Controllability of the semilinear Benjamin-Bona-Mahony equation

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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