Controllability of partially prescribed matrices

Abstract

LetF be an infinite field and letn,p ₁,p ₂,p ₃ be positive integers such thatn =p ₁ +p ₂ +p ₃. Let$C_{1,2} \in F^{p_1 \times p_2 }$,$C_{1,3} \in F^{p_1 \times p_3 }$ and$C_{2,1} \in F^{p_2 \times p_1 }$. In this paper we show that appart from an exception, there always exist$C_{1,1} \in F^{p_1 \times p_1 }$,$C_{2,2} \in F^{p_2 \times p_2 }$ and$C_{2,3} \in F^{p_2 \times p_3 }$ such that the pair

$$(A_1 , A_2 ) = \left( {\left[ {\begin{array}{*{20}c} {C_{1,1} } \\ {C_{2,1} } \\ \end{array} \begin{array}{*{20}c} {C_{1,2} } \\ {C_{2,2} } \\ \end{array} } \right],\left[ {\begin{array}{*{20}c} {C_{1,3} } \\ {C_{2,3} } \\ \end{array} } \right]} \right)$$

is completely controllable. In other words, we study the possibility of the linear system χ (t) =A _1χ(t) +A _2ζ(t) being completely controllable, whenC _1,2,C _1,3 andC _2,1 are prescribed and the other blocks are unknown. We also describe the possible characteristic polynomials of a partitioned matrix of the form

$$C = \left[ {\begin{array}{*{20}c} {C_{1,1} } \\ {C_{2,1} } \\ {C_{3,1} } \\ \end{array} \begin{array}{*{20}c} {C_{1,2} } \\ {C_{2,2} } \\ {C_{3,2} } \\ \end{array} \begin{array}{*{20}c} {C_{1,3} } \\ {C_{2,3} } \\ {C_{3,3} } \\ \end{array} } \right] \in F^{n \times n} ,$$

whereC _1,1,C _2,2,C _3,3 are square submatrices (not necessarily with the same size), whenC _1,2,C _1,3 andC _2,1 are fixed and the other blocks vary.