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Steady-state solutions of nonlinear model of inverted pendulum. (English) Zbl 0993.60051
The authors consider the nonlinear mathematical model of inverted pendulum $$\ddot{x}(t)-(a+\sigma\dot\xi(t))x(t)=u(t)$$ with the control $$u(t)=\int_0^{\infty}dK(\tau)x(t-\tau)$$, the initial conditions $$x(s)=\varphi(s)$$, $$\dot{x}(s)=\dot\varphi(s)$$ and with the stochastic perturbations $$\dot\xi(t)$$ of the white noise type. The nonclassical method of stabilization of this pendulum is exploited. Nonzero steady-state solutions of this system are studied. Conditions under which equilibrium points of the model are stable, unstable or one-sided stable are obtained. Theoretical results are illustrated by numerical examples.

##### MSC:
 60G99 Stochastic processes