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On the controllability of the linearized Benjamin-Bona-Mahony equation. (English) Zbl 1007.93035

This paper is concerned with the boundary controllability properties of the following problem:

u t -u xxt +u x =0,x(0,1),t>0,
u(t,0)=0,u(1,t)=f(t),t>0,
u(0,x)=u 0 (x),x(0,1),

where u 0 H -1 (0,1) and T>0 are fixed. The results can be summarized as follows: a) the system is not spectrally controllable (i.e. no finite linear nontrivial combination of eigenvectors can be driven to zero in finite time by using a control fL 2 (0,1)); b) the system is approximately controllable in L 2 (0,T) (i.e. the set of reachable states at time T is dense in L 2 (0,1) when f runs L 2 (0,1)); c) the system is N-partially controllable to zero (i.e., given N>0 there is a control fL 2 (0,T) such that the projection of the solution of the system over the finite-dimensional space generated by the first N eigenvectors is equal to zero at time t=T). The method relies on the study of the sequences biorthogonal to the family of exponentials of the eigenvalues of the operator.

MSC:
93C20Control systems governed by PDE
93B05Controllability
78M05Method of moments (optics)