Abstract:
In the present paper we study stability of solutions to systems
of quasi-linear delay differential equations of neutral type
$$
\frac{d}{dt}(y(t) + Dy(t-\tau)) = Ay(t) + By(t-\tau) + F(t,y(t),y(t-\tau)),
\quad t > \tau,
$$
where
$A$, $B$, $D$
are $n \times n$ numerical matrices,
$\tau > 0$ is a delay parameter,
$F(t,u,v)$
is a real-valued vector-function satisfying Lipschitz condition
with respect to
$u$
and
$F(t,0,0) = 0$.
Stability conditions of the zero solution to the systems are obtained,
uniform estimates for the solutions on the half-axis
$\{t>\tau\}$ are established.
In the case of asymptotic stability these estimates give
the decay rate of the solutions at infinity.
Keywords:quasi-linear differential equations of neutral type, asymptotic stability, attraction domain, uniform estimates for solutions, modified Lyapunov–Krasovskii functional.