{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T12:44:52Z","timestamp":1774269892396,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T00:00:00Z","timestamp":1735603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The classical problem of stabilization of the controlled inverted pendulum is considered in the case of stochastic perturbations of the type of Poisson\u2019s jumps. It is supposed that stabilized control depends on the entire trajectory of the pendulum. Linear and nonlinear models of the controlled inverted pendulum are considered, and the stability of the zero and nonzero equilibria is studied. The obtained results are illustrated by examples with numerical simulation of solutions of the equations under consideration.<\/jats:p>","DOI":"10.3390\/axioms14010029","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T12:37:19Z","timestamp":1735648639000},"page":"29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["About Stabilization of the Controlled Inverted Pendulum Under Stochastic Perturbations of the Type of Poisson\u2019s Jumps"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7354-1383","authenticated-orcid":false,"given":"Leonid","family":"Shaikhet","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ariel University, Ariel 40700, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,31]]},"reference":[{"key":"ref_1","first-page":"714","article-title":"Dynamical stability of a pendulum when its point of suspension vibrates, and pendulum with a vibrating suspension","volume":"Volume 2","year":"1965","journal-title":"Collected Papers of P.L. Kapitza"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0020-7462(72)90025-X","article-title":"Stability of the inverted pendulum subjected to almost periodic and stochastic base motion\u2014An application of the method of averaging","volume":"7","author":"Mitchell","year":"1972","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1137\/1030140","article-title":"Stability of the inverted pendulum\u2014A topological explanation","volume":"30","author":"Levi","year":"1988","journal-title":"SIAM Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1119\/1.17011","article-title":"Stability and Hopf bifurcations in an inverted pendulum","volume":"60","author":"Blackburn","year":"1992","journal-title":"Am. J. Phys."},{"key":"ref_5","first-page":"239","article-title":"A pendulum theorem. Proceedings of the Royal Society of London, Seria A","volume":"443","author":"Acheson","year":"1993","journal-title":"Math. Phys. Eng. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1038\/366215b0","article-title":"Upside-down pendulums","volume":"366","author":"Acheson","year":"1993","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1137\/1037044","article-title":"Stabilization of the inverted linearized pendulum by high frequency vibrations","volume":"37","author":"Levi","year":"1995","journal-title":"Siam Rev."},{"key":"ref_8","first-page":"203","article-title":"Steady-state solutions of nonlinear model of inverted pendulum","volume":"5","author":"Borne","year":"1999","journal-title":"Theory Stoch. Process."},{"key":"ref_9","first-page":"501","article-title":"Stabilization of inverted pendulum by control with delay","volume":"9","author":"Borne","year":"2000","journal-title":"Dyn. Syst. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/S0167-6911(00)00025-6","article-title":"Stabilization of the inverted pendulum around its homoclinic orbit","volume":"40","author":"Lozano","year":"2000","journal-title":"Syst. Control. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/02681110010001289","article-title":"Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the stability of the harmonic oscillator and the inverted pendulum","volume":"16","author":"Imkeller","year":"2001","journal-title":"Dyn. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1088\/0143-0807\/25\/6\/003","article-title":"Effective Hamiltonian and dynamic stability of the inverted pendulum","volume":"25","author":"Mata","year":"2004","journal-title":"Eur. J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/3-540-26444-2_13","article-title":"Multiple time scale numerical methods for the inverted pendulum problem","volume":"Volume 44","author":"Sharp","year":"2005","journal-title":"Multiscale Methods in Science and Engineering"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1155\/DDNS.2005.215","article-title":"Stability of difference analogue of linear mathematical inverted pendulum","volume":"2005","author":"Shaikhet","year":"2005","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"155","DOI":"10.20965\/jaciii.2006.p0155","article-title":"Genetic algorithm on line controller for the flexible inverted pendulum problem","volume":"10","author":"Dadios","year":"2006","journal-title":"J. Adv. Comput. Intell. Intell. Inform."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"762","DOI":"10.1016\/j.jappmathmech.2006.11.010","article-title":"The stability of an inverted pendulum when there are rapid random oscillations of the suspension point","volume":"70","author":"Ovseyevich","year":"2006","journal-title":"Int. J. Appl. Math. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1049\/iet-cta:20060338","article-title":"Design of nonlinear controller for bi-axial inverted pendulum system","volume":"1","author":"Chang","year":"2007","journal-title":"Iet Control. Theory Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1142\/S0219493708002263","article-title":"Stabilizing with a hammer","volume":"8","year":"2008","journal-title":"Stochastics Dyn."