\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 34, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/34\hfil Estimates for solutions]
{Estimates for solutions to a class of nonlinear
time-delay systems of neutral type}

\author[G. V. Demidenko, I. I. Matveeva \hfil EJDE-2015/34\hfilneg]
{Gennadii V. Demidenko, Inessa I. Matveeva}

\address{  Gennadii V. Demidenko \newline
Laboratory of Differential and Difference Equations,
Sobolev Institute of Mathematics,
4, Acad. Koptyug avenue, Novosibirsk 630090, Russia.\newline
Department of Mechanics and Mathematics,
Novosibirsk State University,
2, Pirogov street, Novosibirsk 630090, Russia}
\email{demidenk@math.nsc.ru}

\address{Inessa I. Matveeva \newline
Laboratory of Differential and Difference Equations,
Sobolev Institute of Mathematics,
4, Acad. Koptyug avenue, Novosibirsk 630090, Russia.\newline
Department of Mechanics and Mathematics,
Novosibirsk State University,
2, Pirogov street, Novosibirsk 630090, Russia}
\email{matveeva@math.nsc.ru}

\thanks{Submitted November 3, 2014. Published February 6, 2015.}
\subjclass[2000]{34K20}
\keywords{Time-delay systems; neutral type; exponential stability;
\hfill\break\indent  Lyapunov-Krasovskii functional}

\begin{abstract}
 We consider nonlinear time-delay systems of neutral type
 with constant coefficients in the linear terms,
 $$
 \frac{d}{dt}\big(y(t) + D y(t-\tau)\big)
 = A y(t) + B y(t-\tau) + F(t, y(t), y(t-\tau)).
 $$
 We obtain estimates characterizing the exponential decay of solutions
 at infinity, and dependending on the norms of the powers of $D$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks



\section{Introduction}

There is large number of works devoted to the study of delay differential
equations, see for instance
\cite{AgBeBrDo, AzMaRa, ElNo, GuKhaChe, Ha, Kha, KhuSha,KoMy, Ko, MiNi}.
The question of asymptotic stability is very important from the theoretical
and practical viewpoints, because delay differential equations arise in many
applied problems when describing the processes whose rates of change 
are defined by present and previous states; see \cite{Er, Ku, MacDo} 
and the bibliography therein.

This article presents a continuation of  our work on stability of solutions 
to delay differential  equations \cite{DeMa05, DeMa06, DeMa07, De09, DeKoSk, DeVoSk, DeMa14-1,
 DeMa14-2, MaShche, Ma}.
We consider the system of nonlinear delay differential equations
\begin{equation}
\frac{d}{dt}\big(y(t) + D y(t-\tau)\big)
= A y(t) + B y(t-\tau) + F(t, y(t), y(t-\tau)), \quad t > 0, \label{e1.1}
\end{equation}
where  $A$, $B$, $D$ are constant $(n\times n)$ matrices, $\tau > 0$
is the time delay, and $F(t,u,v)$
is a real-valued vector function satisfying the Lipschitz condition
with respect to $u$, and the inequality
\begin{equation}
\|F(t,u,v)\| \le q_1\|u\| + q_2\|v\|, \label{e1.2}
\end{equation}
for some constants $q_1,  q_2 \ge 0$.
When $D \neq 0$ this system is called one of neutral type \cite{ElNo}.

Our aim is to obtain new estimates on the exponential decay of solutions 
to \eqref{e1.1} without
finding roots of characteristic quasipolynomials defined by the 
linear part of \eqref{e1.1} (when $F(t,u,v) \equiv 0$).
In recent years, the study in this direction has developed rapidly. 
For constant coefficients, there are a lot of works for linear delay
differential equations including equations of neutral type.
It should be noted that various Lyapunov-Krasovskii functionals
are used for obtaining exponential estimates
(see the bibliography in \cite{Kha}).

The case of nonlinear equations 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 for attraction sets 
of nonlinear equations.
The natural problem is to obtain such estimates
by means of the Lyapunov-Krasovskii functionals
used for exponential stability analysis of equations defined by the linear part.
To the best of our knowledge, the first constructive estimates of
attraction sets for the  system
\begin{equation}
\frac{d}{dt}y(t) = A y(t) + B y(t-\tau) + F(t, y(t), y(t-\tau)),
\label{e1.3}
\end{equation}
using Lyapunov-Krasovskii functionals associated with
the exponentially stable linear system
\begin{equation}
\frac{d}{dt}y(t) = A y(t) + B y(t-\tau),
\label{e1.4}
\end{equation}
were obtained in \cite{DeMa05, DeMa06, DeMa07, MeAgNi}.

To study the asymptotic stability of solutions to \eqref{e1.4},
 the authors in  \cite{DeMa05} proposed to use the Lyapunov-Krasovskii
functional
\begin{equation}
\langle H y(t),y(t) \rangle
+ \int^t_{t-\tau} \langle K(t-s)y(s), y(s) \rangle \, ds,
\label{e1.5}
\end{equation}
where the real matrices $H$ and $K(s)$ satisfy
\begin{equation}
\begin{gathered}
H = H^* > 0, \quad
K(s) = K^*(s) \in C^1[0,\tau], \\
K(s) > 0, \quad \frac{d}{ds}K(s) < 0, \quad s \in [0,\tau],
\end{gathered}\label{e1.6}
\end{equation}
where $H>0$ means that $H$ is postive definite.


