Absztrakt: | Two stability problems for nonautonomous differential systems
are investigated. In the first part, sufficient conditions are
given for the asymptotic stability of the zero solution of nonautonomous
functional-differential equations (FDE's) with finite delay,
of the form $$x'(t)=F(t,x_t), \quad F(t,0) \equiv 0.$$ $F$ is
continuous and maps bounded sets into bounded sets. In the second
part the problem of stabilization (to asymptotic stability) of
the equilibrium of a nonlinear oscillator by intermittent damping
is considered. The oscillator is governed by a second order nonlinear
ordinary differential equation (ODE) $$x'' +h(t,x,x')x'+f(x)=0.$$
The main features of the two problems making reasonable a common
treatment are the following:\par
a) The equations in both models
are nonautonomous, even strongly nonautonomous, in the sense
that the time functions (the varying parameters, coefficients)
involved in the equations are not supposed to be periodic or
to have any tendency at $t\to \infty$. Consequently, such methods
as the invariance principle, limiting equations, etc. cannot
be applied. What is more, the time functions in the equations
(e.g. the damping coefficient) may be unbounded.\par
b) The
stability effects in the systems are controlled only over an
infinite sequence of time intervals (e.g. intermittent damping).\par
c)
The derivative of the Lyapunov functionals with respect to the
equations are not negative definite, they are only negative semidefinite.}
\RV{V.Dragan (Bucure\c{s}ti) |