Abstract
It is shown that the problem of the asymptotic stabilization of a given position of the Lagrange top for any control from a sufficiently wide class does not permit the existence of a single uniformly asymptotically stable equilibrium, even with possible impacts of the top against the horizontal plane; i.e., the global stabilization of the system is impossible. In particular, we show that it is impossible to globally stabilize the top by moving its pivot point along the horizontal plane.
Similar content being viewed by others
References
Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Some Problems on Theory of Motion Stability), Moscow: Gosudarstvennoe Izd. Fiziko-Matematicheskoi Literatury, 1959.
Reissig, R., Sansone, G., and Conti, R., Qualitative Theorie nichtlinearer Differentialgleichungen, Roma: Cremonese, 1963.
Filippov, A.F., Differential equations with discontinuous right-hand side, Mat. Sb., 1960, vol. 51 (93), no. 1, pp. 99–128.
Wazewski, T., Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Pol. Math., 1947, vol. 20, pp. 279–313.
Polekhin, I., Forced oscillations of a massive point on a compact surface with a boundary, Nonlinear Anal.: Theory, Methods Appl., 2015, vol. 128, pp. 100–105.
Polekhin, I., On forced oscillations in groups of interacting nonlinear systems, Nonlinear Anal.: Theory, Methods Appl., 2016, vol. 135, pp. 120–128.
Bolotin, S.V. and Kozlov, V.V., Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney’s inverted pendulum problem, Izv. Ross. Akad. Nauk. Ser. Mat., 2015, vol. 79, no. 5, pp. 894–901.
Polekhin, I., On topological obstructions to global stabilization of an inverted pendulum, Syst. Control Lett., 2018, vol. 113, pp. 31–35.
Polekhin, I., A topological view on forced oscillations and control of an inverted pendulum, Proc. 3rd Conference on Geometric Science of Information, GSI 2017, in Lecture Notes in Computer Science, Nielsen, F. and Barbaresco, F., Eds., Cham: Springer, 2017, vol. 10589, pp. 329–335.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.Yu. Polekhin, 2018, published in Prikladnaya Matematika i Mekhanika, 2018, Vol. 82, No. 5, pp. 599–604.
About this article
Cite this article
Polekhin, I.Y. On the Impossibility of Global Stabilization of the Lagrange Top. Mech. Solids 53 (Suppl 2), 71–75 (2018). https://doi.org/10.3103/S002565441805014X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S002565441805014X