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research
Exact controllability of a second order integro-differential equation with a pressure term
Authors
M.M. Cavalcanti
V.N.D. Cavalcanti
A. Rocha
J.A. Soriano
Publication date
1 January 1998
Publisher
University of Szeged, Hungary
Doi
Cite
Abstract
This paper is concerned with the boundary exact controllability of the equation
u
′
′
−
Δ
u
−
∫
0
t
g
(
t
−
σ
)
Δ
u
(
σ
)
d
σ
=
−
∇
p
u''-\Delta u-\int_0^t g(t-\sigma)\Delta u(\sigma) d\sigma=-\nabla p
u
′′
−
Δ
u
−
∫
0
t
​
g
(
t
−
σ
)
Δ
u
(
σ
)
d
σ
=
−
∇
p
where
Q
Q
Q
is a finite cilinder
Ω
×
]
0
,
T
[
\Omega\times]0,T[
Ω
×
]
0
,
T
[
,
Ω
\Omega
Ω
is a bounded domain of
R
n
R^n
R
n
,
u
=
(
u
1
(
x
,
t
)
,
…
,
u
n
(
x
,
t
)
)
u=(u_1(x,t),\ldots,u_n(x,t))
u
=
(
u
1
​
(
x
,
t
)
,
…
,
u
n
​
(
x
,
t
))
,
x
=
(
x
1
,
…
,
x
n
)
x=(x_1,\ldots,x_n)
x
=
(
x
1
​
,
…
,
x
n
​
)
are
n
n
n
-dimensional vectors and
p
p
p
denotes the pressure term. The result is obtained by applying HUM (Hilbert Uniqueness Method) due to J. L. Lions. The above equation is a simple model of dynamical elasticity equations for incompressible materials with memory
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