The usage  of \eqref{e1.5} allowed us to obtain estimates for the
 exponential decay of solutions to the linear system \eqref{e1.4}.
The authors in  \cite{DeMa05, DeMa06}  considered  \eqref{e1.3}, with
$$
\|F(t,u,v)\| \le q_1\|u\|^{1+\omega_1} + q_2\|v\|^{1+\omega_2},
\quad
q_1 \ge 0, \; q_2 \ge 0, \; \omega_1 \ge 0, \; \omega_2 \ge 0.
$$
Using the  functional in \eqref{e1.5}, 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. 
Using a generalization of the functional in \eqref{e1.5},
analogous results were obtained for linear and nonlinear systems of
delay differential equations with periodic coefficients in the linear terms,
see \cite{DeMa05, DeMa06, DeMa07, MaShche, Ma}.

To study the exponential stability of solutions to the systems
of linear differential equations
\begin{equation}
\frac{d}{dt}(y(t) + Dy(t-\tau)) = A y(t) + B y(t-\tau)
\label{e1.7},
\end{equation}
the first author in \cite{De09} introduced the Lyapunov-Krasovskii functional
\begin{equation}
\begin{aligned}
V(\varphi) &= \langle H\big(\varphi(0) + D\varphi(- \tau)\big),
(\varphi(0) + D\varphi(- \tau)) \rangle\\
&\quad + \int^0_{-\tau} \langle K(-s)\varphi(s), \varphi(s) \rangle \, ds,
\quad \varphi(s) \in C[-\tau, 0]\,,
\end{aligned} \label{e1.8}
\end{equation}
where the matrices $H$ and $K(s)$ satisfy \eqref{e1.6}.
In particular, the following result was obtained.

\begin{theorem} \label{thm1}
Suppose that there exist matrices $H$ and $K(s)$ satisfying \eqref{e1.6}
and that the matrix
\begin{equation}
C = -\begin{pmatrix}
HA+A^*H+K(0) & HB+A^*HD \\
B^*H+D^*HA & D^*HB+B^*HD-K(\tau)
\end{pmatrix}
\label{e1.9}
\end{equation}
is positive definite.
Then the zero solution to \eqref{e1.7} is exponentially stable.
\end{theorem}

Using the functional in \eqref{e1.8},
the study of exponential stability of solutions to \eqref{e1.1}
was conducted in \cite{De09, DeKoSk, DeVoSk, DeMa14-1, DeMa14-2}.
There, conditions for exponential stability of the zero solution,
estimates for the exponential decay of solutions at infinity,  
and estimates of attraction sets of the zero solution were obtained.

Note that in \cite{De09, DeKoSk} the estimates of exponential decay
 of solutions to \eqref{e1.1} were obtained when $\|D\| < 1$
(here and thereafter we use the spectral norm of matrices).
In \cite{DeVoSk}, for the linear case ($F(t,u,v) \equiv 0$)
analogous estimates were established when the spectrum of the matrix
$D$ belongs to the unit disk
$\{\lambda \in {\mathbb C}: |\lambda| < 1\}$.
However, in the case of $\|D\| < 1$,
the estimates are weaker in comparison with the estimates obtained in \cite{De09}.
More precise exponential estimates for the linear systems were obtained
in \cite{DeMa14-1, DeMa14-2}.
Moreover, in \cite{DeMa14-2} the authors established estimates of exponential decay
of solutions of the linear time-delay systems of neutral type
with periodic coefficients.

In this article we consider the nonlinear time-delay system \eqref{e1.1} 
 when the spectrum of the matrix $D$ belongs to the unit disk.
Our aim is to obtain estimates characterizing exponential decay of solutions
at infinity dependending on the norms $\|D^j\|$.



\section{Estimates of solutions}

Consider the initial value problem for \eqref{e1.1},
\begin{equation}
\begin{gathered}
\frac{d}{dt} (y(t) + Dy(t-\tau)) = Ay(t) + By(t-\tau) + F(t,y(t), y(t-\tau)),
\quad t > 0, \\
y(t)=\varphi(t), \quad t \in [-\tau,0], \\
y(0+) = \varphi(0),
\end{gathered} \label{e2.1}
\end{equation}
where $\varphi(t) \in C^{1}[-\tau,0]$
is a given vector function.

Suppose that the conditions of Theorem \ref{thm1} are satisfied.
Using the matrices $H$ and $K(s)$, we introduce 
\begin{gather}
S =\begin{pmatrix}
- HA - A^*H - K(0) & HAD + K(0)D - HB \\
D^*A^*H + D^*K(0) - B^*H & K(\tau) - D^*K(0)D
\end{pmatrix}, \label{e2.2}
\\
q =  \Big(q_1 + \sqrt{q^2_1 + (q_1\|D\|+q_2)^2}\Big)\|H\|, \label{e2.3} \\
\begin{aligned}
R &= - HA - A^*H - K(0) - qI 
- (HAD + K(0)D - HB) \Big[K(\tau) \\
&\quad- D^*K(0)D - qI\Big]^{-1} (HAD + K(0)D - HB)^*,
\end{aligned}\label{e2.4}
\end{gather}
where $I$ is the unit matrix.
It is not hard to verify that the matrix $C$
in \eqref{e1.9} is positive definite if and only if the matrix
$S$ is positive definite.
Note that
$R$
is positive definite if the matrix
$S-qI$
is positive definite.


