About stability of dynamical systems with integrally small perturbations

Abstract

The presented paper deals with the problems of stability of motion of dynamical systems when integrally small perturbations forces affect the system in the finite time interval. It is assumed that the perturbation force is selected fro m a rather broad class of functions, of a class of generalized functions (they may be impulse functions). The need for such studies is dictated by the practice, when you have to take into account the continuous perturbation or impulse effects that can qualitatively change the behavior of the object. Feature of research is that we are not going to follow the "letter" of the classical notion of stability but propose a modification of the concept, tailored applications. So, we admit the possibility of strong perturbations of the object at a finite time interval. If after a specified period of time perturbations in the motion of the object will be small and will remain little longer, then these movement is called stable under small perturbations of the integral. Obviously, such notion of stability is different from the Lyapunov stability, since it permits "fluctuation" instability at a certain interval. However, this instability is not fatal to the nature of the object, since replaced by the steady state of motion.