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\markboth{\hfil Stability of solutions \hfil EJDE--2002/94}
{EJDE--2002/94\hfil Michael I. Gil' \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 94, pp. 1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
Stability of solutions for nonlinear nonautonomous differential-delay \\
equations in Hilbert spaces
%
\thanks{ {\em Mathematics Subject Classifications:} 34G20, 34K20, 34K99.
\hfil\break\indent
{\em Key words:} nonlinear differential-delay equations in Hilbert spaces,
absolute stability, \hfil\break\indent
input-output stability, Aizerman-Myshkis problem.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted September 19, 2002. Published October 31, 2002.
\hfil\break\indent
Partially supported by the Kamea Fund}}
\date{}
\author{Michael I. Gil'}
\maketitle
\begin{abstract}
We consider nonlinear non-autonomous differential-delay
equations having separated linear and sublinear parts.
We assume that the Green functions of the linear part is
selfadjoint and positive definite to obtain solution estimates,
explicit conditions for the absolute stability, and
input-output stability. Moreover, it is shown that the suggested
conditions characterize the equations that satisfy the
generalized Aizerman - Myshkis hypothesis.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{defn}[theorem]{Definition}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section{Introduction and definitions}
Let $H$ be a real separable Hilbert space with a scalar product $(.,.)$, the norm $\|.\|_H=(.,.)^{1/2}$,
and the unit operator $I$, cf. \cite{ahi}.
As usually, $L^2(\omega, H)$ is the space of $H$-valued functions defined on a set $\omega\subseteq \mathbb{R}$
and equipped with the norm
$$
\|w\|_{L^2(\omega)}=[\int_\omega \|w(x)\|^2_H dx]^{1/2}\;\; (w\in L^2(\omega, H)).
$$
Put $R_+=[0,\infty)$ and $R_h=[-h,\infty)$ for a positive $h<\infty$. Let
$A(t)$ and $B(t)\;(t\in R_+)$ be variable selfadjoint generally unbounded operators in $H$
with the dense constant domains $D_A, D_B$, respectively. Besides,
$D_A\subseteq D_B$.
Let $\mu$ be a nondecreasing left-continuous bounded scalar function defined
on $[0, h]$ with the property $\mu(0)=0$.
In the present paper we establish solution estimates, explicit conditions
for the absolute stability and input-output one
of the equation
$$
\dot u + A(t)u + B(t)\int_ 0^h u(t-\tau) d\mu(\tau) = F(t, u(.))\;\;(t> 0,\;\dot u=du/dt)
\eqno(1.1)
$$
where $F: R_+\times L^2(R_h, H)\rightarrow H$ is {\em a causal nonlinearity} in the sense that
$$
F(t, u_1(.))=F(t, u_2(.))\mbox{ if } u_1(\tau)=u_2(\tau)
$$
for all $-h\leq \tau\leq t$ and $u_1, u_2\in L^2(R_h, H)$.
Take the initial condition
$$
u(t)=\Phi (t)\;\;(-h\leq t\leq 0)
\eqno(1.2)
$$
with a given function $\Phi : [-h, 0]\rightarrow D_A$ continuous in the norm of $H$.
Note that in the available literature, the stability of proble (1.1), (1.2) is investigated
mainly under the condition $A(t)\equiv A$. In addition,
it is assumed that the terms containing delays are bounded, cf.
\cite{mur,pao}. In the papers
\cite{gi:98b,gi02b} the equations with variable operators
were investigated, but the terms containing delays are assumed to be bounded.
About linear non-autonomous equations with unbounded terms containing delays
see \cite[p. 184]{pr} and references given therein.
The very interesting papers \cite{fa1} and \cite{fa2} on nonlinear non-autonomous equations should be noted.
But the nonlinearities considered in these papers
are different from the nonlinearities considered in this paper.
Moreover, to the best of our knowledge, the absolute and input-output stability
of abstract nonlinear differential equations were not investigated
in the available literature although these notions are very important in theory of systems,
cf. \cite{vid}.
In addition, in Section 5 below we separate the linear parts of equations of the type (1.1) that satisfy the
generalized Aizerman - Myshkis hypothesis.
\begin{defn}\rm
Let $E_s$ $(s\in\mathbb{R})$ be an orthogonal resolution of the identity in $H$.
We will say that $E_s$ is a simple resolution of identity (s.r.i.), if
there is a (generating) vector $g\in H$, such that for any $v\in H$,
$$
v=\int_{-\infty}^\infty \tilde v_sdE_sg,
$$
where $\tilde v_s$ is an $(E_sg, g)$-measurable scalar-valued function.
