Applied Mathematics, 2011, 2, 661-665
doi:10.4236/am.2011.26087 Published Online June 2011 (http://www.SciRP.org/journal/am)
Uniform Exponential Stabilization for Flexural Vibrations
of a Solar Panel
Prasanta Kumar Nandi1, Ganesh Chandra Gorain2, Samarjit Kar1
1Department of Mat hem at i cs, National Institute of Technology, Durgapur , India
2Department of Mat hem at i cs, J. K. College, Purulia, India
Received February 16, 2011; re vi s ed M ar ch 1, 2011; accepted March 4, 2011
Abstract
Here we study a problem of stabilization of the flexural vibrations or transverse vibrations of a rectangular
solar panel. The dynamics of vibrations is governed by the fourth order Euler-Bernoulli beam equation. One
end of the panel is held by a rigid hub and other end is totally free. Due to attachment of the hub, its dynam-
ics leads to a non-standard equation. The exponential stabilization of the whole system is achieved by apply-
ing an active boundary control force only on the rigid hub. The result of uniform stabilization is obtained by
means of an explicit form of exponential energy decay estimate.
Keywords: Solar Panel, Hybrid System, Flexural Vibrations, Uniform Exponential Stabilization, Energy
Decay Estimate
1. Introduction
Motivated by ambitious space programs, mathematical
investigation on stabilization of vibrating space struc-
tures is an active area of research among others. The
pioneering work was first started since early seventies in
a study aimed at achieving energy decay rates for wave
equation exterior to a bounded obstacle. Later the idea
has been extended in various mathematical problems
related to the vibrations of flexible structures like beams,
plates or slender elements capable of withstanding finite
deformation. The most important problem for these
problems is to suppress the vibrations to assure a good
performance of the overall system.
During the last few decades, the use of flexible struc-
tures is on the rise. The vibrations of flexible structures
are usually non-linear in practice. The vibrations of
flexible structures are the problem of a dynamical system
mathematically governed by partial differential equations,
particularly, the second order wave equation and the
fourth-order Euler-Bernoulli beam equation. Stabilization
for the wave equation in a bounded domain have been
investigated by several authors (cf. G. Chen [1,2], J.
Lagnese [3,4], J. L. Lions [5], V. Komornik [6] and the
references therein). Similarly, those governed by the
fourth-order Euler-Bernoulli beam equation have been
treated by G. Chen, and J. Zhou [7], O. Morgul [8] and A.
M. Krall [9]. Hybrid system of flexible structures con-
sists of a coupled elastic part and a rigid part. The hybrid
system in which a lumped mass is present at one end,
have been treated earlier by G. Chen, M. C. Delfour, A.
M. Krall and G. Payre [10], W. Littman, L. Markus [11]
and B. Rao [12]. The most usual practical approach to
stabilize the problems of such type is to apply the control
force or the stabilizer on the free end of the elastic part
instead of the rigid part.
The mathematical theory of stabilization of distributed
parameter system has become a great interest in view of
its application in various flexible structures. The energy
decay estimate has earlier been studied by several
authors (cf. G. Chen [1,2], J. Lagnese [3,4], J. L. Lions
[5], V. Komornik and E. Zuazua [13]). Recently, G. C.
Gorain [14] treated the case of internally damped wave
equations for the so called Kelvin-Voigt model of vis-
coelsticity together with undamped boundary conditions
(without considering any boundary feedback) to obtain a
uniform exponential energy decay estimate. Several hy-
brid model of the dynamics of torsional vibrations of a
flexible structure hoisted by a rigid hub at one end have
been studied by T. Fukuda, F. Arai, H. Hosogai and N.
Yajima [15]. In engineering literature, a common app-
roach to treat the above problems is to decompose the
vibrations into normal modes and retain the first few
modes to reduce the problem into a finite dimensional

P. K. NANDI ET AL. 662
state space representation. The question of uniform
stabilization or point-wise stabilization of Euler-Ber-
noulli beams or serially connected beams has been stu-
died by a number of authors (cf. J. L. Lions [5], G. Chen,
M. C. Delfour, A. M. Krall and G. Payre [10], K. Ammari
and M. Tuesnak [16], K. Liu and Z. Liu [17], K. Nagaya
[18], R. Rebarbery [19] etc.).
2. Mathematical Formulation of the Problem
We consider a uniform rectangular flexible solar panel
hoisted by a rigid hub at one end. The panel is of length
, unit width, having uniform mass density per unit
length, which is rigidly attached by a lumped mass h
(hub) at one end and that is totally free at the other end.
Our aim is to stabilize the vibrations of the overall
system by applying a suitable stabilizer or damper on the
rigid hub, when it is initially set in motion.
Lmm
Referring to the schematic Figure 1, if


h
y
t is the
transverse displacement of the rigid hub and


,
p
y
xt is
that of the panel at the position
x
along the span of the
panel relative to the hub at time , then the total trans-
verse deflection can be written as
t
 
