Consider the linear time-invariant systems. Then if
we have
Controllability means that every vector in 
is achievable from
by using control 
, i.e., for every state vector 
, there
is a control 
, such that equation (1.4) holds.
Note:
Originally controllability was defined as a property requiring that for
any intial state 
, there is a control 
, defined and
integrable on 
such that
The above definitions are equivalent because by defining 
we have
The matrix 
is invertible for any 
and any 
. Hence,
given any 
there is an 
such that
(and vice versa). This shows the equivalence
of the two definitions.
The left hand side of equation (1.6) can be interpreted
as a linear operator from the space of control functions to the state
space 
. It is convenient to take the space of control functions
as 
, where
that is, the space of square integrable (in the Lebesgue sense) vector
functions , with values in 
(i.e., for each 
, 
). Let
is a linear operator from the space into
the state space . Mathematically, the controllability
question is whether the range 
covers , i.e.,
Physically, controllability means that given any desired system state,
a control exists which will steer the system from the equilibrium
to the desired state. For example, in a simple system
consisting of a mass and force as control, controllability means
that any combination of position and velocity of the mass is
reachable from the zero position-zero velocity state.
Proof: Controllability is characterized by the following
condition, for 
which implies 
. This is because controllability means
that the set of all attainable vectors corresponding
to all possible choices of control is the whole space . The
only vector othogonal to the whole space is the zero vector.
Now (1.7) is equivalent to
for otherwise, choosing 
,
we would obtain
Differentiating (1.8) with respect to 
at 
,
we have
hence
Vice versa, from (1.9) and the Caley-Hamilton theorem
This is obvious for 
. For 
, by
recursively using the identity resulting from the Caley-Hamilton
theorem
we have
where
are some coefficients which depend on 
.
This proves (1.10). Now
Therefore (1.8) and (1.10) hold.
Because (1.7) and (1.9) are equivalent, the
statement (1.7) 
is equivalent
to (1.9) . This concludes
the proof.