Besides $v_s$ will be called the $Eg$-coordinate function of $v$.
Similarly, let $E_s$ be a s.r.i. and
$W$ be a normal operator defined by
$$
W=\int_{-\infty}^\infty \tilde w_sdE_s
$$
where $\tilde w_s$ is an $E$-measurable scalar-valued function.
Then $\tilde w_s$ will be called the
$E$-coordinate function of operator $W$.
\end{defn}
As it is well-known \cite[Section 83, Theorem 2]{ahi}, any selfadjoint operator
with the simple spectrum has a s.r.i.
Note also that we write that $\tilde w_s$ is an $E$-measurable, since
the measurability of $\tilde w_s$ with respect to the measure $(E_sg, g)$ does not
depend on a generating vector $g$ \cite[Section 83]{ahi}.
\section{Statement of the main result}
It is assumed that
$$
A(t)=\int_{-\infty}^\infty a_s(t)dE_s\mbox{ and }
B(t)=\int_{-\infty}^\infty b_s(t)dE_s,
\eqno(2.1)
$$
where $E_s$ is a s.r.i. and $a_s(t), b_s(t)$ are real
functions continuous in $t\in R_+$ for almost all $s\in \mathbb{R}$ and
$E_s$-measurable in $s$ for all $t\geq 0$.
In addition, {\em $a_s(t)$ is positive}.
Moreover, there are nonnegative constants $q$ and $l_0$, such that
$$
\|F (., v(.))\|_{L^2(R_+)}\leq q\|v\|_{L^2(R_h)}+ l_0\;\;(v\in L^2(R_h, H)).
\eqno(2.2)
$$
Let us consider the equation
$$
\dot w(t) + A(t)w(t) + B(t)\int_ 0^h w(t-\tau) d\mu(\tau) = 0\;\;(t> t_1\geq 0)
\eqno(2.3)
$$
Below we will check that under conditions pointed below, problem
(2.3), (1.2) has a solution $\phi $ differentiable almost everywhere on $R_+$. Moreover,
we will show that (2.3) has the Green function $G(t, t_1)\;\;(t, t_1\geq -h)$.
That is $G$ is the operator-valued function whose values are
bounded in $H$ operators satisfying (2.3) almost everywhere on $R_+$
and the initial conditions
$$
G( t_1, t_1)=I\mbox{ and }G( t, t_1) = 0\;\;(t_1-h\leq t < t_1,\;- h\leq t_1< \infty ).
\eqno(2.4)
$$
{\em Let $G$ be the Green function of equation (2.3) and
$\phi $ be a solution of problem (1.4), (1.2). Then
a continuous function $u$ defined on $R_H$, satisfying the equation
$$
u(t)= \phi (t) + \int_0^t G(t, s)F(s, u(.))ds\;\;(t\geq 0)
\eqno(2.5)
$$
and condition (1.2) will be called the mild solution of problem (1.1), (1.2)}.
The existence of mild solutions is assumed.
About various existence results see \cite{pao,gi02b}, etc.
Finally denote by $\gamma_E$ the set of points of the growth of $E_s$ and put
$$
a_s^-=\inf_{t\geq 0} a_s(t)\mbox{ and } b_s^-= \inf_{t\geq 0} b_s(t).
$$
Now we are in a position to formulate the main result of the paper
\begin{theorem}
Let the conditions (2.1), (2.2),
$$
\inf_{s\in\gamma_E} a_s^- + \mu(h) \;b_s^- > q
\eqno(2.6)
$$
and
$$
\beta:= \sup_{s\in\gamma_E} \frac{\mu(h) \;|b_s^-|}{a_s^-} <1
\eqno(2.7)
$$
hold. In addition, let the Green function $G(t, t_1)$ to equation (2.3) be positive definite
for all $t\geq t_1\geq 0$.
Then there is a constant $c_1> 0$, independent of the initial conditions,
such that any solution $u$ of problem (1.1), (1.2) satisfis the inequality
$$
\|u\|_{L^2(R_+)}\leq c_1(\|B_0\Phi \|_{L^2[-h, 0]}+\|\Phi (0)\|_H+ l_0),
\eqno(2.8)
$$
where
$$
B_0=\int_{-\infty}^\infty b_s^-dE_s.
$$
\end{theorem}
The proof of this theorem is divided into a series of lemmas presented in the next two sections.