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.3934\/dcds.2009.24.1335","article-title":"Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations","volume":"24","author":"Shaikhet","year":"2009","journal-title":"Discret. Contin. Dyn. Syst.-A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/S1007-0214(10)70025-0","article-title":"Modeling and simulation of a flexible inverted pendulum system","volume":"14","author":"Tang","year":"2009","journal-title":"Tsinghua Sci. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1016\/j.actaastro.2010.04.015","article-title":"Dynamic characteristic prediction of inverted pendulum under the reduced-gravity space environments","volume":"67","author":"Li","year":"2010","journal-title":"Acta Astronaut."},{"key":"ref_22","unstructured":"Shaikhet, L. (2011). Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer Science & Business Media. Available online: https:\/\/link.springer.com\/book\/10.1007\/978-0-85729-685-6."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Shaikhet, L. (2013). Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer Science & Business Media. Available online: https:\/\/link.springer.com\/book\/10.1007\/978-3-319-00101-2.","DOI":"10.1007\/978-3-319-00101-2"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Belhaq, M. (2015). Hysteretic nonlinearity in inverted pendulum problem. Structural Nonlinear Dynamics and Diagnosis, Springer.","DOI":"10.1007\/978-3-319-19851-4"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1007\/s11071-015-2186-y","article-title":"Elastic inverted pendulum with backlash in suspension: Stabilization problem","volume":"82","author":"Semenov","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.jtbi.2018.08.027","article-title":"A simple extension of inverted pendulum template to explain features of slow walking","volume":"457","author":"Biswas","year":"2018","journal-title":"J. Theor. Biol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/s00419-017-1323-0","article-title":"Coupled inverted pendulums: Stabilization problem","volume":"88","author":"Semenov","year":"2018","journal-title":"Arch. Appl. Mech."},{"key":"ref_28","first-page":"1","article-title":"Spring-loaded inverted pendulum goes through two contraction-extension cycles during the single-support phase of walking","volume":"8","author":"Antoniak","year":"2019","journal-title":"Biol. Open"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1134\/S1064230719020023","article-title":"Observer design for an inverted pendulum with biased position sensors","volume":"58","author":"Aranovskiy","year":"2019","journal-title":"J. Comput. Syst. Sci. Int."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1399","DOI":"10.1080\/00207721.2019.1615575","article-title":"Robust stabilisation of rotary inverted pendulum using intelligently optimised nonlinear self-adaptive dual fractional-order PD controllers","volume":"50","author":"Saleem","year":"2019","journal-title":"Int. J. Syst. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1177\/1077546320923436","article-title":"Stabilization of coupled inverted pendula: From discrete to continuous case","volume":"27","author":"Semenov","year":"2021","journal-title":"J. Vib. Control"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Gikhman, I.I., and Skorokhod, A.V. (1972). Stochastic Differential Equations, Springer.","DOI":"10.1007\/978-3-642-88264-7_7"},{"key":"ref_33","unstructured":"Gikhman, I.I., and Skorokhod, A.V. (1979). The Theory of Stochastic Processes, v.III, Springer."},{"key":"ref_34","unstructured":"Kolmanovskii, V.B., and Nosov, V.R. (1986). Stability of Functional Differential Equations, Academic Press."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Kolmanovskii, V.B., and Myshkis, A.D. (1992). Applied Theory of Functional Differential Equations, Kluwer Academic.","DOI":"10.1007\/978-94-015-8084-7"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Kolmanovskii, V.B., and Myshkis, A.D. (1999). Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic.","DOI":"10.1007\/978-94-017-1965-0"},{"key":"ref_37","unstructured":"Mao, X. (1994). Exponential Stability of Stochastic Differential Equations, Marcel Dekker."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.cnsns.2018.12.008","article-title":"Stability of the neoclassical growth model under perturbations of the type of Poisson\u2019s jumps: Analytical and numerical analysis","volume":"72","author":"Shaikhet","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"109068","DOI":"10.1016\/j.aml.2024.109068","article-title":"About stabilization by Poisson\u2019s jumps for stochastic differential equations","volume":"153","author":"Shaikhet","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Khasminskii, R.Z. (2012). Stochastic Stability of Differential Equations, Springer.","DOI":"10.1007\/978-3-642-23280-0"},{"key":"ref_41","first-page":"21","article-title":"Stability of difference analogues of nonlinear integro-differential equations: A survey of some known results","volume":"16","author":"Shaikhet","year":"2024","journal-title":"Res. Commun. Math. Math. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:58:07Z","timestamp":1760115487000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,31]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010029"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010029","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,31]]}}}