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A topological view on forced oscillations and control of an inverted pendulum. (English) Zbl 1427.70048
Nielsen, Frank (ed.) et al., Geometric science of information. Third international conference, GSI 2017, Paris, France, November 7–9, 2017. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 10589, 329-335 (2017).
Summary: We consider a system of a planar inverted pendulum in a gravitational field. First, we assume that the pivot point of the pendulum is moving along a horizontal line with a given law of motion. We prove that, if the law of motion is periodic, then there always exists a periodic solution along which the pendulum never becomes horizontal (never falls). We also consider the case when the pendulum with a moving pivot point is a control system, in which the mass point is constrained to be strictly above the pivot point (the rod cannot fall “below the horizon”). We show that global stabilization of the vertical upward position of the pendulum cannot be obtained for any smooth control law, provided some natural assumptions.
For the entire collection see [Zbl 1374.94006].

MSC:
70K20 Stability for nonlinear problems in mechanics
34H15 Stabilization of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
94A17 Measures of information, entropy
94C30 Applications of design theory to circuits and networks
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