We also will check
that the Green function to equation (2.3) is positive definite,
provided
$$
e\,h\;\sup_{s\in\gamma_E, t\geq 0} b_s(t)\int_0^h exp \Big[\int_{t-\tau}^t a_s(t_1)dt_1\Big]\;d\mu(\tau)<1\;\;
\eqno(2.9)
$$
(see Corollary 3.5 below). Now Theorem 2.1 implies
\begin{cor} Let conditions (2.1), (2.2), (2.6), (2.8) and (2.9) hold.
Then inequality (2.8) is valid for any solution $u(t)$ to problem (1.1), (1.2).
\end{cor}
\section{ Preliminaries}
Let $\sigma_h$ be
the $\sigma$-algebra of the Borel sets of $[0, h]$
and $\nu(t, .), \nu_+(t, .)$ nonnegative measure defined
on $\sigma_h$ and continuously dependent on $t\geq 0$.
Consider the equations
$$
\dot x (t) + \int_ 0^h x(t-\tau) \nu (t, d\tau)=0\;(t \geq 0)
\eqno(3.1)
$$
and
$$
\dot y (t) + \int_ 0^h y(t-\tau) \nu_+ (t, d\tau)=0\;(t\geq 0).
\eqno(3.2)
$$
Denote by $G_\nu(t, t_1)$ and $G^+_{\nu}(t, t_1)$
the Green function to equations (3.1) and (3.2), respectively.
So they are the solutions of (3.1) and (3.2), respectively, with the initial conditions
$$
G_\nu^+(t_1, t_1)=G_\nu (t_1 ,t_1)=1, \quad G_{\nu}(t, t_1)=G_\nu^+(t, t_1)=0, \quad (t_1-h\leq t t_1\geq 0)$ and
$$
\nu_+(t, \tau)\geq \nu(t, \tau)\;\;
(\tau\in \sigma_h,\;t\geq 0).
$$
Then
$$
G_{\nu}(t, t_1)\geq G_\nu^+ (t, t_1)\geq 0\;(t > t_1\geq 0).
\eqno(3.4)
$$
\end{lemma}
\paragraph{Proof.} From (3.1) with $x(t)=G_\nu(t, 0)$ it follows
$$
\dot x(t) +\int_ 0^h x(t-\tau) \nu_+ (t, d\tau) = f(t)
\eqno(3.5)
$$
where
$$
f(t)=\int_ 0^h x(t-\tau) (\nu_+(t, d\tau)-\nu(t, d\tau)).
$$
According to the initial conditions (3.3), for a sufficiently small $t_0>h $,
$$
x(t)\geq 0\mbox{ and }f(t)\geq 0\;\; (0\leq t\leq t_0).
$$
Hence, by virtue of the Variation of Constants Formula, we get
$$
x(t)=G^+_{\nu}(t, 0)+\int_0^t G^+_{\nu}(t, s)f(s)ds\geq G^+_{\nu}(t, 0)\;(0\leq t\leq t_0).
$$
Extending this inequality to the whole half-line, we arrive at the required result if $t_1=0$.
Similarly inequality (3.4) can be proved in the general case. As claimed. Q.E.D.
It is simple to check that according to (2.1), equation (2.3) has Green's function
represented by
$$
G(t, t_1)=\int_{-\infty}^\infty G_s(t, t_1) dE_s
$$
where $G_s(t, t_1)$ is Green's function to the equation
$$
\dot u(t) + a_s(t) u(t)+ b_s(t)\int_0^h u(t-\tau)d\mu\;(t\geq 0).
\eqno(3.6)
$$
Due to the previous lemma we have
\begin{cor}
Let $G_s(t, t_1)\geq 0$. Then
$$
G_s(t, t_1)\leq W_s(t-t_1)\;\;(t\geq t_1\geq 0),
$$
where $W_s(t)$ is the Green function to the equation
$$
\dot u(t) + a^-_s u(t)+ b^-_s\int_0^h u(t-\tau)d\mu=0\;(t\geq 0).
\eqno(3.7)
$$
\end{cor}
To establish the positivity conditions, let us consider the equation
$$
\dot u(t) + c_0 u(t-h)=0\;\;(t\geq 0)
\eqno(3.8)
$$
with a real constant $c_0$.
\begin{lemma} Let the condition $eh c_0 < 1$
hold. Then the Green function to equation (3.8) is nonnegative.
\end{lemma}
For the proof see for instance \cite{gi:00}.
