First we give two definitions of stability to be used later on.
Definition 2.1.
The zero solution of the system of equations of neutral type (1.1) is called exponentially stable in the metric if there exist constants , and such that, for an arbitrary solution of (1.1), the inequality
holds for .
Definition 2.2.
The zero solution of the system of equations of neutral type (1.1) is called exponentially stable in the metric if it is stable in the metric and, moreover, there exist constants , , and such that, for an arbitrary solution of (1.1), the inequality
holds for .
We will give estimates of solutions of the linear system (1.1) on the interval using the functional (1.9). Then it is easy to see that an inequality
holds on . We will use an auxiliary dimensional matrix:
depending on the parameter and the matrices , , . Next we will use the numbers
The following lemma gives a representation of the linear neutral system (1.1) on an interval in terms of a delayed system derived by an iterative process. We will adopt the customary notation where is an integer, is a positive integer, and denotes the function considered independently of whether it is defined for the arguments indicated or not.
Lemma 2.3.
Let be a positive integer and . Then a solution of the initial problem (1.1), (1.2) is a solution of the delayed system
for where and .
Proof.
For the statement is obvious. If , replacing by , system (1.1) will turn into
Substituting (2.7) into (1.1), we obtain the following system of equations:
where . If , replacing by in (2.7), we get
We do one more iteration substituting (2.9) into (2.8), obtaining
for . Repeating this procedure times, we get the equation
for coinciding with (2.6).
Remark 2.4.
The advantage of representing a solution of the initial problem (1.1), (1.2) as a solution of (2.6) is that, although (2.6) remains to be a neutral system, its righthand side does not explicitly depend on the derivative for depending only on the derivative of the initial function on the initial interval .
Now we give a statement on the stability of the zero solution of system (1.1) and estimates of the convergence of the solution, which we will prove using LyapunovKrasovskii functional (1.9).
Theorem 2.5.
Let there exist a parameter and positive definite matrices , , such that matrix is also positive definite. Then the zero solution of system (1.1) is exponentially stable in the metric . Moreover, for the solution of (1.1), (1.2) the inequality
holds on where .
Proof.
Let . We will calculate the full derivative of the functional (1.9) along the solutions of system (1.1). We obtain
For , we substitute its value from (1.1) to obtain
Now it is easy to verify that the last expression can be rewritten as
or
Since the matrix was assumed to be positive definite, for the full derivative of LyapunovKrasovskii functional (1.9), we obtain the following inequality:
We will study the two possible cases (depending on the positive value of ): either
is valid or
holds.

(1)
Let (2.18) be valid. From (2.3) follows that
We use this expression in (2.17). Since , we obtain (omitting terms and )
or
Due to (2.18) we have
Integrating this inequality over the interval , we get

(2)
Let (2.19) be valid. From (2.3) we get
We substitute this expression into inequality (2.17). Since , we obtain (omitting terms and )
or
Since (2.19) holds, we have
Integrating this inequality over the interval , we get
Combining inequalities (2.24), (2.29), we conclude that, in both cases (2.18), (2.19), we have
and, obviously (see (1.9)),
We use inequality (2.30) to obtain an estimate of the convergence of solutions of system (1.1). From (2.3) follows that
or (because for nonnegative and )
The last inequality implies
Thus inequality (2.12) is proved and, consequently, the zero solution of system (1.1) is exponentially stable in the metric .
Theorem 2.6.
Let the matrix be nonsingular and . Let the assumptions of Theorem 2.5 with and be true. Then the zero solution of system (1.1) is exponentially stable in the metric . Moreover, for a solution of (1.1), (1.2), the inequality
where
holds on .
Proof.
Let . Then the exponential stability of the zero solution in the metric is proved in Theorem 2.5. Now we will show that the zero solution is exponentially stable in the metric as well. As follows from Lemma 2.3, for derivative , the inequality
holds if We estimate and using (2.12) and inequality . We obtain
Since
inequality (2.38) yields
Because , we can estimate
Then
Now we get from (2.40)
Since
the last inequality implies
The positive number can be chosen arbitrarily large. Therefore, the last inequality holds for every . We have obtained inequality (2.35) so that the zero solution of (1.1) is exponentially stable in the metric .