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 righthand 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 discretetime 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 scalarvalued bounded sequences with the property
and are positive scalarvalued 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 discretetime 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 discretetime 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 closedloop 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.