Due to Lemmas 3.1 and 3.3 we can assert that
the Green function to the equation
$$
\dot u(t) + b(t)\int_0^h u(t-\tau)d\mu(\tau)=0\;\;(\;t\geq 0)
$$
with a bounded real function $b(t)$ is positive, provided
$$
eh\sup_{t} b(t)\mu(h) < 1.
\eqno(3.9)
$$
Now let us consider the equation
$$
\dot u(t) + a(t)u(t) + b(t)\int_0^h u(t-\tau)d\mu(\tau)=0\;\;(\;t\geq 0).
\eqno(3.10)
$$
with bounded real functions $a(t), b(t)$.
Substituting in this equation the equality,
$$
u(t)=v(t)exp\Big[-\int_0^t a(t_1)dt_1 \Big]
$$
we have the equation
$$
\dot v(t) + b(t)\int_0^h exp\Big[\int_{t-\tau}^t a(t_1)dt_1\Big]v(t-\tau) d\mu (\tau)=0.
$$
Now (3.9) and Lemma 3.1 imply
\begin{lemma} Let the condition
$$
e\;h\sup_{t\geq 0} b(t)\int_0^h exp\Big[\int_{t-\tau}^t a(t_1)dt_1\Big]\;d\mu(\tau)<1
$$
hold. Then the Green function to equation (3.10) is nonnegative.
\end{lemma}
Since the Green function $G_s(t, t_1)$ to equation (3.6) is the $E_s$-coordinate function to
the Green function $G_s(t, t_1)$ to equation (2.3), the latter result implies
\begin{cor} Let condition (2.9)
hold. Then the Green function to equation (2.3) is positive definite.
\end{cor}
\section{Proof of Theorem 2.1}
Consider the equation
$$
\dot v(t) + A(t)v(t)+ B(t)\int_ 0^h v(t-\tau) d\mu(\tau)=f(t)
\eqno(4.1)
$$
with a given $f\in L^2(R_+, H)$.
Since $E_s$ is a m.r.i., there is a vector $g\in H$ with $\|g\|=1$,
such that function $f$ and {\em the solution $w$ to equation (4.1)
with the zero initial condition} can be represented as
$$
f(t)=\int_{-\infty}^\infty \tilde f_s(t)dE_sg
\eqno(4.2)
$$
and
$$
w(t)=\int_{-\infty}^\infty y_s(t)dE_sg.
\eqno(4.3)
$$
Here $\tilde f_s(t)$ and $y_s(t)$ are
the $Eg$-coordinate functions of $f(t)$ and $w(t)$, respectively.
\begin{lemma} Let $f$ be defined by (4.2) and
$$
\int_0^\infty |\tilde f_s(t)|^2 dt\leq c_0\;\;(s\in\gamma_E)
$$
where constant $c_0$ does not depend on $s$. Then
$$
\|f\|^2_{L^2(R_+)}\leq c_0.
$$
\end{lemma}
\paragraph{Proof.}
We have
$$
\int_0^\infty \|f(t)\|_H^2 dt=
\int_0^\infty \int_{-\infty}^\infty |\tilde f(t)|^2 d(E_sg, g) \;dt=
$$
$$
\int_{-\infty}^\infty \int_0^\infty |\tilde f_s(t)|^2 dt\; d(E_sg, g)
\leq c_0 \int_{-\infty}^\infty d\;(E_sg, g)=c_0.
$$
As claimed. Q.E.D.
Due to (2.1) and (4.3)
$$
\dot y_s(t) + a_s(t) y_s(t) + b_s(t) \int_ 0^h y_s(t-\tau) d\mu(\tau)=\tilde f_s(t).
$$
Hence, under the zero initial condition we have
$$
y_s(t)=\int_0^t G_s (t, t_1) \tilde f_s(t_1)d\tau.
$$
If $G_s(t, t_1)\geq 0$, due to Corollary 3.2,
$$
|y_s(t)|\leq \int_0^t W_s (t-t_1) |\tilde f_s(t_1)|d\tau.
$$
Consequently,
$$
\int_0^\infty |y_s(t)|^2 dt\leq \int_0^\infty |\tilde f_s(t)|^2 dt [\int_0^\infty W_s (t) dt]^2.
$$
Clearly the Laplace transform to $W_s (t)$ is
$$
\int_0^\infty e^{-\lambda t} W_s(t)dt=(\lambda + a_s^-+b_s^- \int_ 0^h e^{-\lambda\tau} d\mu(\tau) )^{-1}.
$$
Hence due to (2.6)
$$
\theta_\mu:=\sup_{s\in\gamma_E}\int_0^\infty W_s(t)dt=\sup_{s\in\gamma_E}(a_s^- + \int_ 0^h d\mu(\tau) b_s^-)^{-1} <1/q<\infty.
