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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: $u_t - u_{xxt} + u_x = 0, \;\;x \in (0,1), t > 0,$ $u(t,0) = 0, u(1,t) = f(t), \;\;t > 0,$ $u(0,x) =u_0(x), \;\;x \in (0,1),$ where $$u_0 \in H^{-1}(0,1)$$ 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 $$f \in L^2(0,1)$$); 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 $$f \in L^2(0,T)$$ 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:
 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 78M05 Method of moments applied to problems in optics and electromagnetic theory
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