*(English)*Zbl 1007.93035

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

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.