\eqno(4.4)
$$
Thus,
$$
\int_0^\infty |y_s(t)|^2 dt\leq \theta_\mu^2 \int_0^\infty |\tilde f_s(t)|^2 dt.
$$
Now Lemma 4.1 implies
$$
\|w\|_{L^2(R_+)}\leq \theta_\mu\|f\|_{L^2(R_+)}.
$$
But due to (2.5)
$$
w(t)=\int_0^t G(t, t_1)f(t_1) dt_1.
$$
Thus, we have proved
\begin{lemma} Under the condition (2.6), let $G$ be positive definite. Then
$$
\|\int_0^t G(t, t_1)f(t_1) dt_1\|_{L^2(R_+)}\leq \theta_\mu\|f\|_{L^2(R_+)},
$$
where $\theta_\mu$ is defined by (4.4).
\end{lemma}
We need also the following
\begin{lemma} Under the condition (2.7), let the Green function $W_s$ to the scalar equation
(3.7) be non-negative and
$$
c:= \inf_{s\in\gamma_E} (a_s+\mu(h)b_s) >0.
\eqno(4.5)
$$
Then
$$
\sup_{s\in\gamma_E}\int_0^\infty W_s^2(t)dt < \infty.
$$
\end{lemma}
\paragraph{Proof.}
If $b_s>0$, then
from (3.7) and $W_s(t)\geq 0$ it follows that $\dot W_s(t)\leq 0$. So $W_s(t)\leq 1$ $(t\geq 0)$.
Hence due to (4.4),
$$
\int_0^\infty W_s^2(t)dt\leq \int_0^\infty W_s(t)dt\leq \theta_\mu.
$$
So in this case the result is proved. Let now $b_s<0$.
Recall that the Laplace transform to $W_s (t)$ is
$$
k_s(\lambda):=\Big(\lambda + a_s^-+b_s^- \int_ 0^h e^{-\lambda\tau} d\mu(\tau) \Big)^{-1}.
$$
For all real $\omega$,
\begin{eqnarray*}
|i\omega + a_s^-+b_s^- \int_ 0^h e^{-i\omega \tau} d\mu(\tau) |
&\geq& [\omega^2 + (a_s^-)^2]^{1/2}-|b_s^- |\mu(h)\\
&\geq& a_s^- -|b_s^- |\mu(h)\geq c.
\end{eqnarray*}
In addition,
$$
|i\omega + a_s^-+b_s^- \int_0^h e^{-i\omega \tau} d\mu(\tau) |\geq
[\omega^2 + (a_s^-)^2]^{1/2}-|b_s^- |\mu(h)\geq
$$
$$
[\omega^2 + (a_s^-)^2]^{1/2}- a_s\beta\geq c_3|\omega|\;\; (|\omega|\geq c).
$$
Hence,
$$
\sup_s\int_{-\infty}^{-c} + \int_{c}^\infty |k_s(i\omega)|^2 d\omega\leq 2c_3^{-1} \int_{c}^\infty
\omega^{-2}d\omega <\infty
$$
and
$$
\int_{-c}^{c} |k_s(i\omega)|^2 d\omega\leq 2 c \;c^{-2}=2c^{-1}.
$$
This result and the Parseval equality
prove the required result. Q.E.D.
\begin{lemma} Under the conditions (4.5), (2.7),
let the Green function $W_s$ to the scalar equation (3.7) be non-negative. Then
any solution $\tilde y_s$ of problem (3.7),
$$
\tilde y_s(t)=\tilde \Phi_s(t)\;\;(-h\leq t\leq 0)
\eqno(4.6)
$$
with a continuous $\tilde \Phi_s$, satisfies the inequality
$$
\|\tilde y_s\|_{L^2(R_+)}\leq c_4 (|b_s|\|\tilde \Phi_s\|_{L^2[-h, 0]}+|\tilde \Phi_s(0)|)
$$
where constant $c_4$ does not depend on $s\in \mathbb{R}$.
