Controllability of partially prescribed matrices

Abstract

Let F be an infinite field and let n, p 1 , p 2 , p 3 be positive integers such thatIn this paper we show that appart from an exception, there always exist C 1,1 ∈ F p 1 ×p 1 , C 2,2 ∈ F p 2 ×p 2 and C 2,3 ∈ F p 2 ×p 3 such that the pairis completely controllable. In other words, we study the possibility of the linear system · χ (t) = A 1 χ(t) + A 2 ζ(t) being completely controllable, when C 1,2 , C 1,3 and C 2,1 are prescribed and the other blocks are unknown.We also describe the possible characteristic polynomials of a partitioned matrix of the form