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Controllability of the semilinear Benjamin-Bona-Mahony equation

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

Journal of Mathematics and Applications

2012

Vol. 35

39--51

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 [formula/wzor] where a ≥ 0 and b > 0 are constans, Ω is a domain in IRN, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0, τ;L2(Ω)) and the nonlinear function ƒ:[0, τ] x IR x IR → IR is smooth enough and there are c, d, e ∈ IR, with c ≠ -1, ea + b > 0 such that [formula/wzor] where Qr = [0, τ] x IR x IR. We prove that for all τ > 0 and any nonempty open subset ω of Ω the system the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the system from an initial state z0 to an ε-neighborhood of the final state z1 on time > 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.

Interior controllability of the Benjamin-Bona-Mahony equation

Leiva H., Merentes N., Sanchez J.

Journal of Mathematics and Applications

2010

Vol. 33

51-59

In this paper we prove the interior approximate controllability of the following Generalized Benjamin-Bona-Mahony type equation (BBM) with homogeneous Dirichlet boundary conditions [formula/wzór] where a(mniejszy-równy) and b > 0 are constants, Ω is a domain in IR(N), ω is an open nonempty subset of Ω denotes the characteristic function of the set ω and the distributed control [formula/wzór]. We prove that for all r>0 and any nonempty open subset ω of Ω the system is approximately controllable on [0, r]. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time. As a consequence of this result we obtain the interior approximate controllability of the heat equation by putting a = 0 and b = 1.