Abstract
A general class of the nonlinear time-varying systems of Itô stochastic differential equations is considered. Two problems on the partial stability in probability are studied as follows: 1) the stability with respect to a given part of the variables of the trivial equilibrium; 2) the stability with respect to a given part of the variables of the partial equilibrium. The stochastic Lyapunov functions-based conditions of the partial stability in probability are established. In addition to the main Lyapunov function, an auxiliary (generally speaking, vector-valued) function is introduced for correcting the domain in which the main Lyapunov function is constructed. A comparison with the well-known results on the partial stability of the systems of stochastic differential equations is given. An example that well illustrates the peculiarities of the suggested approach is described. Also a possible unified approach to analyze the partial stability of the time-invariant and time-varying systems of stochastic differential equations is discussed.
Similar content being viewed by others
References
Khas’minskii, R.Z., Ustoichivost’ sistem differentsial’nykh uravnenii pri sluchainykh vozmushcheniyakh ikh parametrov, Moscow: Nauka, 1969. Translated under the title Stochastic Stability of Differential Equations, Alphen aan den Rijn: Sijthoff & Noordhoff, 1980.
Kushner, H.J., Stochastic Stability and Control, New York: Academic, 1967. Translated under the title Stokhasticheskaya ustoichivost’ i upravlenie, Moscow: Mir, 1969.
Mao, X.R., Stochastic Differential Equations and Applications, Oxford: Woodhead, 2008, 2nd ed.
Kats, I.Ya. and Krasovskii, N.N., On the Stability of Systems with Random Parameters, J. Appl. Math. Mech., 1960, vol. 24, no. 5, pp. 809–823.
Kats, I.Ya. and Martynyuk, A.A., Stability and Stabilization of Nonlinear Systems with Random Structure, London: Taylor & Francis, 2002.
Mao, X.R. and Yuan, C.G., Stochastic Differential Equations with Markovian Switching, London: Imperial College Press, 2006.
Vorotnikov, V.I., Two Classes of Partial Stability Problems: Unification of the Notions and Common Conditions of Solvability, Dokl. Physics, 2002, vol. 47, no. 5, pp. 377–381.
Vorotnikov, V.I. and Martyshenko, Yu.G., On Partial Detectability of the Nonlinear Dynamic Systems, Autom. Remote Control, 2009, vol. 70, no. 1, pp. 20–32.
Vorotnikov, V.I. and Martyshenko, Yu.G., On the Partial Stability of Nonlinear Dynamical Systems, J. Comp. Syst. Sci. Int., 2010, vol. 49, no. 5, pp. 702–709.
Vorotnikov, V.I., Partial Stability and Control: The State of the Art and Developing Prospects, Autom. Remote Control, 2005, vol. 66, no. 4, pp. 511–561.
Vorotnikov, V.I. and Martyshenko, Yu.G., Stability in a Part of Variables of “Partial” Equilibria of Systems with Aftereffect, Math. Notes, 2014, vol. 96, no. 4, pp. 477–483.
Vorotnikov, V.I. and Martyshenko, Yu.G., On the Partial Stability of Nonlinear Discrete Systems, Mekhatr. Avtomatiz. Upravlen., 2017, vol. 18, no. 6, pp. 371–375.
Ramirez-Llanos, E. and Martinez, S., Distributed and Robust Fair Optimization Applied to Virus Diffusion Control, IEEE Trans. Network Sci. Eng., 2017, vol. 4, no. 1, pp. 41–54.
Sharov, V.F., Stability and Stabilization of Stochastic Systems with Respect to Part of the Variables, Autom. Remote Control, 1978, vol. 39, no. 11, pp. 1629–1636.
Vorotnikov, V.I., Partial Stability and Control, Boston: Birkhauser, 1998.
Potcovaru, G., On the Partial Stability of a Dynamical Systems with Random Parameters, An. Univ. Bucur. Mat., 1999, vol. 48, no. 2, pp. 163–168.
Ignatyev, O., Partial Asymptotic Stability in Probability of Stochastic Differential Equations, Statist. Probab. Lett., 2009, vol. 79, no. 5, pp. 597–601.
Ignatyev, O., New Criterion of Partial Asymptotic Stability in Probability of Stochastic Differential Equations, Appl. Math. Comp., 2013, vol. 219, no. 23, pp. 10961–10966.
Zuev, A.L., Ignatyev, A.O., and Kovalev, A.M., Ustoichivost’ i stabilizatsiya nelineinykh sistem (Stability and Stabilization of Nonlinear Systems), Kiev: Naukova Dumka, 2013.
Rajpurohit, T. and Haddad, W.M., Stochastic Finite-Time Partial Stability, Partial-State Stabilization, and Finite-Time Optimal Feedback Control, Math. Control, Signals, Syst., 2017, vol. 29, no. 2, art. 10.
Rajpurohit, T. and Haddad, W.M., Partial-State Stabilization and Optimal Feedback Control for Stochastic Dynamical Systems, J. Dynam. Syst., Measurement, Control, 2017, vol. 139, no. 9, p. DS–15–1602.
Rumyantsev, V.V. and Oziraner, A.S., Ustoichivost’ i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh (Stability and Stabilization of Motion with Respect to Part of the Variables), Moscow: Nauka, 1987.
Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L., Nelineinoe i adaptivnoe upravlenie slozhnymi dinamicheskimi sistemami (Nonlinear and Adaptive Control of Complex Dynamic Systems), St. Petersburg: Nauka, 2000.
Rumyantsev, V.V., On Motion Stability with Respect to Part of the Variables, Vestn. Mosk. Gos. Univ., Ser. Mat. Mekh. Fiz. Astronom. Khim., 1957, no. 4, pp. 9–16.
Vorotnikov, V.I., On the Theory of Partial Stability, J. Appl. Math. Mech., 1995, vol. 59, no. 4, pp. 525–531.
Kao, Y., Wang, C., Zha, F., and Cao, H., Stability in Mean of Partial Variables for Stochastic Reaction-Diffusion Systems with Markovian Switching, J. Franklin Inst., 2014, vol. 351, no. 1, pp. 500–512.
Socha, L. and Zhu, Q.X., Exponential Stability with Respect to Part of the Variables for a Class of Nonlinear Stochastic Systems with Markovian Switching, Math. Comp. Simul., 2019, vol. 155, pp. 2–14.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 5, pp. 86–98.
Rights and permissions
About this article
Cite this article
Vorotnikov, V.I., Martyshenko, Y.G. On the Partial Stability in Probability of Nonlinear Stochastic Systems. Autom Remote Control 80, 856–866 (2019). https://doi.org/10.1134/S0005117919050059
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117919050059