A Sufficient Condition for Function Space Controllability of a Linear Neutral System

A proof is given for the following conjecture. When ${\operatorname{rank}}[b,A_{ - 1} b, \cdots ,A_{ - 1}^{n - 1} b] = n$, a sufficient condition for function space controllability of $\dot x(t) = A_{ - 1} \dot x(t - h) + A_0 x(t) + A_1 x(t - h) + bu(t)$ is that $K(\lambda )\zeta (e^{ - \lambda h} ) \ne 0$ for all complex $\lambda $, where $K(\lambda )$ is a $n \times n$ polynomial matrix in $\lambda $ constructed from $A_{ - 1} $, $A_0 $, $A_1 $, b and $\zeta (S)$ is the transpose of $[1,S, \cdots ,S^{n - 1} ]$.