\begin{theorem} \label{thm2}
 Let the conditions of Theorem \ref{thm1} be satisfied.
Suppose that the parameters $q_1$, $q_2$ are such that the matrix
$S-qI$ is positive definite.
Let $k > 0$ be the maximal number such that
\begin{equation}
\frac{d}{ds}K(s) + k K(s) \le 0,
\quad s \in [0,\tau]\,.
\label{e2.5}
\end{equation}
Let $r_{\rm min} > 0$ be the minimal eigenvalue of the matrix $R$.
Then, each solution to  \eqref{e2.1} satisfies
\begin{equation}
\|y(t) + Dy(t-\tau)\|
\le \sqrt{\frac{V(\varphi)}{h_{\rm min}}} \exp\Big(-\frac{\gamma t}{2 \|H\|}\Big),
\quad t > 0, \label{e2.6}
\end{equation}
where $V(\varphi)$ is defined by \eqref{e1.8},
$h_{\rm min} > 0$ is the minimal eigenvalue of the matrix $H$, and
\begin{equation}
\gamma = \min\{r_{\rm min}, \ k\|H\|\} > 0.
\label{e2.7}
\end{equation}
\end{theorem}

\begin{proof}
We follow the strategy in \cite{DeMa05}. Let
$y(t)$ be a solution to \eqref{e2.1}.
Using the matrices  $H$ and $K(s)$
indicated in Theorem \ref{thm1},
we consider the Lyapunov-Krasovskii functional defined in \eqref{e1.8}.
Introducing the conventional notation
$$
y_t:  \theta \to y(t+\theta), \quad \theta \in [-\tau,0],
$$
we have
\begin{align*}
V(y_t) 
&= \langle H(y_t(0) + Dy_t(- \tau)), (y_t(0) + Dy_t(- \tau)) \rangle
+ \int^0_{-\tau} \langle K(-\theta)y_t(\theta), y_t(\theta) \rangle \, d\theta \\
&=\langle H(y(t) + Dy(t- \tau)), (y(t) + Dy(t- \tau)) \rangle
+ \int^t_{t-\tau} \langle K(t-s)y(s), y(s) \rangle \, ds.
\end{align*}
The time derivative of this functional is
\begin{align*}
\frac{d}{dt}V(y_t)
& \equiv \langle H (Ay(t)+By(t-\tau)), (y(t)+Dy(t-\tau)) \rangle \\
&\quad + \langle H (y(t)+Dy(t-\tau)), (Ay(t)+By(t-\tau)) \rangle \\
&\quad + \langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle \\
&\quad + \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\quad + \langle K(0)y(t),y(t) \rangle - \langle K(\tau)y(t-\tau),y(t-\tau) \rangle \\
&\quad + \int_{t-\tau}^{t} \big\langle \frac{d}{dt}K(t-s)y(s),y(s) \big\rangle ds.
\end{align*}
Using the matrix  $C$ defined in \eqref{e1.9},
we obtain
\begin{equation}
\begin{aligned}
\frac{d}{dt} V(y_t)
&\equiv -\Big\langle C \begin{pmatrix} y(t) \\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t) \\ y(t-\tau)\end{pmatrix}\Big\rangle \\
&\quad + \langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle\\
&\quad + \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\quad + \int^t_{t-\tau} \big\langle \frac{d}{dt}K(t-s)y(s),y(s) \big\rangle ds.
\end{aligned} \label{e2.8}
\end{equation}
Consider the first summand in the right-hand side of \eqref{e2.8}.
Since
$$
\begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix} =
\begin{pmatrix}
I & -D
\\
0 & I
\end{pmatrix}
\begin{pmatrix} y(t) + D y(t-\tau) \\ y(t-\tau)\end{pmatrix},
$$
it follows that
$$
\Big\langle C \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}\Big\rangle
\equiv \Big\langle S
\begin{pmatrix} y(t) + Dy(t-\tau)\\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t) + Dy(t-\tau)\\ y(t-\tau)\end{pmatrix}\Big\rangle,
$$
where
$$
S =\begin{pmatrix}
I & 0
\\
-D^* & I
\end{pmatrix}
C
\begin{pmatrix}
I & -D
\\
0 & I
\end{pmatrix}
=
\begin{pmatrix}
S_{11} & S_{12}
\\
S^*_{12} & S_{22}
\end{pmatrix}
$$
which is defined in \eqref{e2.2}.

