Existence and impulsive stability for second order retarded differential equations

One of the most important parts of the qualitative theory in differential equations is the stability of solutions. The problem of stabilizing the solutions by imposing proper impulse controls is very important to many areas of the sciences and engineering. It is important, for instance, in pharmacokinetics, biotechnology, economics, chemical technology and others. We consider certain second order delay differential equations and prove that the solutions can be stabilized by imposing proper impulse effects. An application of these equations appears, for instance, in impact theory. An impact is a short-time interaction of bodies and can be considered as an impulse action. In this direction we mention systems of billiard type which can be modelled by second order equations with impulses acting on the first derivative only, since positions of the colliding balls do not change at the moment of impulse action (impact) and their velocities acquire finite increments. Equations with impulses and delay are important, for instance, in models describing colliding viscoelastic bodies. See [2].