Estimates for Solutions to a Class of Time-delay Systems of Neutral Type with Periodic Coefficients and Several Delays

We consider a class of nonlinear time-delay systems of neutral type with periodic coefficients in linear terms and several delays. We establish conditions under which the zero solution is exponentially stable and obtain estimates characterizing exponential decay of solutions at infinity. The conditions are formulated in terms of differential matrix inequalities. All the values characterizing the decay rate are written out in explicit form.


Introduction
There is a large number of works devoted to delay differential equations (for instance, see [1,3,15,[17][18][19][20][21][22][23]28] and the bibliography therein).One of the important questions is asymptotic stability of solutions to delay differential equations.This question is very important from theoretical and practical viewpoints because delay differential equations arise in many applied problems when describing the processes whose speeds are defined by present and previous states (for example, see [16,24,25] and the bibliography therein).
Our aims are to establish conditions under which the zero solution is exponentially stable and obtain estimates characterizing exponential decay of solutions at infinity.To establish conditions of stability, researchers often use various Lyapunov or Lyapunov-Krasovskii functionals.At present, there is a large number of works in this direction; for example, see the bibliographies in the survey [2] and in the book [30] devoted wholly to obtaining conditions of stability by the use of Lyapunov-Krasovskii functionals.However, not every Lyapunov-Krasovskii functional makes it possible to obtain estimates characterizing exponential decay of solutions at infinity.In recent years, the study in this direction has developed rapidly.In the case of constant coefficients, there are a lot of works for linear delay differential equations including equations of neutral type (for example, see [20] and the bibliography therein).
The case of nonlinear equations with variable coefficients in linear terms is of special interest and is more complicated in comparison with the case of linear equations.Along with estimates of exponential decay of solutions, a very important question is deriving estimates of attraction sets for nonlinear equations.The natural problem is to obtain such estimates by means of Lyapunov-Krasovskii functionals used for exponential stability analysis of equations defined by their linear part.To the best of our knowledge, in the case of constant coefficients, the first constructive estimates were established in [5,6,27].For periodic coefficients, the first constructive estimates of attraction sets for the system using a Lyapunov-Krasovskii functional associated with the exponentially stable linear system where the matrix valued functions Here and hereinafter the matrix inequality Q > 0 (or Q < 0) means that the Hermitian matrix Q is positive (or negative) definite.In the case of the T-periodic matrix A(t) such that the zero solution to the system of ordinary differential equations is asymptotically stable, it is not difficult to construct the functional (1.5) by the use of the asymptotic stability criterion of the authors' article [4].Indeed, in accord with this criterion the following boundary value problem for the Lyapunov differential equation is uniquely solvable for every continuous matrix Extend T-periodically the matrix H(t) to the whole half-axis {t > 0} and use it in (1.5), since (1.6) are fulfilled.In view of [5,6] solutions to (1.4) are asymptotically stable if there exists a matrix K(s) satisfying (1.7) and such that the matrix is positive definite.Note that this condition is equivalent to the matrix inequality Obviously, for a wide class of T-periodic matrices B(t), the matrix K(s) can be found in the form The usage of the functional (1.5) allowed us to obtain estimates of exponential decay of solutions to the linear system (1.4).The authors considered in [6,26] nonlinear systems of delay differential equations of the form (1.3), where Using the same functional (1.5), conditions of asymptotic stability of the zero solution were obtained, estimates characterizing the decay rate at infinity were established, and estimates of attraction sets of the zero solution were derived.It should be noted that the estimates are constructive.All the values characterizing the decay rate and attraction sets depend on the matrices H(t) and K(s).As was mentioned above, to construct these matrices it is sufficient to solve the boundary value problem (1.8) for the Lyapunov differential equation with periodic coefficients.The authors in [4] showed that this problem is well-conditioned from the viewpoint of perturbation theory.Therefore we may apply numerical methods for solving this problem to a high degree of accuracy.A survey of computational methods for continuoustime periodic systems can be found in [29].Thus, the proposed approach makes it possible to study numerically exponential stability of solutions to time-delay systems with periodic coefficients in linear terms.
To study exponential stability of solutions to the systems of linear differential equations of neutral type with constant coefficients, the first author in [7] introduced the Lyapunov-Krasovskii functional where H = H * > 0 and the matrix K(s) satisfies (1.7).Using this functional, the study of exponential stability of solutions to systems of the form (1.1) with constant coefficients and one delay was conducted in [7,8,10,14].There, conditions of exponential stability of the zero solution, estimates of exponential decay of solutions at infinity, and estimates of attraction sets of the zero solution were obtained.Some examples given in [14] show effectiveness of the proposed approach.
The usage of the functionals (1.5) and (1.9) leads to the idea of constructing the Lyapunov-Krasovskii functional for the study of exponential stability of solutions to the linear time-delay system of neutral type with periodic coefficients d dt (y(t) + Dy(t − τ)) = A(t)y(t) + B(t)y(t − τ), t > 0. (1.11) Using this functional, the authors in [11] established conditions of exponential stability of the zero solution to (1.11) and derived estimates characterizing exponential decay of solutions at infinity.
In this article we consider the nonlinear time-delay system (1.1) with several delays.As was mentioned above, our aims are to establish conditions of exponential stability of the zero solution to (1.1) and to obtain estimates characterizing exponential decay of solutions to (1.1) at infinity.It should be noted that, in the case of constant coefficients, similar results were established in [9,12,13].In Sections 2, 3 we study the linear time-delay system We formulate the main results for (1.12) in Section 2 and prove them in Section 3. Using these results, we formulate the main results for (1.1) in Section 4 and prove them in Section 5.

