8. Stabilization of the Controlled Inverted Pendulum by a Control with Delay

Chapter 8 devotes to the classical problem of stabilization of the controlled inverted pendulum. The problem of stabilization for the mathematical model of the controlled inverted pendulum during many years is very popular among the researchers. Unlike of the classical way of stabilization in which the stabilized control is a linear combination of the state and velocity of the pendulum here another way of stabilization is proposed. It is supposed that only the trajectory of the pendulum can be observed and stabilized control depends on whole trajectory of the pendulum. Linear and nonlinear models of the controlled inverted pendulum by stochastic perturbations are considered, in particular, under influence of Markovian stochastic perturbations. Via the general method of construction of Lyapunov functionals sufficient conditions for stabilization of zero solution by stochastic perturbations are obtained, nonzero steady-state solutions are investigated. 38 figures show a behavior of the controlled inverted pendulum in the case of stable and unstable equilibrium.