Now we consider the second and the third summands in the right-hand side 
of \eqref{e2.8}.
In view of \eqref{e1.2} we have
\begin{align*}
&\langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle\\
&+ \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\le 2\|H\|\bigl(q_1\|y(t)\| + q_2 \|y(t-\tau)\|\bigr) \|y(t)+Dy(t-\tau)\|\\
&\le 2 q_1 \|H\|\|y(t)+Dy(t-\tau)\|^2 + 2(q_1\|D\|+q_2)\|H\| \|y(t-\tau)\| 
 \|y(t)+Dy(t-\tau)\| \\
&\le q \bigl(\|y(t)+Dy(t-\tau)\|^2 + \|y(t-\tau)\|^2\bigr),
\end{align*}
where  $q$ is given in \eqref{e2.3}.
Hence,
\begin{equation}
\begin{aligned}
&-\Big\langle C \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}\Big\rangle
+ \langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle\\
&+ \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
&\le - \Big\langle
(S-qI) \begin{pmatrix} y(t)+Dy(t-\tau) \\ y(t-\tau)\end{pmatrix},
\begin{pmatrix} y(t)+Dy(t-\tau) \\y(t-\tau)\end{pmatrix}
\Big\rangle.
\end{aligned} \label{e2.9}
\end{equation}
By the conditions of Theorem \ref{thm2}, the matrix $S-qI$
is positive definite. Using the representation
\begin{align*}
S-qI &= \begin{pmatrix}
I & S_{12} (S_{22} - qI)^{-1}
\\
0 & I
\end{pmatrix}
\begin{pmatrix}
S_{11} - qI - S_{12} (S_{22} - qI)^{-1} S^*_{12} & 0
\\
0 & S_{22} - qI
\end{pmatrix} \\
&\quad \times
\begin{pmatrix}
I & 0
\\
(S_{22} - qI)^{-1} S^*_{12} & I
\end{pmatrix},
\end{align*}
we have
\begin{align*}
&\Big\langle
(S-qI) \begin{pmatrix}y(t)+Dy(t-\tau)\\ y(t-\tau)\end{pmatrix},
\begin{pmatrix}y(t)+Dy(t-\tau)\\ y(t-\tau)\end{pmatrix}
\Big\rangle \\
& \ge \langle
[S_{11} - qI - S_{12} (S_{22} - qI)^{-1} S^*_{12}]
(y(t)+Dy (t-\tau)), (y(t)+Dy(t-\tau)) \rangle.
\end{align*}
Since the matrix $S-qI$ is positive definite,  the matrix
$$
R = S_{11} - qI - S_{12} (S_{22} - qI)^{-1} S^*_{12}
$$
is positive definite.
Taking into account \eqref{e2.2}, the matrix $R$ has the form \eqref{e2.4}.
Consequently, from \eqref{e2.9} we obtain
\begin{equation}
\begin{aligned}
&-\Big\langle C \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix},
 \begin{pmatrix} y(t)\\ y(t-\tau)\end{pmatrix}\Big\rangle \\
& + \langle H F(t,y(t),y(t-\tau)), (y(t)+Dy(t-\tau)) \rangle \\
& + \langle H (y(t)+Dy(t-\tau)), F(t,y(t),y(t-\tau)) \rangle \\
& \le - \langle R (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \rangle\\
&\le - r_{\rm min} \|y(t) + D y(t-\tau)\|^2,
\end{aligned}\label{e2.10}
\end{equation}
where $r_{\rm min} > 0$ is the minimal eigenvalue of $R$.
Using the matrix $H$, we have
$$
\|y(t) + D y(t-\tau)\|^2
\ge \frac{1}{\|H\|}
\langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \rangle.
$$
By \eqref{e2.10}, from \eqref{e2.8} we derive
\begin{align*}
\frac{d}{dt} V(y_t)
&\le - \frac{r_{\rm min}}{\|H\|}
\langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \rangle \\
&\quad + \int^t_{t-\tau}
\big\langle \frac{d}{dt}K(t-s)y(s),y(s) \big\rangle ds.
\end{align*}
Using \eqref{e2.5}, we have
\begin{align*}
\frac{d}{dt} V(y_t)
&\le - \frac{r_{\rm min}}{\|H\|}
 \langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \rangle \\
&\quad - k \int^t_{t-\tau}
\langle K(t-s)y(s),y(s) \rangle ds.
\end{align*}
Taking into account the definition of the functional \eqref{e1.8}, we obtain
$$
\frac{d}{dt} V(y_t) \le -\frac{\gamma}{\|H\|} V(y_t),
$$
where
$\gamma = \min\{r_{\rm min}, k\|H\|\} > 0$.
From this differential inequality we obtain the estimate
$$
V(y_t) \le V(\varphi)\exp\Big(-\frac{\gamma t}{\|H\|}\Big).
$$
Clearly,
$$
\|y(t) + Dy(t-\tau)\|^2
\le \frac{1}{h_{\rm min}}
\langle H (y(t) + D y(t-\tau)), (y(t) + D y(t-\tau)) \rangle,
$$
where $h_{\rm min}$ is the minimal eigenvalue of $H$.
Then, using the definition of the functional in \eqref{e1.8}, we have
$$
\|y(t) + Dy(t-\tau)\|
\le \sqrt{\frac{V(y_t)}{h_{\rm min}}}
\le \sqrt{\frac{V(\varphi)}{h_{\rm min}}}
\exp\Big(-\frac{\gamma t}{2 \|H\|}\Big).
$$
The proof is complete.
\end{proof}

In the next theorem, based on \eqref{e2.6}, we prove estimates for
the solution to \eqref{e2.1}. These estismates 
 will be used for proving our main results.
We introduce the following values:
\begin{equation}
\alpha = \sqrt{\frac{V(\varphi)}{h_{\rm min}}},\quad
\beta = \frac{\gamma}{2 \|H\|}, \quad
\Phi = \max_{s \in [-\tau,0]}\|\varphi(s)\|.
\label{e2.11}
\end{equation}


\begin{theorem} \label{thm3}
Let the conditions of Theorem \ref{thm2} be satisfied.
Then, on each segment $t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$,
the solution $y$ to \eqref{e2.1} satisfies 
\begin{equation}
\|y(t)\| \le \alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)} + \|D^{k+1}\|\Phi,
\label{e2.12}
\end{equation}
where $\alpha$, $\beta$, and $\Phi$ are defined in \eqref{e2.11}.
\end{theorem}