,=,, 0, 0.
hp
yxty tyxtxLt (1)
Let us assume that the vibrations undergo only small
deformations, that means,


,yxt L and

,
yxt
x

1, and neglect the gravitational effect and
rotatory inertia of the panel cross sections. Then


,
y
xt
satisfies fourth order Euler-Bernoulli beam equation
 
24
24
,,=0, 0,
yy
mxtDxt xLt
tx


 0, (2)
where

1
32
11
12
DEh 

. The constants , , DE

and are the flexural rigidity, the Youngs’s modulus,
the Poisson’s ratio and the thickness of the panel
respectively.
h
The dynamics at the hub end , where a control
force is applied, yields the differential equation
=0x

Qt
  
3
2
23
0,=0, 0.
p
h
h
y
y
mtD tQtt
tx


 
(3)
The Equation (3) is not a standard Dirichlet or Neu-
mann boundary condition, that is generally found in the
several works of this subject area. Again since

0, 0
p
yt, it follows from (1) that



0,=h
y
tyt
and also we have
 
,= ,
p
y
y

x
t
xx



xt . Hence the Equa-
tion (3) becomes
Figure 1. Schematic of the solar pane l wi th rigid hub.
 
32
32
0,0,=0, 0,
yy
ttQt
xt

 t


 (4)
where =h
m
D
 and 1 =D  . Assuming at , there =0x
is no rotational deflection of the panel relative to the hub
(that means, the panel is built in position with hub at
=0x), we have

0,=0
p
yt
x

, implying

0,=0, 0.
ytt
x

 (5)
Since the panel is assumed to be free at =
x
L, so at
this end
 
23
23
,=0 and ,=0, 0,
yy
LtLt t
xx


 (6)
Let the panel be set to vibrations with arbitrary initial
values
 
01
,0= and ,0=, 0.
y
yx yxx yxxL
t

 (7)
Therefore, the mathematical model to be studied for
flexural or transverse vibrations of a uniform rectangular
flexible solar panel with a rigid hub at one end, is go-
verned by the following system of equations :
 
24
24
,,=0, 0,
yy
mxtDxt xLt
tx
 0,

 
 (8)
 
32
32
0,0,= 0 and 0,=0,
yy y
ttQt t
x xt

 
 
 (9)
 
23
23
,=0 and ,=0, 0,
yy
LtLt t
xx


 (10)
 
01
,0= and,0=, 0.
y
yx yxx yxxL
t

 (11)
Because of the damping character of the control force

P. K. NANDI ET AL.663

Qt, it must be an odd function of velocity, that means,
 
=0,
y
Qt ft
t





,


(12)
where
f
is an odd function of its argument such that
and for every

0=f0

>0uf u


0.u For
example, if

=
f
uu, we have a simple viscous dam-
per.
3. Energy of the System
The total energy at time is defined by

Et t
 
2
2
2
2
0
11
=d
22
for 0.
L
h
yy y
EtmDx mt
t x
t


 
 



 
 
  




2
0,
t
(13)
Differentiating this with respect to
t
and using the
governing Equations (8)-(10), we obtain
 
 

223
2
222
0
2
2
32
32
d=
d
0,0,
=0,0,0,0,
=0,,
L
h
Eyyyy
cm Dx
tt txxt
yy
mt t
tt
yy yy
Dt tt t
tt
x
y
DtQt
t



 







 







d
t
(14)
where the integration is performed by parts and the
boundary conditions (9)-(10) are used. By the help of the
Equation (12), we get
 
d=0,0,=< 0 for 0,
d
Ey y
tft ufut
tt t





 
(15)
where

=0,
y
u
t



t is the velocity at the hub end and
=1D

. The result (15) implies that the energy


Et
of the system (8)-(12) is a non increasing function of
time. Integrating (15), we have , where

0Et E
      
22
10 1
0
11
0=d 0
2
L
h
EmyxDyxxmy





2
2
(16)
is the initial energy of the system. As the energy decays,
our main interest is whether this decays is uniformly
exponential or not. An affirmative answer can be found
in the next section.
4. Uniform Stability Result and Proof
The main result of this paper can be stated in the
following theorem.
Theorem 1. Let


,
y
xt be a solution of the system
(8)-(12) with the initial values

01

,
y
y for which


0<E

, where


0E is defined in (16). Then the
total energy of the system decays uniformly exponen-
tially with time, that means, , satisfies the
relation
0