Main results for (1.12)
In this section we consider the linear time-delay system (1.12).As was mentioned above, the case of the system with one delay (m = 1) was studied in [11].Hereafter we consider m ≥ 2.
Theorem 2.1.Suppose that there exist (n such that the matrix Then the zero solution to (1.12) is exponentially stable.
Remark 2.2.In the case of m = 1, the matrix C(t) defined by (2.4) should be replaced with the matrix (see [11]) .
Consider the initial value problem for (1.12) where ) is a given vector function.Assuming that the conditions of Theorem 2.1 are satisfied, below we provide estimates characterizing exponential decay of solutions to (2.5) as t → ∞.To formulate the results we introduce some notations.If the matrix H(t) satisfies the conditions of Theorem 2.1, then In this case H(t) > 0 on [0, T] (see [4]).Extend T-periodically this matrix to the whole half-axis {t ≥ 0}, keeping the same notation.Using this matrix H(t) and the matrices K j (s), j = 1, . . ., m, satisfying the conditions of Theorem 2.1, we define and (2.7) We put ) It is not hard to show that the spectrum of the matrix D belongs to the unit disk {λ ∈ C : |λ| < 1} if the conditions of Theorem 2.1 are fulfilled; i.e., if the matrix C(t) is positive definite.Hence, D j → 0 as j → ∞.Let l be the minimal positive integer such that D l < 1.In dependence on D l , below in Theorems 2.3-2.5 we establish estimates if respectively.
Theorem 2.3.Let the conditions of Theorem 2.1 be satisfied and Then a solution to the initial value problem (2.5) satisfies the estimate where α, β(t), β + , and Φ are defined in (2.10) and (2.11).
Theorem 2.4.Let the conditions of Theorem 2.1 be satisfied and Then a solution to the initial value problem (2.5) satisfies the estimate where α, β(t), β + , β − , and Φ are defined in (2.10) and (2.11), Theorem 2.5.Let the conditions of Theorem 2.1 be satisfied and Then a solution to the initial value problem (2.5) satisfies the estimate where α, β − , and Φ are defined in (2.10) and (2.11).
We prove Theorems 2.3-2.5 in Section 3. Obviously, Theorem 2.1 immediately follows from these theorems.

