Controllability of pairs of matrices with prescribed entries

Let FF be an infinite field and let (A1,A2)=([a1,1a1,2a2,1a2,2],[a1,3a2,3]),where the entries ai,j∈F,i∈{1,2},j∈{1,2,3}ai,j∈F,i∈{1,2},j∈{1,2,3}. In this paper we establish necessary and sufficient conditions under which it is possible to prescribe some entries of [A1A2],so that the pair (A1,A2)(A1,A2) is completely controllable.