M.E. Semenov, D.V. Shevlyakova, O.I. Kanishcheva, D.V. Grachikov

This article focuses on important and interesting subject - the inverted pendulum where control action is an output of the hysteresis operator. In 1951, P.L. Kapitza proved that the top position of a physical pendulum can be stable (or even asymptotically stable) with vertical harmonic oscillations of the lower suspension point the high frequency and low amplitude. More accurate relations for the selection of the desired mode oscillation fulcrum, were obtained by investigating the stability diagram of the Mathieu equation, which is the equation of a linear oscillator with a harmonic parametric excitation. The article presents a mathematical model of the stabilization of the inverted pendulum with the base formed by the physical system, the piston - cylinder, treated as a hysteretic converter type backlash. We present study dissipative models. The system is dissipative if there exists a region in phase space and agreed with her region in the state space hysteretic converter that decision, which had begun in this area, do not extend to infinity. As a result of numerical experiments obtained the following: 1) the estimates of the constants - the parameters of the pendulum to ensure the dissipative system; 2) establish their inverse relationship; 3) the dependence of the maximum possible length of the solution of the cylinder oscillation frequency of the suspension