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Journal of Mathematics and Applications

Journal of Mathematics and Applications
4/35, DOI: 10.7862/rf.2012.4

Controllability of the semilinear Benjamin-Bona-Mahony equation

Leiva H., Merentes N., Sanchez J. L.

DOI: 10.7862/rf.2012.4

Abstract

In this paper we prove the interior approximate controllability of the following Generalized Semilinear Benjamin-Bona-Mahony type equation (BBM) with homogeneous Dirichlet boundary conditions
$$
 left{
egin{array}{l}
  z_t - aDelta z_t - bDelta z =1_{omega}u(t,x)+f(t,z,u(t,x)), t in (0, au],qquad x in Omega, \
  z(t, x) = 0, t geq 0, qquad x in partialOmega , \
end{array}
ight.
$$
where $a geq 0$ and $b>0$ are constants, $Omega$ is a domain in $R^N$, $omega$ is an open nonempty subset of $Omega$, $1_{omega}$ denotes the characteristic function of the set $omega$, the distributed control $u$ belongs to $L^{2}(0, au; L^{2}(Omega))$ and the nonlinear function $f:[0, au] imes R imes R ightarrow R$ is smooth enough and there are $c,d,e in R$, with $c eq -1$, $ea+b >0$, such that
$$
sup_{(t,z,u) in Q_{ au}} |f(t,z,u) -ez-cu-d | < infty,
$$
where $Q_{ au}= [0, au] imes R imes R$.
We prove that for all $ au>0$ and any nonempty open subset $omega$ of $Omega$ the system is approximately controllable on $[0, au]$. Moreover, we exhibit a sequence of controls steering the system from an initial state $z_0$ to an $epsilon$-neighborhood of the final state $z_1$  on time $ au>0 $. As a consequence of this result we obtain the interior approximate controllability of the semilinear heat equation by putting $a =0$ and $b=1$.}

 

Full text (pdf)

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About this Article

TITLE:
Controllability of the semilinear Benjamin-Bona-Mahony equation

AUTHORS:
Leiva H. (1)
Merentes N. (2)
Sanchez J. L. (3)

AUTHORS AFFILIATIONS:
(1) Departamento de Matematicas, Universidad de Los Andes, Merida 5101- VENEZUELA
(2) Departamento de Matematicas, Facultad de Ciencias, Universidad Central de Venezuela, Caracas 1051 VENEZUELA
(3) Departamento de Matematicas, Facultad d

JOURNAL:
Journal of Mathematics and Applications
4/35

KEY WORDS AND PHRASES:
interior controllability, semilinear BBM equation, strongly
continuous semigroups

FULL TEXT:
http://doi.prz.edu.pl/pl/pdf/jma/4

DOI:
10.7862/rf.2012.4

URL:
http://dx.doi.org/10.7862/rf.2012.4

RECEIVED:
2011-10-10

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