On the complete controllability indexes assignment problem

New solvability conditions for the simultaneous assignment by static-state feedback of both the controllability indexes of (A, B) and the invariant factors of (sI-A, -B) are given. These conditions enhance the fact that this problem is a generalization of the control structure problem of Rosenbrock (1970) and of the controllability indexes assignment problem of Heymann (1981). The choice of a polynomial framework allows one to obtain simpler proofs of the results than those previously published.