Proof of the main results for (1.12)
First, we formulate the auxiliary lemma of the matrix theory used by us further.Here and hereafter we denote by I the unit matrix.
be a Hermitian positive definite matrix with continuous entries.Then the representation holds where , and Q 33 (t) are positive definite.
To prove Theorems 2.3-2.5 we need the auxiliary results obtained below.Lemma 3.2.Let the conditions of Theorem 2.1 be satisfied.Then, for a solution to the initial value problem (2.5), the following inequality holds where V 0 (ϕ) and γ(t) are defined by (2.6) and (2.9), respectively, h min (t) > 0 is the minimal eigenvalue of the matrix H(t).
Proof.We follow the strategy in [5].Let y(t) be a solution to the initial value problem (2.5).
Using the above matrices H(t) and K j (s), j = 1, . . ., m, we consider the Lyapunov-Krasovskii functional Using the matrix C(t) defined by (2.4), we obtain where Consider the first summand in the right-hand side of (3.3).Since where Taking into account the entries of the matrix C(t) in (2.4), we have The lemma is proved.
Proofs of Theorems 2.3-2.5.In the case of one delay, in [11] the analogs of Theorems 2.3-2.5 (see Theorems 2-4 in [11]) were proved in detail by the use of the auxiliary assertions (see Lemmas 2-4 in [11]).In the present paper, using Lemmas 3.2, 3.3 and repeating the reasoning carried out when proving Theorems 2-4 in [11], we derive the required estimates for solutions to the initial value problem (2.5).
Using the proof of Lemma 3.2, we can reformulate the conditions of exponential stability of the zero solution to the system (1.12) as follows.

Main results for (1.1)
In this section we consider the nonlinear time-delay system (1.1).Using the results of Sections 2, 3, we establish conditions of exponential stability of the zero solution to (1.1) and obtain estimates characterizing exponential decay of solutions to (1.1) at infinity.
Remark 4.2.In the case of m = 1, the function q(t) defined by (4.1) and the matrix S(t) in (3.5) should be replaced with , respectively.
Consider the initial value problem for (1.1) where ϕ(t) ∈ C 1 ([−τ 1 , 0]) is a given vector function.This problem has a unique solution because the vector function F(t, u, v 1 , . . ., v m ) satisfies the Lipschitz condition with respect to u and τ j > 0, j = 1, . . ., m (for example, see [15,Ch. 1]).Assuming that the conditions of Theorem 4.1 are satisfied, below we establish estimates characterizing the rate of exponential decay of the solution as t → ∞.
We introduce the matrix It is not hard to verify that the matrix P(t) is positive definite if the matrix (S(t) − q(t)I) is positive definite (for details, see Section 5).Denote by p min (t) > 0 the minimal eigenvalue of the matrix P(t).We put As was mentioned in Section 2, the spectrum of the matrix D belongs to the unit disk {λ ∈ C : |λ| < 1} if the conditions of Theorem 2.1 are fulfilled.Hence, D j → 0 as j → ∞.Let l be the minimal positive integer such that D l < 1.In dependence on D l , we distinguish three cases and establish estimates for solutions to (4.2) if respectively.
Theorem 4.3.Let the conditions of Theorem 4.1 be satisfied and Then a solution to the initial value problem (4.2) satisfies the estimate where α, β(t), β + , and Φ are defined in (2.10) and (4.5).
Theorem 4.4.Let the conditions of Theorem 4.1 be satisfied and Then a solution to the initial value problem (4.2) satisfies the estimate where α, β(t), β + , β − , and Φ are defined in (2.10) and (4.5), Theorem 4.5.Let the conditions of Theorem 4.1 be satisfied and Then a solution to the initial value problem (4.2) satisfies the estimate where α, β − , and Φ are defined in (2.10) and (4.5).
We prove Theorems 4.3-4.5 in Section 5. Obviously, Theorem 4.1 immediately follows from these theorems.

Proof of the main results for (1.1)
As in Section 3, to prove Theorems 4.3-4.5 we need the auxiliary results obtained below.Lemma 5.1.Let the conditions of Theorem 4.1 be satisfied.Then, for a solution to the initial value problem (4.2), the following inequality holds where the entries of the matrix S(t) are defined in (3.6).