{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T12:44:53Z","timestamp":1774269893000,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"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>This paper continues a series of papers by the author devoted to unsolved problems in the theory of stability and optimal control for stochastic systems. A delay differential equation with stochastic perturbations of the white noise and Poisson\u2019s jump types is considered. In contrast with the known stability condition, in which it is assumed that stochastic perturbations fade on the infinity quickly enough, a new situation is studied, in which stochastic perturbations can either fade on the infinity slowly or not fade at all. Some unsolved problem in this connection is brought to readers\u2019 attention. Additionally, some unsolved problems of stabilization for one stochastic delay differential equation and one stochastic difference equation are also proposed.<\/jats:p>","DOI":"10.3390\/axioms14060452","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T06:46:01Z","timestamp":1749451561000},"page":"452","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["About Some Unsolved Problems in the Stability Theory of Stochastic Differential and Difference Equations"],"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":[[2025,6,9]]},"reference":[{"key":"ref_1","unstructured":"Shaikhet, L. (September, January 31). About some unsolved problems of stability theory for stochastic hereditary systems. Proceedings of the Leverhulme International Network: Numerical and Analytical Solution of Stochastic Delay Differential Equations, University of Chester, Chester, UK. Abstracts."},{"key":"ref_2","unstructured":"Shaikhet, L. (2011, January 5\u20137). Unsolved stability problem for stochastic differential equation with varying delay. Proceedings of the Leverhulme International Network: Numerical and Analytical Solution of Stochastic Delay Differential Equations, University of Chester, Chester, UK. Abstracts."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1080\/10236190903489973","article-title":"About an unsolved stability problem for a stochastic difference equation with continuous time","volume":"17","author":"Shaikhet","year":"2011","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1016\/j.aml.2011.10.002","article-title":"Two unsolved problems in the stability theory of stochastic differential equations with delay","volume":"25","author":"Shaikhet","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_5","unstructured":"Shaikhet, L. (2013, January 29\u201331). About an unsolved optimal control problem for stochastic partial differential equation. Proceedings of the XVI International Conference \u201cDynamical System Modeling and Stability Investigations\u201d (DSMSI-2013), Kiev, Ukraine. Abstracts."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Shaikhet, L. (2013). Some Unsolved Problems: Problem 1, Problem 2. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer Science & Business Media.","DOI":"10.1007\/978-3-319-00101-2_1"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Shaikhet, L. (2022). Some unsolved problems in stability and optimal control theory of stochastic systems. Mathematics, 10.","DOI":"10.3390\/math10030474"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Shaikhet, L. (2024). About an unsolved problem of stabilization by noise for difference equations. Mathematics, 12.","DOI":"10.3390\/math12010110"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"395","DOI":"10.15559\/24-VMSTA253","article-title":"Unsolved problem about stability of stochastic difference equations with continuous time and distributed delay","volume":"11","author":"Shaikhet","year":"2024","journal-title":"Mod. Stochastics Theory Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"32571","DOI":"10.3934\/math.20241560","article-title":"About one unsolved problem in asymptotic p-stability of stochastic systems with delay","volume":"9","author":"Shaikhet","year":"2024","journal-title":"AIMS Math."},{"key":"ref_11","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_12","unstructured":"Gikhman, I.I., and Skorokhod, A.V. (1979). The Theory of Stochastic Processes, Springer."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Shaikhet, L. (2013). Lyapunov Functionals and Stability of Stochastic Functional Differential Equations, Springer Science & Business Media.","DOI":"10.1007\/978-3-319-00101-2"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.aml.2018.10.004","article-title":"About stability of delay differential equations with square integrable level of stochastic perturbations","volume":"90","author":"Shaikhet","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_15","first-page":"3651","article-title":"Stability of delay differential equations with fading stochastic perturbations of the type of white noise and Poisson\u2019s jumps","volume":"25","author":"Shaikhet","year":"2020","journal-title":"Discret. Contin. Dyn. Syst. Ser. B"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Shaikhet, L. (2011). Lyapunov Functionals and Stability of Stochastic Difference Equations, Springer Science & Business Media.","DOI":"10.1007\/978-0-85729-685-6"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1080\/002071797223406","article-title":"Discretized LMI set in the stability problem of linear time-delay systems","volume":"68","author":"Gu","year":"1997","journal-title":"Int. J. Control"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1016\/j.automatica.2017.04.015","article-title":"Stabilization by using artificial delays: An LMI approach","volume":"81","author":"Fridman","year":"2017","journal-title":"Automatica"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.sysconle.2018.12.007","article-title":"Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay","volume":"124","author":"Fridman","year":"2019","journal-title":"Syst. Control Lett."},{"key":"ref_20","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_21","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_22","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_23","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_24","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_25","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_26","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_27","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_28","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_29","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_30","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":"J. Appl. Math. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1142\/S0219493708002263","article-title":"Stabilizing with a hammer","volume":"8","year":"2008","journal-title":"Stoch. Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Shaikhet, L. (2025). About Stabilization of the Controlled Inverted Pendulum under Stochastic Perturbations of the Type of Poisson\u2019s Jumps. Axioms, 14.","DOI":"10.20944\/preprints202504.2254.v1"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Khasminskii, R.Z. (2012). Stochastic Stability of Differential Equations, Springer.","DOI":"10.1007\/978-3-642-23280-0"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/452\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:48:41Z","timestamp":1760032121000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/452"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,9]]},"references-count":33,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060452"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060452","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,9]]}}}