\end{lemma}
\paragraph{Proof.} Take into account that
$$
\tilde y_s(t)=W_s(t)\tilde \Phi_s(0)+b_s
\int_{0}^h \int_{-\tau}^0 W_s(t-\tau-z)\tilde \Phi_s(\tau) dz\;d\tau,
$$
see \cite[Section 1.6]{ha}, \cite[Section 8.2]{gi:98a}.
Since
$$
\tilde y_s(t)=W_s(t)\tilde \Phi_s(0)+
\int_{0}^h \int_{-\tau}^0 W_s(t-\tau-z)b_s\tilde \Phi_s(\tau) dz\;d\tau,
\eqno(4.7)
$$
thanks to the previous lemma we have the required result. Q.E.D.
\begin{lemma} Under condition (2.7) and (4.5),
let the Green function $G_s$ to the scalar equation (3.6) be non-negative.
Then any solution $\phi_s$ of problem (3.6), (4.6)
satisfies the inequality
$$
\|\phi_s\|_{L^2(R_+)}\leq c_1 [|\tilde \Phi_s(0)| + |b_s|\|\tilde \Phi_s\|_{L^2[-h, 0]}].
$$
\end{lemma}
\paragraph{Proof.}
Let $\tilde y_s$ be a solution of problem (3.7), (4.6).
Since delays do net depend on time, if $\tilde \Phi_s$ is non-negative, then $\phi_s$ is positive
due to the integral representations of solutions \cite[p.140]{ko}.
Repeating the arguments of Lemma 3.1 we have $\phi_s(t)\leq \tilde y_s(t)$. If
$-\tilde \Phi_s$ is non-negative, then $-\phi_s$ is non-negative and
$|\phi_s(t)|\leq |\tilde y_s(t)|$.
Since the initial function is a difference of two
non-negative functions, we easily have $|\phi_s(t)|\leq 2|\tilde y_s(t)|$.
Now the required result is due to the previous lemma. Q.E.D.
It is simple to check that according to (2.1), problem (2.3), (1.2) has a solution $\phi $
represented by
$$
\phi (t)=\int_{-\infty}^\infty \tilde \phi_s(t) dE_s
$$
where $\tilde \phi_s(t)$ is a solution to problem (3.6), (4.6).
Due to the previous lemma and Lemma 4.1 we have
$$
\|\phi\|_{L^2(R_+)}\leq c_1(\|\Phi (0)\|_{H}+ \|B_0\Phi \|_{L^2[-h, 0]}).
\eqno(4.8)
$$
\paragraph{Proof of Theorem 2.1}
Due to (2.5) and (4.8), Lemma 4.2 and condition (2.2) yield
\begin{eqnarray*}
\|u\|_{L^2(R_+)}&\leq&
c_1(\|\Phi (0)\|_{H}+\|B_0\Phi \|_{L^2[-h, 0]}) + \theta_\mu \|F\|_{L^2(R_+)} \\
&\leq& c_1(\|B_0\Phi \|_{L^2[-h, 0]}+\|\Phi (0)\|_{H})
+ \theta_\mu (q\|u\|_{L^2(R_+)}+l_0).
\end{eqnarray*}
Now condition (2.5) implies the required result. Q.E.D.
\section{Absolute stability and the generalized \\
Aizerman-Myshkis hypothesis}
\begin{defn}
The zero solution of equation (1.1)
is said to be absolutely stable in the class of nonlinearities satisfying
the condition
$$
\|F(., u(.))\|_{L^2(R_+)}\leq q\|u\|_{L^2(R_h)}
\eqno(5.1)
$$
if there exists a positive constant $c_1$
independent of the specific form of function $F$ (but dependent on $q$),
such that the inequality
$$
\|u\|_{L^2(R_+)}\leq c_1( \|\Phi (0)\|_{H}+\|B_0\Phi \|_{L^2[-h, 0]})
$$
holds for any solution $u$ of (1.1) with the initial condition (1.2).
\end{defn}
Let $a, b, c$ be an $n\times n$-matrix, a column-matrix and a row-matrix, respectively.
In 1949 M. A. Aizerman conjectured the following hypothesis:
for the absolute stability of the
zero solution of the equation $\dot{x} = A_0x + b f(c x)$
in the class of nonlinearities $f: {\bf R}^1 \rightarrow {\bf R}^1$,
satisfying $0\leq f(s)/s \leq q$ $(q=const>0, s\in {\bf R}^1, \;s\neq 0)$
it is necessary and sufficient that the linear equation
$\dot{x} = ax + q_1bcx$
be asymptotically stable for any $q_1\in [0,q]$ \cite{aiz}.
