Exact control of linear systems with multiple controls

It is shown how the linear control system
\dot{x} = Ax + Bu
where
A
and
B
are
n \times n
and
n \times m
matrices, respectively, can be driven from an arbitrary initial condition
x(0) = x_{0} \in E^{n}
to an arbitrary final condition
x(T) = x_{f} \in E^{n}
by a vector
u = (u_{i})
of piecewise constant controls
u_{i}(t)
in an arbitrary time period
T
. A simple algorithm for finding
u
is given along with a worked example.