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Controllability of the Benjamin-Bona-Mahony equation. (English) Zbl 1217.93027

Summary: We study the controllability of the Generalized Benjamin-Bona-Mahony equation (BBM) with homogeneous Dirichlet boundary conditions. Under some conditions we shall prove the system is approximately controllable on \([0, t_1]\) if and only if the following algebraic condition holds: \(\mathrm{Rank} B_j = \gamma_j\) , where \(B_j\) maps from \(\mathbb{R}^m\) to \(R(E_j)\), \(\lambda_j\)’s are the eigenvalues of \(-\Delta\) with Dirichlet boundary condition and \(\gamma j\) the corresponding multiplicity, \(E_j\)’s are the projections on the corresponding eigenspace and \(R(E_j)\) denotes the range of \(E_j\).

MSC:

93B05 Controllability
35B41 Attractors
35Q53 KdV equations (Korteweg-de Vries equations)
93C20 Control/observation systems governed by partial differential equations
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