These hypothesis caused the great interest among the
specialists. Counterexamples
were set up that demonstrated it was not, in general, true, (see
\cite{nar}, \cite{vid}, and references therein). Therefore, the
following problem arose: to find the class of
systems that satisfy Aizerman's hypothesis.
The author showed that any system satisfies
Aizerman hypothesis if its impulse function
is non-negative \cite{gi:83}. The similar result was proved for multivariable
systems and distributed ones, cf. \cite{gi:98a}.
On the other hand, A.D. Myshkis \cite{mys} pointed out at the
importance of consideration of the generalized Aizerman problem for retarded
systems. The problem pointed by A.D. Myshkis was considered in \cite{gi:00} and \cite{gi:02}.
In the present paper we will consider the following generalization of
the Aizerman problem:
Put
$$
A_0=\int_{-\infty}^\infty a_s^-dE_s.
$$
{\bf Problem 1}: {\em To separate a class of equations (1.1), such that the
asymptotic stability of the linear equations
$$
\dot u + A_0u+B_0\int_0^h u(t-\tau)d\mu(\tau)=\tilde qu
\eqno(5.2)
$$
with some $\tilde q\in [0, q]$ provides the absolute stability of equation (1.1)
in the class of nonlinearities (5.1)}.
Theorem 2.1 with $l_0=0$ implies
\begin{theorem}
Let conditions (2.1), (2.6) and (2.7) hold. In addition,
let the Green function to equation (2.3) be positive definite.
Then the zero solution to equation (1.1) is absolutely stable in the class of
nonlinearities (5.1).
\end{theorem}
Let us check that Theorem 5.2 separates a class of nonlinearities satisfying Problem 1.
To this end we will show that, if the Green function is positive definite,
the stability of equation (5.2) with $\tilde q = q$ implies
condition (2.6). Indeed, let $v$ be a solution of (5.2). Then
$$
v(t)=\int_0^\infty v_s(t)dP_sg,
$$
where
$ v_s(t)$ is the $Eg$-coordinate function of $v(t)$. According to (2.1) and (5.2) $v_s$ satisfies the equation
$$
\dot v_s(t) + a_s^- v_s(t) + b_s^- \int_0^h v_s(t-\tau)d\mu=qv(t).
$$
Since equation (5.2) is assumed to be asymptotically stable,
the roots of the function
$$
\lambda+ a_s^- + b_s^- \int_0^h e^{-\lambda \tau }d\mu -q
$$
are in the open left half-plane. So
$$
-q +i\omega+ a_s^- + b_s^- \int_0^h e^{-i\omega \tau }d\mu\neq 0
$$
for all $\omega\in\mathbb{R}$. Hence, with $\omega=0$
we get (2.6), as claimed.
\section{Input-output stability }
Let us consider the equation
$$
\dot u + A(t)u + B(t)\int_0^h u(t-\tau) d\mu(\tau) = \Psi(t, u(.), \zeta (.))\;\;(t\geq 0)
\eqno(6.1)
$$
where $\zeta: R_+\rightarrow H$ is a given function (input),
$\Psi: R_+\times L^2(R_h, H)\times L^2(R_+, H)\rightarrow H$ is {\em a causal nonlinearity} in the sense that
$$
\Psi(t, u_1(.), \zeta)=\Psi (t, u_2(.), \zeta)\mbox{ if } u_1(\tau)=u_2(\tau)
$$
for all
$$
\tau\leq t, \zeta\in L^2(R_+, H) \mbox{ and }u_1, u_2\in L^2(R_h, H).
$$
\begin{defn} We will say that equation (6.1) is $L^2$-input-output stable,
if for any $\epsilon>0$, there is a $\delta>0$, such that $\|\zeta\|_{L^2(R_+)}\leq \delta$ implies
$\|u\|_{L^2(R_+)}\leq \epsilon$ for any solution $u$ of (6.1)
under the zero initial condition
$u(t)=0\;\;(t\leq 0)$.
\end{defn}
\begin{theorem}
Let the conditions (2.1), (2.6), (2.7) and
$$
\|\Psi (., v(.), \zeta)\|_{L^2(R_+)}\leq q\|v\|_{L^2(R_h)}+ \mu\|\zeta\|_{L^2(R_+)}+ l_0
$$
$$
(v\in L^2(R_h, H), \zeta\in L^2(R_+, H),\mu=const>0),
\eqno(6.2)
$$
hold. In addition,
let the Green function to equation (2.3) be positive definite.
