PROBLEM WITH IMPULSE ACTION FOR A SINGLE EQUATION OF THE STRINGS FLUCTUATION

Abstract

The development of modern science and technology requires thestudy of problems for differential equations describing processes withshort-term changes, or which are influenced by external forces, theduration of which can be neglected when compiling relevantmathematical models. Such tasks are found in thermophysics,mechanics, chemical technologies, biology, management theory and other branches of science and technology, where they studyprocesses under the influence of short-term forces and calledsystems with impulse action. Such systems contain differentialequations, an equation describing gaps in the first kind at moments ofimpulse action and impulse conditions. The presence of impulseaction substantially changes and complicates the behavior of thetrajectories of such problems, even for relatively simple differentialequations. The study of differential equations with impulse actionbegan with the development of nonlinear mechanics and attracted theattention of many researchers to the possibility to really describe theprocesses in nonlinear systems. In spite of the large number of worksdevoted to various questions of the theory of differential equationswith impulse action, a large number of problems of the qualitativetheory of such equations remain open for many cases. There is theinfluence of impulse action in fixed and non-fixed moments of time. Inthis paper we consider the process of oscillation of a string withinstantaneous increase of energy at moments when the full energy ofa string reaches this critical level. That is, the moments of impulseaction are not predetermined, but regulated by the process itself. Forsuch a task there are conditions for the existence of solutions forwhich the impulse action is carried out an infinite number of times andconstructed such solutions.