Journalof Mathematics and Applications No 35, pp 39-51 (2012) COPYRIGHT c  by Publishing Department Rzesz´ ow University of Technology P.O. Box 85, 35-959 Rzesz´ ow, Poland Controllability of the semilinear Benjamin-Bona-Mahony equation H. Leiva and N. Merentes and J.L. Sanchez Submitted by: J´ozefBana´ s Abstract: In this paper we prove the interior approximate control- lability of the following Generalized Semilinear Benjamin-Bona-Mahony type equation (BBM) with homogeneous Dirichlet boundary conditions  z t − aΔz t − bΔz =1 ω u(t, x)+ f (t,z,u(t, x)), t ∈ (0,τ ], x ∈ Ω, z(t, x)=0, t ≥ 0, x ∈ ∂ Ω, where a ≥ 0 and b> 0 are constants, Ω is a domain in IR 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 ] ×IR×IR → IR is smooth enough and there are c, d, e ∈ IR, with c = −1, ea + b> 0, such that sup (t,z,u)∈Qτ |f (t,z,u) − ez − cu − d| < ∞, where Q τ = [0,τ ] × IR × IR. 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 ǫ-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. AMS Subject Classification: Primary 93B05, Secondary 93C25. Key Words and Phrases: interior controllability, semilinear BBM equation, strongly continuous semigroups This work was supported by MCTI, CDCHT-ULA under projects ConCiencia-3837, C-1667-09- 05-AA and by BCV