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\begin{center}
{\bf SUMMARY} \\[1mm]
of Research project 96-0915 ''Qualitative properties of
impulsive differential equations with applications to control
theory and mathematical biology''
\vskip1mm
\end{center}

Period: \ October 1, 1997 -- September 30, 1999.
\vskip1mm

Project Co-ordinator \ professor Mokhtar Kirane,
Antenne de l'Universite de Picardie Jules Verne, Departament of
Mathematics and Informatics;
(Lamfa Upres A 6119 \ \ 33, Rue Saint Leu \ \ 80039 Amiens France) \\
Phone: +33 3 44 06 88 57; \ \ fax:   +33 3 44 06 88 50 \\
E-mail: mokhtar.kirane@u-picardie.fr

An important and significant results for the future development and
apllications of the theory of impulsive systems have been obtained.

Partial differential equations of parabolic type as well as partial
differential equations with impulse are studied. New conditions of
existence of their global solutions are obtained and their
stability are found. By using the Laypunov function approach the
sufficient conditions of existence of uniform bounded solutions of
a system of two partial differential equations of
reaction-diffusion type with impulse are obtained. Formulae for
exact solutions of initial boundary problems for linear partial
differential equations of hyperbolic and parabolic type were
obtained and their analysis were made. Sufficient conditions for
the problem to have periodic solutions are found.

The system of ordinary differential equations defined on the
direct product of a torus and Euclidean space subjected to impulse
action on a submanifold of the torus of codimension 1 is studied.
Conditions of the existence of invariant sets of the system are
obtained under assumption that its linearized system posses the
property of exponential dichotonomy. By means of Poincar\`e mapping
the existence of periodic solutions of the system of differential
equations with impulses has been studied and their stability are
considered. The problem on existence and stability of periodic solutions
of perturbed differential equation of general form with impulses is
studied under the assumption that the generating equation has the
family of quasiperiodic solutions.

Various applications of the numerical-analytical methods to
impulsive systems of general form have been considered. It is found
sharp conditions under which nonlinear boundary-value problem for
the impulsive system can be investigated by means of 
numerical-analytical method.

Periodic boundary value problem for a functional differential
equation with impulses at fixed times have been studied. A new
maximal principle has been proved for a general problem.  Delay
impulsive differential equations are studied and sufficient
conditions for the existence of periodic solutions systems have
been obtained. The algorithm of construction approximate solution
of delay impulsive systems with small perturbation is developed.

Some new comparison principles have been proposed and applied to
study the existence of solutions to the first order periodic
impulsive problem and to a second order boundary value problem for
ordinary differential equations with impulses.

Necessary and sufficient conditions of controllability for the
boundary value problem for integro-differential equations with
impulse and for a system of impulsive differential equations with
non-fixed moments of impulses have been obtained. 
\newpage

Key references: \\
1. N.I.~Ronto, A.M.~Samoilenko, S.I.~Trofimchuk, Theory of
numerical-analytic method: achievements and new trends of
development.IV.  Ukrain. Math. J. 50 (1998), no. 12, p.1656-1672. \\
2. M.~Akhmetov and R.~Sejilova, The control of boundary value
problem for linear impulsive integro-differential system. J. Math.
Anal. Appl. 1999, v.236, p.312-326. \\
3. M.~Frigon and D.~O'Regan, First order impulsive initial and
periodic problems with variable moments. J. Math. Anal. Appl.
233(1999), no.2, p.730-739. \\
4. A.M.~Samoilenko, V.Hr.~Samoilenko and V.V.~Sobchuk, Periodic
solutions of equations of mathematical pendulum with impulse
influence. Ukrain. Math. J. 51(1999), no.6, p.837-846. \\
5. A.~Cabada, J.~Nieto, D.~Franco and S.~Trofimchuk, A
generalization of the monotone method for second order periodic
boundary value problem with impulses at fixed points, Dynamics of
Continuous, Discrete and Impulsive systems. (To appear). \\
6. M.~Akhmetov and M.~Kirane, Control problems of quasilinear
integro-differential equations. SIAM, Appl. Math. (Submitted). \\

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