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Stochastic stabilization and destabilization. (English) Zbl 0820.93071

This paper presents generalizations of previous results on the (de)stabilization by noise of systems to the nonlinear case. Continuous- time finite-dimensional nonlinear systems having a certain contractive property are considered and shown to be almost surely asymptotically stabilizable when perturbed by Brownian motion whose intensity can be adjusted. Also, it is shown that any system whose dimension is larger than one can be destabilized in the almost sure sense by proper choice of the intensity of the perturbing Brownian motion.

MSC:

93E15 Stochastic stability in control theory
93C10 Nonlinear systems in control theory
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[1] Arnold, L., A new example of an unstable system being stabilized by random parameter noise, Inform. Comm. Math. Chem., 133-140 (1979) · Zbl 0424.93059
[2] Arnold, L., Stabilization by noise revisited, Z. Zngew. Math. Mech., 70, 235-246 (1990) · Zbl 0765.93076
[3] Arnold, L.; Crauel, H.; Wihstutz, V., Stabilization of linear systems by noise, SIAM J. Control Optim., 21, 451-461 (1983) · Zbl 0514.93069
[4] Bellman, R.; Bentsman, J.; Meerkov, S., Stability of fast periodic systems, IEEE Trans. Automat. Control, AC-30, 289-291 (1985) · Zbl 0557.93055
[5] Bucher, C. G.; Lin, Y. K., Effect of spanwise correlation of turbulence field on the motion stability of long-span bridges, J. Fluids Structures, 2, 437-451 (1988) · Zbl 0677.73047
[6] Has’minskii, R. Z., Stochastic Stability of Differential Equations (1980), Sijthoff and Noordhoff · Zbl 0276.60059
[7] Liptser, R., A strong law of large numbers for local martingales, Stochastics, 3, 217-228 (1980) · Zbl 0435.60037
[8] Meerkov, S., Condition of vibrational stabilizability for a class of non-linear systems, IEEE Trans. Automat. Control, AC-27, 485-487 (1982) · Zbl 0491.93034
[9] Metivier, M., Semimartingales (1982), Walter de Gruyter · Zbl 0503.60054
[10] Roberts, J. B.; Spanos, P. D., Stochastic averaging: An approximate method of solving random vibration problems, Internat. J. Non-linear Mech., 21, 111-134 (1986) · Zbl 0582.73077
[11] