On the controllability of the linearized Benjamin-Bona-Mahony equation (Q2706170)

This paper is concerned with the boundary controllability properties of the following problem:NEWLINE\[NEWLINE u_t - u_{xxt} + u_x = 0, \;\;x \in (0,1), t > 0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(t,0) = 0, u(1,t) = f(t), \;\;t > 0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0,x) =u_0(x), \;\;x \in (0,1),NEWLINE\]NEWLINE 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.