As is known, the problem of the inverted pendulum plays the central role in the control theory. In particular, the problem of the inverted pendulum (as a test model) presents many challenging problems to the control designs. Because of their nonlinear nature, the pendulums have preserved their usefulness and now they are used to illustrate many of the ideas emerging in the field of a nonlinear control. Typical examples of that are the feedback stabilization, variable structure control, passivitybased control, back-stepping and forwarding, nonlinear observers, friction compensation, and nonlinear model reduction. The challenges of the control made the inverted pendulum systems a classic tool for the control laboratories. It should also be pointed out that the problem of stabilization of such a system is a classical problem of the dynamics and control theory. Moreover, the model of the inverted pendulum is widely used as a standard for testing of the control algorithms (for PID controllers, neural networks, fuzzy control, etc.). In this paper, the authors investigate the elastic inverted pendulum with a hysteretic nonlinearity (a backlash) in the suspension point. Namely, the problems of stabilization and optimization of such a system are considered. The algorithm, which ensures an effective procedure for finding of the optimal parameters, is presented and applied to the considered system. The results of the numerical simulations, namely the phase portraits and the dynamics of Lyapunov function, are also presented and discussed.