Kyiv Taras Shevchenko National University,

64 Volodymyrska Str.,

01017 Kyiv,

UKRAINE

E-mail: cva@skif.kiev.ua

_{Matrix impulsive differential equations with impulses
at variable times}

**Abstract:**

The main purpose of the report is consideration of the properties
of solutions of impulsive matrix equations with impulses occurring at the
moment of time when the phase point intersects given hypersurfaces in the
extended phase space. These equations are more complicated than the equations
with impulses at fixed times, which have been considered in our previous
report [1]. In what follows, we will be considering impulsive equations
the solutions of which intersect each hypersurface only once. Essential
difference between equation with impulses at variable times and equation
with fixed times of an impulsive effect is that solutions of first equation,
in general, do not depend on initial conditions continuously in such
a way that this continuity be uniform on a finite interval. In our investigation
the stability of solution of matrix impulsive equations with impulses at
variable times have been proved. Direct Lyapunov method for studying
of stability of solutions of impulsive equations with impulses at variable
times have been observed. Chesary's method for finding solutions of these
equations have been considered. The questions of existence of a periodic
solutions for these equations have been studied. The approximation of these
solutions have been constructed.

- Chiricalov V.A. Matrix impulsive system of differential equations with bilinear main part, Proc. of Inst. of Mathematics of NAS of Ukraine, 2004, V.50, part 3, pp.1282-1287.