The main result of this paper states sufficient conditions for the existenc
e of a completion A(c) of an n x n partial upper triangular matrix A, such
that the pair (A(c), B) has prescribed controllability indices, being B an
n x m matrix. If A is a partial Hessenberg matrix some conditions may be dr
opped. An algorithm that obtains a completion A(c) of A such that pair (A(c
), e(k)) is completely controllable, where ek is a unit vector, is used to
proof the results.