Automatica 50 (2014) 2225–2233
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Automatica
journal homepage: www.elsevier.com/locate/automatica
Stabilization of the Euler–Bernoulli plate with variable coefficients by
nonlinear internal feedback
✩
Shun Li
a,b
, Peng-Fei Yao
b,1
a
School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, PR China
b
Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing, PR China
a r t i c l e i n f o
Article history:
Received 2 July 2012
Received in revised form
13 April 2014
Accepted 27 April 2014
Available online 13 August 2014
Keywords:
Energy decay
Euler–Bernoulli plate
Internal feedback
Variable coefficients
Riemannian geometry
a b s t r a c t
We consider the energy decay for solutions of the Euler–Bernoulli plate equation with variable coeffi-
cients where a nonlinear internal feedback acts in a suitable subregion of the domain. The Riemannian
geometric method is used to deal with variable coefficient problems. When the feedback region has a
structure similar to that for the wave equation with constant coefficients, we establish the stabilization of
the system in the case of fixed boundary conditions. Several energy decay rates are established according
to various growth restrictions on the nonlinear feedback near the origin and at infinity. We further show
that, unlike for the case of constant coefficients, choices of such feedback regions depend not only on the
type of boundary conditions but also on the curvature of a Riemannian metric, based on the coefficients
of the system.
© 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Let Ω ⊂ R
n
be an open bounded domain with smooth bound-
ary Γ and let A be the operator on C
∞
0
(R
n
) defined by
Au = div A(x)∇u, u ∈ C
∞
0
(R
n
), x ∈ R
n
, (1)
where div is the divergence operator of R
n
in the standard Eu-
clidean metric, A(x) = (a
ij
(x)) is a symmetric, positive definite ma-
trix with smooth entries a
ij
(x), i, j = 1, . . . , n for each x ∈ R
n
, and
∇ is the gradient operator.
We consider the decay estimates for solutions of the following
system
u
tt
+ A
2
u + χ
G
(x)h(u
t
) = 0 in Ω × (0, ∞), (2)
u =
∂u
∂ν
A
= 0 on Γ × (0, ∞), (3)
u(x, 0) = u
0
(x), u
t
(x, 0) = u
1
(x) in Ω, (4)
✩
This work is supported by the National Natural Science Foundation of China,
Grant no. 61174083. The material in this paper was not presented at any conference.
This paper was recommended for publication in revised form by Associate Editor
George Weiss under the direction of Editor Miroslav Krstic.
E-mail addresses: lishun@amss.ac.cn (S. Li), pfyao@iss.ac.cn (P.-F. Yao).
1
Tel.: +86 1082541888; fax: +86 1082541832.
where G ⊆ Ω is a subset, χ
G
(x) is the characteristic function of the
subset G, and h : R → R is a non-decreasing continuous function
such that h(0) = 0. In (3) ν
A
= A(x)ν where ν is the unit normal
of Γ pointing toward the exterior of Ω and
∂u
∂ν
A
is the directional
derivative of u with respect to ν
A
.
The energy of system (2)–(4) is defined by
E(t) =
1
2
Ω
[u
2
t
+ (Au)
2
]dx. (5)
We mention that in Yao (2011), G is called an escape region,
which is not appropriate, and the escape region should be Ω \ G.
Roughly speaking, an open set G ⊂ Ω is a control region for plate
system (2)–(4) if the energy E(t) uniformly decays in some way
as time t goes to infinity, which will be precisely defined later; see
Definition 2.1. Moreover, if G is a control region, then the set Ω \G is
an escape region for system (2)–(4). This is because waves in Ω \G
can go out without any control. The structure of a control region
reflects the effect of the internal feedback. We seek geometric con-
ditions on G. We note that G = Ω is one of the choices for such
purposes (see Theorems 2.1 and 2.2). However, we are particularly
interested in the case where G is not the whole domain Ω and is as
small as possible in a technical sense. Such a structure of control re-
gions was first given by Liu (1997) for the wave equation with con-
stant coefficients and was used then by Alabau-Boussouira (2005,
2006). This structure was extended to the wave equation with vari-
able coefficients by Feng and Feng (2004) and used for the quasi-
linear wave equation by Zhang and Yao (2008). In addition, it is
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