Now, we are ready to establish the main results of the paper, which will be valid for the system (1.1)-(1.2) with slowly varying coefficients.
Consider in
the equation
subject to the initial conditions (1.2), where
is a given integer and
is a variable
matrix.
Proposition 3.1.
Suppose that
-
(a)
-
(b)
-
(c)
Then the zero solution of system (3.1)–(1.2) is globally exponentially stable. Moreover, any solution of (3.1) satisfies the inequality
where
Proof.
Rewrite (3.1) in the form
with a fixed nonnegative integer
. The variation of constants formula yields
Taking
, we have
Hence,
Thus,
Hence,
From this inequality we obtain
But, the right-hand side of this inequality does not depend on
. Thus, it follows that
This proves the global stability of the zero solution of (3.1)–(1.2).
To establish the global exponential stability of (3.1)–(1.2), we take the function
with
small enough, where
is a solution of (3.1).
Substituting (3.13) in (3.1), we obtain
where
Applying the above reasoning to (3.14), according to inequality (3.3), it follows that
is a bounded function. Consequently, relation (3.13) implies the global exponential stability of the zero solution of system (3.1)–(1.2).
Computing the quantities
and
, defined by
is not an easy task. However, in this section we will improve the estimates to these formulae.
Proposition 3.2.
Assume that (a) and (b) hold, and in addition
where
is the spectral radius of
for each
If
then the zero solution of system (3.1)–(1.2) is globally exponentially stable.
Proof.
Let us turn now to inequality (3.3). Firstly we will prove the inequality
where
Consider
By Theorem A, we have
But
Hence,
Proceeding in a similar way, we obtain
These relations yield inequality (3.19). Consequently,
where
Relation (3.26) proves the global stability of the zero solution of system (3.1)–(1.2). Establishing the exponential stability of this equation is enough to apply the same arguments of the Proposition 3.1.
Theorem 3.3.
Under the assumption (a), let
be stabilizable for each fixed
with respect to a matrix function
satisfying the following conditions:
-
(i)
-
(ii)
and
-
(iii)
If,
then system (1.1)-(1.2) is globally exponentially stabilizable by means of the feedback law (2.1).
Proof.
Rewrite (1.4) in the form
where 
According to (i), (ii), and (iii), the conditions (b) and (3.17) hold. Furthermore, condition (3.28) assures the existence of a matrix function
such that condition (3.18) is fulfilled. Thus, from Proposition 3.2, the result follows.
Put
where the minimum is taken over all
matrices
satisfying (i), (ii), and (iii).
Corollary 3.4.
Suppose that (a) holds, and the pair
is stabilizable for each fixed
If
then the system (1.1)-(1.2) is globally exponentially stabilizable by means of the feedback law (2.1).
Now, consider in
the discrete-time control system
subject to the same initial conditions (1.2), where
and
are constant matrices. In addition, one assumes that the pair
is stabilizable, that is, there is a constant matrix
such that all the eigenvalues of
are located inside the unit disk. Hence,
In this case,
and
Thus,
Hence, Theorem 3.3 implies the following corollary.
Corollary 3.5.
Let
be a stabilizable pair of constant matrices, with respect to a constant matrix
satisfying the condition
Then system (3.32)-(1.2), under condition (a), is globally exponentially stabilizable by means of the feedback law (2.1).
Example 3.6.
Consider the control system in
:
where
and
, subject to the initial conditions
where
is a given function with values in
,
are positive scalar-valued bounded sequences with the property
and
are positive scalar-valued sequences with
In the present case, the pair
is controllable. Take
Then
where
and 
By inequality
it follows that
Assume that
Since
and
are constants, by (3.37) we have
Hence, according to (3.28),
If
and
are small enough such that for some
and
we have
then by Theorem 3.3, system (3.35)-(3.36), under conditions (3.37) and (3.38), is globally exponentially stabilizable.
In the same way, Theorem 3.3 can be extended to the discrete-time control system with multiple delays
where
(
) are variable
matrices, 
Denote
Theorem 3.7.
Let
be stabilizable for each
with respect to a matrix function
satisfying the conditions (i), (ii), and (iii). In addition, assume that
If
then system (3.45)-(3.46) is globally exponentially stabilizable by means of the feedback law (2.1). Moreover, any solution of (3.45)-(3.46) satisfies the inequality
As an application, one consider, the stabilization of the nonlinear discrete-time control system
where
is a given nonlinear function satisfying
for some positive numbers
and 
One recalls that nonlinear control system (3.51)-(3.52) is stabilizable by a feedback control
where
is a matrix, if the closed-loop system
is asymptotically stable.
Theorem 3.8.
Under (3.53), let
be stabilizable for each
with respect to a matrix function
satisfying conditions (i), (ii), and (iii). In addition, assume that
If
then system (3.51)-(3.52) is globally exponentially stabilizable by means of the feedback law (2.1).
Proof.
Rewrite (3.54) in the form
where 
Thus, by reasoning as in Theorem 3.3, and using the estimates established in Proposition 3.2, the result follows.