This paper is concerned with the boundary controllability properties of the following problem:
where and 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 ); b) the system is approximately controllable in (i.e. the set of reachable states at time is dense in when runs ); c) the system is -partially controllable to zero (i.e., given there is a control such that the projection of the solution of the system over the finite-dimensional space generated by the first eigenvectors is equal to zero at time ). The method relies on the study of the sequences biorthogonal to the family of exponentials of the eigenvalues of the operator.