Unsolved problem about stability of stochastic difference equations with continuous time and distributed delay
Volume 11, Issue 4 (2024), pp. 395–402
Pub. online: 11 April 2024
Type: Research Article
Open Access
Open Access
Received
26 January 2024
26 January 2024
Revised
11 March 2024
11 March 2024
Accepted
25 March 2024
25 March 2024
Published
11 April 2024
11 April 2024
Abstract
Despite the fact that the theory of stability of continuous-time difference equations has a long history, is well developed and very popular in research, there is a simple and clearly formulated problem about the stability of stochastic difference equations with continuous time and distributed delay, which has not been solved for more than 13 years. This paper offers to readers some generalization on this unsolved problem in the hope that it will help move closer to its solution.
References
Blizorukov, M.G.: On the construction of solutions of linear difference systems with continuous time. Differ. Uravn. 32(1), 127–128 (1996), (in Russian), translated in Differ. Equ. 32(1), 133–134 (1996). MR1432957
Damak, S., Di Loreto, M., Lombardi, W., Andrieu, V.: Exponential ${L_{2}}$-stability for a class of linear systems governed by continuous-time difference equations. Automatica 50(12), 3299–3303 (2014). MR3284169. https://doi.org/10.1016/j.automatica.2014.10.087
Damak, S., Di Loreto, M., Mondie, S.: Stability of linear continuous-time difference equations with distributed delay: Constructive exponential estimates. Int. J. Robust Nonlinear Control 25(17), 3195–3209 (2015). MR3419673. https://doi.org/10.1002/rnc.3249
Di Loreto, M., Damak, S., Mondie, S.: Stability and stabilization for continuous-time difference equations with distributed delay. In: Delays and Networked Control Systems, pp. 17–36, 2 (2016). MR3559212. https://doi.org/10.1007/978-3-319-32372-5
Gikhman, I.I., Skorokhod, A.V.: The theory of stochastic processes, Springer, Berlin, Germany (1974). Volume I, 1975, Volume II, 1979, Volume III. MR0651014
Gil’, M., Cheng, S.: Solution estimates for semilinear difference-delay equations with continuous time. Discrete Dyn. Nat. Soc. 2007(1) (2007). Article ID 82027, 8 pages. MR2346522. https://doi.org/10.1155/2007/82027
Korenevskii, D.G.: Criteria for the stability of systems of linear deterministic and stochastic difference equations with continuous time and with delay. Mat. Zametki 70(2), 213–229 (2001), (in Russian), translated in Math. Notes 70(1–2), 192–205 (2001). MR1882411. https://doi.org/10.1023/A:1010202824752
Ma, Q., Gu, K., Choubedar, N.: Strong stability of a class of difference equations of continuous time and structured singular value problem. Automatica 87, 32–39 (2018). MR3733898. https://doi.org/10.1016/j.automatica.2017.09.012
Melchor-Aguilar, D.: Exponential stability of some linear continuous time difference systems. Syst. Control Lett. 61, 62–68 (2012). MR2878688. https://doi.org/10.1016/j.sysconle.2011.09.013
Melchor-Aguilar, D.: Exponential stability of linear continuous time difference systems with multiple delays. Syst. Control Lett. 62(10), 811–818 (2013). MR3084925. https://doi.org/10.1016/j.sysconle.2013.06.003
Melchor-Aguilar, D.: Further results on exponential stability of linear continuous time difference systems. Appl. Math. Comput. 219(19), 10025–10032 (2013). MR3055719. https://doi.org/10.1016/j.amc.2013.03.051
Peics, H.: Representation of solutions of difference equations with continuous time. J. Qual. Theory Differ. Equ. 21, 1–8 (2000). Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999). MR1798671
Pelyukh, G.P.: Representation of solutions of difference equations with a continuous argument. Differ. Uravn. 32(2), 256–264 (1996), (in Russian), translated in Differ. Equ. 32(2), 260–268 (1996). MR1435097
Pepe, P.: The Liapunov’s second method for continuous time difference equations. Int. J. Robust Nonlinear Control 13(15), 1389–1405 (2003). MR2027371. https://doi.org/10.1002/rnc.861
Rocha, E., Mondie, S., Di Loreto, M.: On the Lyapunov matrix of linear delay difference equations in continuous time. IFAC-PapersOnLine 50(1), 6507–6512 (2017). https://doi.org/10.1016/j.ifacol.2017.08.1048
Rocha, E., Mondie, S., Di Loreto, M.: Necessary stability conditions for linear difference equations in continuous time. IEEE Trans. Autom. Control 2018, 4405–4412 (2018). MR3891471. https://doi.org/10.1109/tac.2018.2822667
Shaikhet, L.: Lyapunov functionals construction for stochastic difference second kind Volterra equations with continuous time. Adv. Differ. Equ. 2004(1), 67–91 (2004). MR2059203. https://doi.org/10.1155/S1687183904308022
Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer Science & Business Media (2011). MR3015017. https://doi.org/10.1007/978-0-85729-685-6
Shaikhet, L.: About an unsolved stability problem for a stochastic difference equation with continuous time. J. Differ. Equ. Appl. 17(3), 441–444 (2011). ifirst article 1–4, 1563–5120 (2010), first published on 05 march 2010. MR2783358. https://doi.org/10.1080/10236190903489973
Shaikhet, L.: About stability of difference equations with continuous time and fading stochastic perturbations. Appl. Math. Lett. 98, 284–291 (2019). https://www.sciencedirect.com/science/article/pii/S0893965919302733?via. MR3979526. https://doi.org/10.1016/j.aml.2019.06.029
Shaikhet, L.: Behavior of solution of stochastic difference equation with continuous time under additive fading noise. Discrete Contin. Dyn. Syst., Ser. B 27(1), 301–310 (2022). MR4354194. https://doi.org/10.3934/dcdsb.2021043
Shaikhet, L.: Some unsolved problems in stability and optimal control theory of stochastic systems. MDPI Math. 10(3), 474, 1-10 (2022). Special Issue “Models of Delay Differential Equations-II”. https://doi.org/10.3390/math10030474
Shaikhet, L.: About an unsolved problem of stabilization by noise for difference equations. MDPI Math. 12(1), 110, 1–11 2024. https://www.mdpi.com/2227-7390/12/1/110
Sharkovsky, A.N., Maistrenko, Yu.L.: Difference equations with continuous time as mathematical models of the structure emergences. In: Dynamical Systems and Environmental Models, Eisenach, 1986, Math. Ecol. pp. 40–49. Akademie-Verlag, Berlin (1987). MR0925661. https://doi.org/10.1515/9783112484685-007
Zhang, Y.: Robust exponential stability of uncertain impulsive delay difference equations with continuous time. J. Franklin Inst. 348(8), 1965–1982 (2011). MR2841890. https://doi.org/10.1016/j.jfranklin.2011.05.014
