Controllability of partially prescribed matrices

Let F be an infinite field and let n,p(1),p(2),p(3) be positive integers such that n = p(1) + p(2) + p(3). Let C-1,C-2 is an element of F-p1xp2, C-1,C-3 is an element of F-p1xp3 and C-2,C-1 is an element of F-p2xp1. In this paper we show that appart from an exception, there always exist C-1,C-1 is an element of F-p1xp1, C-2,C-2 is an element of F-p2xp2 and C-2,C-3 is an element of F-p2xp3 such that the pair (A(1), A(2)) = ([(C1,1)(C2,1) (C1,2)(C2,2)], [(C1,3)(C2,3)]) is completely controllable. In other words, we study the possibility of the linear system (chi) over dot (t) = A(1 chi)(t) + A(2 zeta)(t) being completely controllable, when C-1,C-2, C-1,C-3 and C-2,C-1 are prescribed and the other blocks are unknown. We also describe the possible characteristic polynomials of a partitioned matrix of the form [GRAPHICS] where C-1,C-1, C-2,C-2, C-3,C-3 are square submatrices (not necessarily with the same size), when C-1,C-2, C-1,C-3 and C-2,C-1 are fixed and the other blocks vary.