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
or
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
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
.