Ett


et Et M


 (17)
for some finite reals >1
M
and >0

, both being
independent of the time .
t
The theorem will be proved after some preliminary
steps. First, we require the following inequality.
For any real number >0

, we have by the Cauchy-
Schwartz’s inequality


2
2
1.
2
fgf g


 2 (18)
Next we consider the following lemma:
Lemma 1. For every solution of the system
(8)-(12), the time derivative of the functional (cf. G.
C. Gorain [20], G. C. Gorain and S. K. Bose [21])
defined by
),( txy G

0
=d for 0
Lyy
Gtm xxt
tx
 

 (19)
satisfies
  
22
d1
12 ,0,.
d2
h
Gy y
LmL tmtEt
tt t







(20)
Proof: If we differentiate (19) with respect to
t
and
using the governing Equation (8), we obtain
 

24
4
0
22
2
2
2
0
d= d
d
11
= ,0,
22
d,
L
h
L
Gyyyy
xmD x
ttxtx x
yy
LmL tmt
tt
y
Et Dx
x

 























(21)
where the integration is done by parts and the boundary
conditions (9)-(10) are used. The Lemma 1 then follows
immediately from (21).
Proof of Theorem 1: Proceeding as in G. C. Gorain
[20] and G. C. Gorain and S. K. Bose [21], we define
energy like Lyapunov functional by
V




= for 0,Vt EtGtt 

 (22)
where >0

is a fixed constant. Differentiating (22)
with respect to
t
, and using (15) and (21) we obtain

P. K. NANDI ET AL. 664
 
 
2
2
d1
0,0, ,
d2
1
0,.
2h
Vy yy
tftLmLt
tt tt
y
mtEt
t


 
 
 
 
 
 








(23)
We choose a feedback controller or stabilizer in such
way that satisfies mathematically
 

22
11
,0
22
0,0,,
h
yy
LmL tmt
tt
yy
tf t
tt












,



(24)
where

is a finite positive constant independent of
time . t
Hence, using the above relation (24), we can write (23)
as
 
d10, 0,
d
Vyy
tft Et
ttt
 


 


.
(25)
Since

is small, we may assume that
1
0< < 

(26)
so that the differential relation (25) reduces to

d0 for 0.
d
VEt t
t
   (27)
Now applying the Inequality (18), we have from (19),

22
22
2
0
2πd,
π4
L
Lm yy
Gtm Dx
Dt x L


 


 

 






(28)
Again using Wirtinger’s inequality
2222
22
00
4
dd
π
LL yLy
x
x
xx
 


 

 

(29)
the above Inequality (28) can be expressed as
 
2
4,
π
Lm
Gt Et
D
 (30)
that means,
 
22
44
for 0.
ππ
Lm Lm
Et GtEtt
DD
 (31)
So defined by (22) can be estimated as V
 
2
44
11
ππ
Lm Lm
Et VtEt
D

 

 
 
 
2
.
D
(32)
Since

is small, we may further assume that
2
π
0< <.
4
D
m L
 (33)
Then it follows from (32) that for every
. Invoking the Inequality (32), the relation (27)
leads to the differential inequality

>0Vt
0t

d0,
d
VVt
t


 (34)
where
2
=>0
4
1π
Lm
D








.
(35)
Multiplying (34) by et

and integrating from 0 to ,
we obtain
t


e0 for 0.
t
Vt Vt
 

 (36)
Applying again the Inequality (32) in (36), we get


e0
t
Et ME
 


, (37)
where
22
44
=11 >1.
π
Lm Lm
MDD











(38)
Hence the theorem.
5. Conclusions
Here we have achieved the uniform boundary stabili-
zation of flexural vibrations of a rectangular solar panel
which is held by a rigid hub at one end and is totally free
at the other. We have also estimated directly the ex-
ponential energy decay rate

that is explicitly found
in (35). Again, considering

as an explicit function of

, we have
2
2
d4
=1 >14,
dπ
Lm
D








(39)
in view of (33). Hence the exponential decay rate as a
function of

will be maximum for largest admissible
value ,

an upper bound of which can be determined
by considering the coupled relations (26) and (33) si-
multaneously. The determination of exact value is
restricted by the lack of explicit knowledge in general of
the parameter

appearing in the literature. The mo-
tivation of considering the vibrations of this type of hy-
brid system arises from many practical problems such as
spacecraft with flexible attachment, robot with flexible
links and thin plates of different mechanical system. The
significant result in this paper is that the solution of the
system governed by (8)-(12) converges uniformly to zero
as time .
t

P. K. NANDI ET AL. 665
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