\begin{proof}
Obviously, taking into account \eqref{e2.11},
by \eqref{e2.6} for $t \in [0,\tau)$ we have the inequality
$$
\|y(t)\| \le \alpha  e^{-\beta t} + \|D y(t-\tau)\|
\le \alpha  e^{-\beta t} + \|D\| \Phi
$$
which gives us \eqref{e2.12} for $k=0$.
Let $t \in [k\tau, (k+1)\tau)$, $k =1, 2, \dots$.
It is not hard to show the sequence of the inequalities
\begin{align*}
\|y(t)\|& \le \alpha  e^{-\beta t} + \|D y(t-\tau)\| \\
&\le \alpha  e^{-\beta t} + \|D y(t-\tau) + D^2 y(t-2\tau)\|
+ \|D^2 y(t-2\tau) + D^3 y(t-3\tau)\| + \dots \\
& \quad + \|D^k y(t-k\tau) + D^{k+1} y(t-(k+1)\tau)\|
+ \|D^{k+1} y(t-(k+1)\tau)\| \\
&\le \alpha  e^{-\beta t} + \|D\|\,\|y(t-\tau) + D y(t-2\tau)\|
+ \|D^2\|\,\|y(t-2\tau) + D y(t-3\tau)\| \\
&\quad + \dots  + \|D^k\|\,\|y(t-k\tau) + D y(t-(k+1)\tau)\|
+ \|D^{k+1}\|\,\|y(t-(k+1)\tau)\|.
\end{align*}
By \eqref{e2.6} we derive the estimate
\begin{align*}
\|y(t)\| 
&\le \alpha  e^{-\beta t} + \alpha \|D\| e^{-\beta (t-\tau)}
+ \alpha \|D^2\| e^{-\beta (t-2\tau)} + \dots \\
&\quad + \alpha\|D^k\| e^{-\beta (t-k\tau)}+ \|D^{k+1}\| \Phi
\end{align*}
which implies \eqref{e2.12}.
The proof is complete.
\end{proof}

Next we obtain estimates for solutions to \eqref{e2.1} on the whole 
half-line $\{t > 0\}$.
Analogy as in \cite{De09}, we distinguish three cases
allowing us to obtain more precise estimates.
Since the spectrum of the matrix $D$ belongs to the unit disk
$\{\lambda \in {\mathbb C}: |\lambda| < 1\}$, it follows that
$\|D^j\| \to 0$ as $j \to \infty$.
Let $l > 0$ be the minimal integer such that
$\|D^l\| < 1$. In Theorems \ref{thm4}--\ref{thm6} below we establish estimates if
$$
\|D^l\| < e^{-l\beta \tau}, \quad
\|D^l\| = e^{-l\beta \tau}, \quad
e^{-l\beta \tau} < \|D^l\| < 1,
$$
respectively, where
$ \beta = \frac{\gamma}{2 \|H\|}$, with
$\gamma$ defined in \eqref{e2.7}.


\begin{theorem} \label{thm4}
Assume that
\begin{equation}
\|D^l\| < e^{-l\beta \tau}.
\label{e2.13}
\end{equation}
Then the solution to the initial value problem \eqref{e2.1} satisfies
\begin{equation}
\|y(t)\|
\le \Big[ \alpha \bigl(1-\|D^l\|e^{l\beta \tau}\bigr)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \{\|D\|e^{\beta\tau}, \dots, \|D^l\| e^{l\beta\tau}\} \Phi
\Big] e^{-\beta t}, \label{e2.14}
\end{equation}
for $t > 0$, where
$\alpha$, $\beta$, and $\Phi$
are defined in \eqref{e2.11}.
\end{theorem}

\begin{proof}
Using \eqref{e2.12}, on each segment
$t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$,
one can write the inequality
$$
\|y(t)\| \le
\Big[\alpha \sum^k_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^{k+1}\| e^{(k+1)\beta\tau} \Phi \Big]
e^{-\beta t}.
$$
In view of the condition on $\|D^l\|$,
we obtain the estimate on the whole half-line
$\{t > 0\}$,
\begin{equation}
\|y(t)\| \le
\Bigl[\alpha \sum^\infty_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \left\{\|D\|e^{\beta\tau}, \dots, \|D^l\| e^{l\beta\tau}\right\} \Phi
\Bigl] e^{-\beta t}.
\label{e2.15}
\end{equation}
Consider the series $ \sum^\infty_{j=0} \|D^j\| e^{j\beta\tau}$.
Obviously,
\begin{align*}
&\sum^\infty_{j=0} \|D^j\| e^{j\beta\tau}\\
&=\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+\sum^{2l-1}_{j=l} \|D^j\| e^{j\beta\tau}
+\sum^{3l-1}_{j=2l} \|D^j\| e^{j\beta\tau} + \dots \\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+\|D^l\| e^{l\beta\tau} \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ (\|D^l\| e^{l\beta\tau})^2 \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \dots \\
&= \Big(1 + \|D^l\| e^{l\beta\tau} + (\|D^l\| e^{l\beta\tau})^2 + \dots \Big)
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Since $\|D^l\|e^{l\beta\tau}<1$, by \eqref{e2.13},
we have
$$
\sum^\infty_{j=0} \|D^j\| e^{j\beta\tau}
\le \big(1 - \|D^l\| e^{l\beta\tau} \big)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
Using this inequality, from~\eqref{e2.15} we derive the required
estimate~\eqref{e2.14}.
\end{proof}


\begin{theorem} \label{thm5}
Assume that
\begin{equation}
\|D^l\| = e^{-l\beta \tau}. \label{e2.16}
\end{equation}
Then the solution to the initial value problem \eqref{e2.1} satisfies 
\begin{equation}
\begin{aligned}
\|y(t)\| &\le \Bigl[
\alpha \big(1+\frac{t}{l\tau}\big)
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} 
+ \max \{1, \|D\|e^{\beta\tau}, \dots,\\
&\quad  \|D^{l-1}\| e^{(l-1)\beta\tau}\} \Phi
\Bigr] e^{-\beta t}, \quad t > 0,
\end{aligned} \label{e2.17}
\end{equation}
where
$\alpha$, $\beta$, and $\Phi$
are defined in \eqref{e2.11}.
\end{theorem}

