Stabilizability of a class of stochastic bilinear hybrid systems. (English) Zbl 1228.60068

Using the Lyapunov technique, sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, bilinear hybrid systems with a Markovian or any switching rule are discussed, and a closed loop controller is found. In addition, sufficient conditions for exponential mean-square stabilizability are formulated using the Lie algebra approach, and an open-loop controller is designed. The results are illustrated by examples and simulations.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
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[1] Agrachev, A.A.; Liberzon, D., Lie-algebraic stability criteria for switched systems, SIAM J. control optim., 40, 253-269, (2001) · Zbl 0995.93064
[2] Bensoubaya, M.; Ferera, A.; Iggidr, A., A jurdjevic-quinn type theorem for stochastic nonlinear control systems, IEEE trans. automat. control, 45, 93-98, (2000) · Zbl 0976.93085
[3] Blom, H.A.P.; Lygeros, J., Stochastic hybrid systems, (2006), Springer Berlin
[4] Boukas, E.K., Stabilization of stochastic nonlinear hybrid systems, Int. J. innov. comput., inf. control, 1, 131-141, (2005) · Zbl 1085.93026
[5] Boukas, E.K., Stochastic switching systems, (2006), Birkhäuser Boston · Zbl 1090.93048
[6] Brockett, R.W., Lie algebras and Lie groups in control theory, (), 43-82
[7] Bruni, C.; Dipillo, G.; Koch, G., Bilinear systems: an appealing class of “nearly linear” systems in theory and applications, IEEE trans. automat. control, 19, 334-348, (1974) · Zbl 0285.93015
[8] Casandras, C.G.; Lygeros, J., Stochastic hybrid systems, (2007), Lavoisier, pp. 2000-2010
[9] Dragan, V.; Morozan, T., Stability and robust stabilization to linear stochastic systems described by stochastic differential equations with Markovian jumping and multiplicative white noise, Stoch. anal. appl., 20, 33-92, (2002) · Zbl 1136.60335
[10] Elliott, D.L., Bilinear control systems, (2009), Springer New York
[11] Fleming, W.H.; Rishel, R.W., Deterministic and stochastic optimal control, (1975), Springer New York · Zbl 0323.49001
[12] Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, (1992), Springer New York · Zbl 0713.60085
[13] Florchinger, P., A stochastic jurdjevic-quinn theorem, SIAM J. control optim., 41, 83-88, (2002) · Zbl 1014.60062
[14] Florchinger, P., A stochastic version of jurdjevic-quinn theorem, Stoch. anal. appl., 12, 473-480, (1994) · Zbl 0810.60051
[15] Khasminskii, R.Z.; Zhu, C.; Yin, G., Stability of regime-switching diffusions, Stochastic process. appl., 117, 1037-1051, (2007) · Zbl 1119.60065
[16] Kushner, H.J., Stochastic stability and control, (1967), Academic Press New York · Zbl 0178.20003
[17] Liberzon, D., Switching in systems and control, (2003), Birkhäuser Boston · Zbl 1036.93001
[18] Luesink, R.; Nijmeijer, H., On the stabilization of bilinear systems via constant feedback, Linear algebra appl., 122/123/124, 457-474, (1989)
[19] Mao, X.; Yin, G.G.; Yuan, C., Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 43, 264-273, (2007) · Zbl 1111.93082
[20] Mohler, R.R., Bilinear control processes, (1973), Academic Press New York · Zbl 0204.46205
[21] Mohler, R.P.; Kolodziej, W.J., An overview of stochastic bilinear control processes, IEEE trans. syst. man cybern., 10, 913-918, (1980) · Zbl 0475.93054
[22] Samelson, H., Notes on the Lie algebra, (1969), Van Nostrand Reinhold New York · Zbl 0209.06601
[23] Seroka, E.; Socha, L., Lie algebra approach in the study of the stability of stochastic linear hybrid systems, J. theoret. appl. mech., 49, 31-50, (2011)
[24] E. Seroka, L. Socha, Stability of a class of stochastic linear hybrid systems, in: Proceedings of the American Control Conference, Baltimore, 2010, pp. 924-929.
[25] Stengel, R.F., Stochastic optimal control, (1986), Wiley-Interscience New York · Zbl 0666.93126
[26] Wang, Z.; Qiao, H.; Burnham, K.J., On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters, IEEE trans. automat. control, 47, 640-646, (2002) · Zbl 1364.93672
[27] Zhai, G.; Liu, D.; Imae, J.; Koboyashi, T., Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems, IEEE trans. circuits syst. I. regul. pap., 53, 152-156, (2006)
[28] G. Zhai, X. Xu, H. Lin, D. Liu, An extension of Lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems, in: Proc. 2006 IEEE International Conference on Networking, Sensing and Control (ICNSCʼ06), 2006, pp. 362-367.
[29] Yong, J.; Zhou, X.Y., Stochastic controls, (1999), Springer New York
[30] Yuan, C.; Lygeros, J., Stabilization of a class of stochastic differential equations with Markovian switching, Systems control lett., 54, 819-833, (2005) · Zbl 1129.93517
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