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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

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

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 [formulas] where a ≥ 0 and b > 0 are constants, Ω is a domain in R^N, ω is an open nonempty subset of Ω, $1_ω denotes the characteristic function of the set ω, the distributed control u belongs to L^2(0,τ; L^2(Ω)) and the nonlinear function f: [0,τ] × R × R → R is smooth enough and there are c,d,e ∈ R, with c ≠ -1, ea+b > 0, such that [formula], where Q_τ = [0,τ] × R × R. We prove that for all τ > 0 and any nonempty open subset ω of Ω the system is approximately controllable on [0,τ]. Moreover, we exhibit a sequence of controls steering the system from an initial state z_0 to an ε-neighborhood of the final state z_1  on time τ > 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.

 

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About this Article

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

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

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

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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