
and
Thomas
,
J.
, “
The second frequency spectrum of Timoshenko beams
,”
J. Sound Vib.
51
,
123
137
(
1977
).
2.
Alabau-Boussouira
,
F.
, “
Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control
,”
Nonlinear Differ. Equations Appl.
14
,
643
669
(
2007
).
3.
Amar-Khodja
,
F.
,
Benabdallah
,
A.
,
Muñoz Rivera
,
J. E.
, and
Racke
,
R.
, “
Energy decay for Timoshenko systems of memory type
,”
J. Differ. Equations
194
,
82
115
(
2003
).
4.
Borichev
,
A.
and
Tomilov
,
Y.
, “
Optimal polynomial decay of functions and operator semigroups
,”
Math. Ann.
347
,
455
478
(
2009
).
5.
Cao
,
X.-G.
,
Liu
,
D.-Y.
, and
Xu
,
G.-Q.
, “
Easy test for stability of laminated beams with structural damping and boundary feedback controls
,”
J. Dyn. Control Syst.
13
,
313
336
(
2007
).
6.
Feng
,
B.
,
Ma
,
T. F.
,
Monteiro
,
R. N.
, and
Raposo
,
C. A.
, “
Dynamics of laminated Timoshenko beams
,”
J. Dyn. Differ. Equations
2017
,
1
.
7.
Feng
,
D.-X.
,
Shi
,
D.-H.
, and
Zhang
,
W.
, “
Boundary feedback stabilization of Timoshenko beam with boundary dissipation
,”
Sci. China, Ser. A: Math.
41
,
483
490
(
1998
).
8.
Guesmia
,
A.
, “
Some well-posedness and general stability results in Timoshenko systems with infinite memory and distributed time delay
,”
J. Math. Phys.
55
,
081503
(
2014
).
9.
Guesmia
,
A.
and
Messaoudi
,
S. A.
, “
General energy decay estimates of Timoshenko systems with frictional versus viscoelastic damping
,”
Math. Methods Appl. Sci.
32
,
2102
2122
(
2009
).
10.
Hansen
,
S. W.
, “
Control and estimation of distributed parameter systems: Non-linear phenomena
,”
Int. Ser. Numer. Anal.
118
,
143
170
(
1994
).
11.
Hansen
,
S. W.
and
Spies
,
R.
, “
Structural damping in a laminated beams due to interfacial slip
,”
J. Sound Vib.
204
,
183
202
(
1997
).
12.
Kim
,
J. U.
and
Renardy
,
Y.
, “
Boundary control of the Timoshenko beam
,”
SIAM J. Control Optim.
25
,
1417
1429
(
1987
).
13.
Lo
,
A.
and
Tatar
,
N.-E.
, “
Stabilization of laminated beams with interfacial slip
,”
Electron. J. Differ. Equations
129
,
1
14
(
2015
).
14.
Lo
,
A.
and
Tatar
,
N.-E.
, “
Exponential stabilization of a structure with interfacial slip
,”
Discrete Contin. Dyn. Syst.
36
,
6285
6306
(
2016
).
15.
Liu
,
W.
and
Zhao
,
W.
, “
Exponential and polynomial decay for a laminated beam with Fourier’s type heat conduction
,”
Preprints
2017
, 2017020058.
16.
Messaoudi
,
S. A.
and
Mustafa
,
M. I.
, “
On the stabilization of the Timoshenko system by a weak nonlinear dissipation
,”
Math. Methods Appl. Sci.
32
,
454
469
(
2009
).
17.
Muñoz Rivera
,
J. E.
and
Racke
,
R.
, “
Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability
,”
J. Math. Anal. Appl.
276
,
248
276
(
2002
).
18.
Muñoz Rivera
,
J. E.
and
Racke
,
R.
, “
Global stability for damped Timoshenko systems
,”
Discrete Contin. Dyn. Syst.
9
,
1625
1639
(
2003
).
19.
Muñoz Rivera
,
J. E.
and
Sare
,
H. D. F.
, “
Stability of Timoshenko systems with past history
,”
J. Math. Anal. Appl.
339
,
482
502
(
2008
).
20.
Pazy
,
A.
,
Semigroups of Linear Operators and Applications to Partial Differential Equations
(
Springer-Verlag
,
New York
,
1983
).
21.
Raposo
,
C. A.
, “
Exponential stability for a structure with interfacial slip and frictional damping
,”
Appl. Math. Lett.
53
,
85
91
(
2016
).
22.
Timoshenko
,
S. P.
, “
On the correction for shear of the differential equation for transverse vibrations of prismatic bars
,”
Philos. Mag. Ser. 6
41
,
744
746
(
1921
).
23.
Timoshenko
,
S. P.
and
Gere
,
J. M.
,
Mechanics of Materials
(
D. Van Nostrand Company, Inc.
,
New York
,
1972
).
24.
Said-Houari
,
B.
and
Kasimov
,
A.
, “
Decay property of Timoshenko system in thermoelasticity
,”
Math. Methods Appl. Sci.
35
,
314
333
(
2012
).
25.
Sare
,
H. D. F.
and
Racke
,
R.
, “
On the stability of damped Timoshenko systems: Cattaneo versus Fourier law
,”
Arch. Ration. Mech. Anal.
194
,
221
251
(
2009
).
26.
Soufyane
,
A.
and
Wehbe
,
A.
, “
Uniform stabilization for the Timoshenko beam by a locally distributed damping
,”
Electron. J. Differ. Equations
29
,
1
14
(
2003
).
27.
Wang
,
J.-M.
,
Xu
,
G.-Q.
, and
Yung
,
S.-P.
, “
Exponential stabilization of laminated beams with structural damping and boundary feedback controls
,”
SIAM J. Control Optim.
44
,
1575
1597
(
2005
).
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