Then equation (6.1) is $L^2$-input-output stable.
Moreover, there is a constant $c_1 > 0$, such that
$$
\|u\|_{L^2(R_+)}\leq c_1(\|\zeta\|_{L^2(R_+)}+ l_0)
$$
for any solution $u$ of (6.1) with the zero initial condition.
\end{theorem}
Indeed, condition (6.2) implies inequality (2.2) with
$\|\zeta\|_{L^2(R_+)}+ l_0$ instead of $l_0$. Now the result is due to Theorem 2.1.
\section{Example}
First note that condition (5.1) holds, in particular, if
$F(t,u(.)) =F_0(u(t), u(t-h))$ where $F_0: H^2\rightarrow H$ is a function
satisfying
$$
\|F(z_1, z_2)\|_{H}\leq q_0\|z_0\|_{H}+q_1\|z_1\|_{H}\;\;(q_0, q_1=const;\;z_0, z_1\in H).
\eqno(7.1)
$$
Indeed, in this case
$$
\|F_0(u(t), u(t-h))\|_{L^2(R_+)}\leq
q_0\|u\|_{L^2(R_+)}+q_1\|u(t-h)\|_{L^2(R_+)}\leq
$$
$$
(q_0+q_1)\|u\|_{L^2(R_h)}.
\eqno(7.2)
$$
So condition (3.1) holds with $q=q_0+q_1$.
Let us consider the equation
$$
\frac{\partial u(t, x)}{\partial t} =
g(t)\frac{\partial^2 u(t, x)}{\partial x^2} - c(t) \frac{\partial^2u(t-h, x)}{\partial x^2} +
F_1(x, u(t, x), u(t-h, x))
$$
$$
\;\;(0 < x < 1,\;t\geq 0)
\eqno(7.3)
$$
where $g(t), c(t)$ are continuous bounded positive functions, $F_1$ is a scalar continuous
function defined on $[0, 1]\times \mathbb{R}^2$ and
having the property
$$
|F_1(x, z_0, z_1)|\leq q_0 |z_0|+q_1|z_1|
\;\;(x\in [0, 1], z_0, z_1\in \mathbb{R}).
\eqno(7.4)
$$
According to (7.2), condition (7.4) implies (5.1) with $q=q_0+q_1$.
Take some selfadjoint boundary conditions.
Let $S$ be the operator defined by $Su=-d^2u/dx^2\;\;(0 < x < 1)$
with
the taken boundary conditions and the positive eigenvalues
$\lambda_k$ $(k=1, 2, ... )$ numerated in
the increasing order with the multiplicities taken into account.
Then (7.1) can be written as (1.1) with $A(t)=g(t)S, B(t)=-c(t)S$ and
$$
a_s(t)\equiv a_k(t)=g(t)\lambda_k, b_s(t)\equiv b_k(t)=-c(t)\lambda_k .
$$
So
$$
a_s^-\equiv a_k^-=\lambda_k \inf_{t\geq 0} g(t);\;
b_s^-\equiv b_k^-= -\lambda_k \sup_{t\geq 0} c(t).
$$
Then condition (2.6) takes the form
$$
\lambda_1(\inf_{t\geq 0} g(t)-\sup_{t\geq 0} c(t))> q.
\eqno(7.5)
$$
Condition (2.7) is fulfilled, provided for a $\beta<1$,
$$
\beta\inf_{t\geq 0} g(t)\geq \sup_{t\geq 0} c(t).
\eqno(7.6)
$$
Condition (2.9) is always holds.
Due to Theorem 2.1, under conditions (7.5)-(7.6), any solution to equation (7.3)
is in $L^2(R_+, H)$, provided condition (7.4) holds. Moreover,
due to Theorem 5.2, the zero solution to equation (7.3) is absolutely stable in the class of
nonlinearities (7.4).
In particular, if we take the Dirichlet boundary conditions
$u(t, 0)=u(t, 1)=0\;\;(t\geq 0)$,
then $\lambda_k=\pi^2 k^2\;\;(k=1, 2, ...)$.
{\em Acknowledgment}
I am very grateful to the late Professor
M. A. Aizerman for his interest in
and approval of my investigations.
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\noindent\textsc{Michael I. Gil'}\\
Department of Mathematics \\
Ben Gurion University of the Negev \\
P.0. Box 653, Beer-Sheva 84105, Israel\\
E-mail: gilmi@black.bgu.ac.il
\end{document}