\begin{proof}
By Theorem \ref{thm3}, the solution to \eqref{e2.1} satisfies \eqref{e2.12}
 on each segment $t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$.
Consequently,
$$
\|y(t)\| \le
\Big[\alpha \sum^k_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^{k+1}\| e^{(k+1)\beta\tau} \Phi \Big]
e^{-\beta t}.
$$
Taking into account condition \eqref{e2.16} on $\|D^l\|$,
we obtain
\begin{equation}
\|y(t)\| \le
\Big[\alpha \sum^k_{j=0} \|D^j\| e^{j\beta\tau}
+ \max \{1, \|D\|e^{\beta\tau}, \dots,
\|D^{l-1}\| e^{(l-1)\beta\tau}\} \Phi \Big]
e^{-\beta t}. \label{e2.18}
\end{equation}
If $k \le l-1$, then \eqref{e2.17} follows from~\eqref{e2.18} for $t\in [0, l\tau)$.

Let $l \le k \le 2l-1$; i.e., $ 1 \le \frac{t}{l\tau} < 2$.
Consider the sum $ \sum^k_{j=0} \|D^j\| e^{j\beta\tau}$.
Clearly,
\begin{align*}
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^k_{j=l} \|D^j\| e^{j\beta\tau} \\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^l\| e^{l\beta\tau} \sum^{k-l}_{j=0} \|D^j\| e^{j\beta\tau}\\
&=\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{k-l}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Then we have
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \frac{t}{l \tau}\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
By this inequality,
\eqref{e2.17} follows from~\eqref{e2.18} for $t\in [l\tau, 2l\tau)$.

Let $ml \le k \le (m+1)l-1$, $m = 2, 3, \dots$; i.e.,
$ m \le \frac{t}{l\tau} < m+1$.
Consider the sum
$ \sum^k_{j=0} \|D^j\| e^{j\beta\tau}$.
It is not difficult to see that
\begin{align*}
&\sum^k_{j=0} \|D^j\| e^{j\beta\tau}\\
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{2l-1}_{j=l} \|D^j\| e^{j\beta\tau}
+ \dots
+ \sum^{k}_{j=ml} \|D^j\| e^{j\beta\tau} 
\\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^l\| e^{l\beta\tau} \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \dots
+ \|D^{ml}\| e^{ml\beta\tau} \sum^{k-ml}_{j=0} \|D^j\| e^{j\beta\tau}
\\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \dots
+ \sum^{k-ml}_{j=0} \|D^j\| e^{j\beta\tau}\\
&\le (1+m)\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Consequently,
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le
\big(1 + \frac{t}{l \tau} \big)\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
In view of this estimate, \eqref{e2.17} follows from~\eqref{e2.18} for
$t\in [ml\tau, (m+1)l\tau)$.
Owing to arbitrariness of $m$,
\eqref{e2.17} is valid for all $t>0$.
\end{proof}

\begin{theorem} \label{thm6}
Assume that
\begin{equation}
e^{-l\beta \tau} < \|D^l\| < 1.
\label{e2.19}
\end{equation}
Then the solution to the initial value problem \eqref{e2.1} satisfies 
\begin{equation}
\begin{aligned}
\|y(t)\|
&\le  \Bigr[
\alpha \|D^l\| e^{l\beta\tau}(\|D^l\| e^{l\beta\tau} - 1)^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\quad + \|D^l\|^{\frac{1}{l} - 1}
\max \{1, \|D\|, \dots, \|D^{l-1}\| \} \Phi \Bigr]
\exp\big( \frac{t}{l\tau} \ln\|D^l\| \big),
\end{aligned} \label{e2.20}
\end{equation}
for $t > 0$,
where $\alpha$, $\beta$, and $\Phi$ are defined in \eqref{e2.11}.
\end{theorem}

\begin{proof}
In view of Theorem \ref{thm3}, a solution to \eqref{e2.1} satisfies \eqref{e2.12}
on each segment
$t \in [k\tau, (k+1)\tau)$, $k = 0, 1, \dots$.

