Comments on "Exact control of linear systems with multiple controls"

Given a linear system
\dot{x} = Ax + Bu
, where
A
and
B
are
n \times n
and
n \times m
matrices, with
m \leq n
and
B
is of full rank, Farlow's theorem in the paper gives a necessary condition for the existence of coefficients for
nm
rectangular pulse functions that constitute the control vector, so that the state of the system is driven to zero in a finite given time. It is shown in this note that rank adj
(A - s_{i}I)
, where
A
has
n
distinct eigenvalues
s_{i}, i = 1, 2,..., n
, is one. As a result, there are only
n
linearly independent equations in Farlow's condition for the
nm
coefficients and as such only
n
of the coefficients are determined uniquely. The other
n(m -1)
coefficients can be chosen to achieve other desirable characteristics of the system response.