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 right-hand 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 Lyapunov-Krasovskii 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

Since the matrix

was assumed to be positive definite, for the full derivative of Lyapunov-Krasovskii functional (1.9), we obtain the following inequality:

We will study the two possible cases (depending on the positive value of

): either

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

)

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

)

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

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

Because

, we can estimate

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
.