At first we consider the first summand in the right-hand side of \eqref{e2.12}.
For $k\le l-1$ we obviously have
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
Let $ml \le k \le (m+1)l-1$, $m = 1, 2, 3, \dots$.
Clearly,
\begin{align*}
&\sum^k_{j=0} \|D^j\| e^{j\beta\tau}\\
&= \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \sum^{2l-1}_{j=l} \|D^j\| e^{j\beta\tau}
+ \dots
+ \sum^{k}_{j=ml} \|D^j\| e^{j\beta\tau} \\
&\le \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \|D^l\| e^{l\beta\tau} \sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}
+ \dots
+ \|D^{ml}\| e^{ml\beta\tau} \sum^{k-ml}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\le \big[1 + \|D^l\|e^{l\beta\tau} + \dots + \|D^l\|^m e^{ml\beta\tau}\big]
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Consequently,
\begin{align*}
&\sum^k_{j=0} \|D^j\| e^{j\beta\tau}\\
&\le \|D^l\|^m e^{ml\beta\tau}
[1 + (\|D^l\|e^{l\beta\tau})^{-1} + \dots +
(\|D^l\| e^{l\beta\tau})^{-m}]
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\le \|D^l\|^m e^{ml\beta\tau}
[1 + (\|D^l\|e^{l\beta\tau})^{-1} + \dots +
(\|D^l\| e^{l\beta\tau})^{-m} + \dots]
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
Since $\|D^l\|e^{l\beta\tau}>1$ owing to \eqref{e2.19}, 
$$
\sum^k_{j=0} \|D^j\| e^{j\beta\tau}
\le \|D^l\|^m e^{ml\beta\tau}
\big[1 - (\|D^l\|e^{l\beta\tau})^{-1}\big]^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
$$
Taking into account that
$m l \tau \le t < (m+1) l \tau$,
we obtain
\begin{align*}
\sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)}
&\le \|D^l\|^m e^{-\beta (t-ml\tau)}
\big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\big]^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau} \\
&\le \|D^l\|^{\frac{t}{l\tau}}
\big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\big]^{-1}
\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}.
\end{align*}
As result, we derive the estimate for the first summand in \eqref{e2.12} 
for every $k$,
\begin{equation}
\begin{aligned}
&\alpha \sum^k_{j=0} \|D^j\| e^{-\beta (t-j\tau)}\\
&\le \alpha \big[1 - \left(\|D^l\|e^{l\beta\tau}\right)^{-1}\big]^{-1}
\Big(\sum^{l-1}_{j=0} \|D^j\| e^{j\beta\tau}\Big)
\exp\big(\frac{t}{l \tau} \ln \|D^l\| \big).
\end{aligned}\label{e2.21}
\end{equation}

We now consider the second summand in the right-hand side of \eqref{e2.12}.
Obviously, for $0 \le k \le l-2$, we have
$$
\|D^{k+1}\| \le \max \{\|D\|, \dots, \|D^{l-1}\| \}.
$$
Let $ml-1 \le k \le (m+1)l-2$, $m = 1, 2, \dots$.
Hence,
$$
\|D^{k+1}\| \le \|D^l\|^m \|D^{k+1-ml}\|
\le \|D^l\|^m \max \left\{1, \|D\|, \dots, \|D^{l-1}\| \right\}.
$$
Since
$\|D^l\|<1$ and $t < ((m+1)l-1)\tau$,
$$
\|D^l\|^m \le \|D^l\|^{\frac{t-(l-1)\tau}{l\tau}}
= \|D^l\|^{\frac{1}{l}-1}\exp\Big(\frac{t}{l \tau} \ln \|D^l\| \Big).
$$
Owing to arbitrariness of $m$, we infer that
$$
\|D^{k+1}\|
\le \|D^l\|^{\frac{1}{l}-1}
\max \left\{1, \|D\|, \dots, \|D^{l-1}\| \right\}
\exp\big(\frac{t}{l \tau} \ln \|D^l\| \big)
$$
for every $k$.
Taking into account the estimate~\eqref{e2.21} for the first summand
in the right-hand side of~\eqref{e2.12}, we derive~\eqref{e2.20}.
\end{proof}

We remark that the results obtained above give us the assertions on robust 
stability for \eqref{e1.7}. Indeed, consider uncertain systems of the form
\begin{equation}
\frac{d}{dt}\left(y(t) + D y(t-\tau)\right) = A y(t) + B y(t-\tau)
+ \Delta A(t) y(t)
+ \Delta B(t) y(t-\tau),
\label{e2.22}
\end{equation}
where $\Delta A(t)$ and $\Delta B(t)$ are unknown
$(n\times n)$ matrices such that
$$
\|\Delta A(t)\| \le q_1,
\quad
\|\Delta B(t)\| \le q_2.
$$
Obviously, in this case the vector function
$$
F(t,u,v) = \Delta A(t) u + \Delta B(t) v
$$
satisfies \eqref{e1.2}.
Then Theorem \ref{thm2} gives us the conditions of robust exponential
stability for \eqref{e1.7}.
From Theorems \ref{thm4}--\ref{thm6} we have the estimates of exponential decay of
solutions to \eqref{e2.22}.


\section{Illustrative examples}

Consider the system \eqref{e1.1}, where
$$
D =\begin{pmatrix}
-0.1 & 0
\\
0 & -0.1
\end{pmatrix},
\quad
A = \begin{pmatrix}
-3 & -2
\\
1 & 0
\end{pmatrix},
\quad
B = \begin{pmatrix}
0 & a
\\
a & 0
\end{pmatrix},
$$
 $a$ is a real parameter,
$F(t,u,v)$ is a real-valued vector function satisfying the Lipschitz
 condition with respect to $u$
and the inequality \eqref{e1.2}.

First we consider the linear case ($F(t,u,v) \equiv 0$); i.e. $q_1 = q_2 = 0$.
In \cite{ParkWon}, in the case of arbitrary positive $\tau$,
stability was shown for $|a| < 0.4$. The same system was studied in \cite{LiuXu},
where stability was established for $|a| < 0.533$. In \cite{BaDiKhuRy} exponential
stability was shown for $|a| \le 0.6213$.
Moreover, in the case of $a = 0.6213$ and $\tau = 1$,
the following estimate for solutions was obtained
\[
\|y(t)\| \le \Big(c_1\|y(0)\| + c_2 \sup_{-1\le s \le 0}\|y(s)\|
+ c_3 \sup_{-1\le s \le 0}\|\frac{d}{ds}y(s)\| \Big)e^{-0.00001559\,t/2},
\]
with $c_j > 0$.
In the same case, using our results, we establish the following inequality
\begin{equation}
\|y(t)\| \le d \max_{-1\le s \le 0}\|y(s)\| e^{-0.147\,t/2},
\quad d > 0. \label{e3.1}
\end{equation}
Indeed, we choose the matrices $H$ and $K(s)$ as follows
$$
H =\begin{pmatrix}
0.3 & 0.2
\\
0.2 & 0.8
\end{pmatrix},
\quad
K(s) = e^{-ks} K_0,
\quad
k=0.147,
\quad
K_0 =
\begin{pmatrix}
0.8 & 0.2
\\
0.2 & 0.2
\end{pmatrix}.
$$
Obviously, these matrices satisfy \eqref{e1.6} and \eqref{e2.5}.
Since the matrix
$$
C = \begin{pmatrix}
0.6 & 0.2 & -0.19426 & -0.16639
\\
0.2 & 0.6 & -0.55704 & -0.16426
\\
-0.19426 & -0.55704 & 0.7154872 & 0.2410018
\\
-0.16639 & -0.16426 & 0.2410018 & 0.1975108
\end{pmatrix}
$$
is positive definite, then by Theorem \ref{thm1} the zero solution to the system is
exponentially stable.
To establish \eqref{e3.1} we need to calculate, for $q = 0$, the matrix $R$,
its minimal eigenvalue
$r_{\rm min}$, $\|H\|$, and
$ \beta=\frac{1}{2}\min\{\frac{r_{\rm min}}{\|H\|}, k\}$.
In our case
\begin{gather*}
R = \begin{pmatrix}
0.4741402 & 0.1208201
\\
0.1208201 & 0.1704705
\end{pmatrix},
\quad
r_{\rm min} = 0.1282659, \\
\|H\| = 0.8701562, \quad
\beta = \frac{1}{2}\min\{0.1474056, 0.147\}=\frac{0.147}{2}.
\end{gather*}
Since $\|D\| < e^{-\beta \tau}$,
by Theorem \ref{thm4} we have \eqref{e3.1}.

It should be noted that, using the same matrices $H$ and $K_0$, 
it is not hard to
establish exponential stability in the case of arbitrary positive $\tau$ for
$-0.9 \le a \le 0.78$.
It is enough to take, for example,
$k = 0.015/\tau$.
Changing slightly $H$ and $K_0$, the boundaries for $a$ may be enlarged.

We now consider the case of $F(t,u,v) \not\equiv 0$. Let $a= 0.6213$,
 $\tau = 1$, $q_1=0.01$, $q_2=0.02$.
As mentioned above, in \cite{DeKoSk} the authors established estimates 
of exponential decay for solutions of systems of the form \eqref{e1.1} 
in the case of $\|D\| < 1$.
Using \cite[Theorem 2]{DeKoSk}, one can write down the inequality
\begin{equation}
\|y(t)\| \le d_1 \max_{-1\le s \le 0}\|y(s)\| e^{-\beta_1\,t},
\quad d_1 > 0, \label{e3.2}
\end{equation}
where
$$
\beta_1 = \frac{1}{2}\min\Big\{\frac{c_{\rm min}}{(1+\|D\|^2)\|H\|}
-\frac{\big( q_1+\|D\| q_2+ \sqrt{(1+\|D\|^2)(q_1^2+q_2^2)} \big)}
{(1+\|D\|^2)}, k\Big\},
$$
$c_{\rm min}$ is the minimal eigenvalue of $C$ defined by \eqref{e1.9}.
Choosing the same matrices $H$, $K_0$, and $k = 0.1$, we have
\begin{gather*}
C = \begin{pmatrix}
    0.6     &   0.2     & - 0.19426   & - 0.16639   \\
    0.2     &   0.6     & - 0.55704   & - 0.16426   \\
  - 0.19426 & - 0.55704 &   0.7487219 &   0.2493105 \\
  - 0.16639 & - 0.16426 &   0.2493105 &   0.2058195
\end{pmatrix}, \\
c_{\rm min} = 0.0660526, \quad
\beta_1 = \frac{1}{2}\min\{0.0410265, 0.1\} = \frac{0.0410265}{2}.
\end{gather*}
At the same time, by Theorem \ref{thm4} we have the estimate
\begin{equation}
\|y(t)\| \le d_2 \max_{-1\le s \le 0}\|y(s)\| e^{-\beta\,t},
\quad d_2 > 0, \label{e3.3}
\end{equation}
where $\beta = 0.0991651/2$.
Indeed, in our case
$$
R = \begin{pmatrix}
0.4247753 & 0.1334548 \\
0.1334548 & 0.1389063
\end{pmatrix},
\quad
r_{\rm min} = 0.0862891.
$$
Consequently,
$$
\beta = \min\{\frac{r_{\rm min}}{\|H\|}, k\}
= \frac{1}{2}\min\{0.0991651, 0.1\}=\frac{0.0991651}{2}.
$$
Obviously, \eqref{e3.3} is more strong than \eqref{e3.2} because $\beta$
characterizing the exponential decay rate of the solutions to \eqref{e1.1}
at infinity is larger than $\beta_1$.

All the numerical computations were performed by using Scilab 5.5.1.


\subsection*{Acknowledgments}
The authors are grateful to the anonymous referee for the helpful comments and
suggestions.

The authors were supported by the Russian Foundation for Basic Research
(project no.~13-01-00329) and the Siberian Branch of the Russian Academy 
of Sciences
(interdisciplinary project no.~80).

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\end{document}