# American Institute of Mathematical Sciences

December  2018, 10(4): 411-417. doi: 10.3934/jgm.2018015

## On motions without falling of an inverted pendulum with dry friction

 Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

* Corresponding author: Ivan Polekhin

Received  March 2017 Revised  September 2018 Published  November 2018

Fund Project: This work was supported by the Russian Science Foundation under Grant No. 14-50-00005

An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizon are presented. The proof is based on the fact that solutions of the corresponding differential inclusion are right-unique and continuously depend on initial conditions, which is also shown in the paper.

Citation: Ivan Polekhin. On motions without falling of an inverted pendulum with dry friction. Journal of Geometric Mechanics, 2018, 10 (4) : 411-417. doi: 10.3934/jgm.2018015
##### References:
 [1] B. Bardin and A. Markeyev, The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension, Journal of Applied Mathematics and Mechanics, 59 (1995), 879-886. doi: 10.1016/0021-8928(95)00121-2. [2] S. V. Bolotin and V. V. Kozlov, Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney inverted pendulum problem, Izvestiya: Mathematics, 79 (2015), 894-901. doi: 10.4213/im8413. [3] E. I. Butikov, On the dynamic stabilization of an inverted pendulum, American Journal of Physics, 69 (2001), 755-768. [4] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. [5] A. P. Ivanov, Bifurcations in systems with friction: Basic models and methods, Regular and Chaotic Dynamics, 14 (2009), 656-672. doi: 10.1134/S1560354709060045. [6] P. Kapitsa, Pendulum with vibrating axis of suspension (in Russian), Uspekhi fizicheskich nauk, 44 (1954), 7-20. [7] I. Y. Polekhin, Examples of topological approach to the problem of inverted pendulum with moving pivot point (in Russian), Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 10 (2014), 465-472. [8] I. Polekhin, Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Analysis: Theory, Methods & Applications, 128 (2015), 100-105. doi: 10.1016/j.na.2015.07.022. [9] I. Polekhin, On forced oscillations in groups of interacting nonlinear systems, Nonlinear Analysis: Theory, Methods & Applications, 135 (2016), 120-128. doi: 10.1016/j.na.2016.01.021. [10] I. Polekhin, A topological view on forced oscillations and control of an inverted pendulum, Systems Control Lett., 113 (2018), 31-35. doi: 10.1016/j.sysconle.2018.01.005. [11] I. Polekhin, On topological obstructions to global stabilization of an inverted pendulum, Geometric Science of Information. GSI 2017, Lecture Notes in Comput. Sci., 10589, Springer, Cham, (2017), 329-335. [12] V. Popov, Contact Mechanics and Friction: Physical Principles and Applications, Springer Science & Business Media., 2010. [13] R. Reissig, G. Sansone and R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Edizioni Cremonese, 1963. [14] A. Seyranian and A. Seyranian, The stability of an inverted pendulum with a vibrating suspension point, Journal of Applied Mathematics and Mechanics, 70 (2006), 754-761. doi: 10.1016/j.jappmathmech.2006.11.009. [15] R. Srzednicki, On periodic solutions in the Whitney's inverted pendulum problem, arXiv preprint, arXiv:1709.08254, 2017. [16] T. Ważewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Annales De La Societe Polonaise De Mathematique, 20 (1947), 279-313 (1948).

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##### References:
 [1] B. Bardin and A. Markeyev, The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension, Journal of Applied Mathematics and Mechanics, 59 (1995), 879-886. doi: 10.1016/0021-8928(95)00121-2. [2] S. V. Bolotin and V. V. Kozlov, Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney inverted pendulum problem, Izvestiya: Mathematics, 79 (2015), 894-901. doi: 10.4213/im8413. [3] E. I. Butikov, On the dynamic stabilization of an inverted pendulum, American Journal of Physics, 69 (2001), 755-768. [4] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. [5] A. P. Ivanov, Bifurcations in systems with friction: Basic models and methods, Regular and Chaotic Dynamics, 14 (2009), 656-672. doi: 10.1134/S1560354709060045. [6] P. Kapitsa, Pendulum with vibrating axis of suspension (in Russian), Uspekhi fizicheskich nauk, 44 (1954), 7-20. [7] I. Y. Polekhin, Examples of topological approach to the problem of inverted pendulum with moving pivot point (in Russian), Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 10 (2014), 465-472. [8] I. Polekhin, Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Analysis: Theory, Methods & Applications, 128 (2015), 100-105. doi: 10.1016/j.na.2015.07.022. [9] I. Polekhin, On forced oscillations in groups of interacting nonlinear systems, Nonlinear Analysis: Theory, Methods & Applications, 135 (2016), 120-128. doi: 10.1016/j.na.2016.01.021. [10] I. Polekhin, A topological view on forced oscillations and control of an inverted pendulum, Systems Control Lett., 113 (2018), 31-35. doi: 10.1016/j.sysconle.2018.01.005. [11] I. Polekhin, On topological obstructions to global stabilization of an inverted pendulum, Geometric Science of Information. GSI 2017, Lecture Notes in Comput. Sci., 10589, Springer, Cham, (2017), 329-335. [12] V. Popov, Contact Mechanics and Friction: Physical Principles and Applications, Springer Science & Business Media., 2010. [13] R. Reissig, G. Sansone and R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Edizioni Cremonese, 1963. [14] A. Seyranian and A. Seyranian, The stability of an inverted pendulum with a vibrating suspension point, Journal of Applied Mathematics and Mechanics, 70 (2006), 754-761. doi: 10.1016/j.jappmathmech.2006.11.009. [15] R. Srzednicki, On periodic solutions in the Whitney's inverted pendulum problem, arXiv preprint, arXiv:1709.08254, 2017. [16] T. Ważewski, Sur un principe topologique de l'examen de l'allure asymptotique des intégrales des équations différentielles ordinaires, Annales De La Societe Polonaise De Mathematique, 20 (1947), 279-313 (1948).
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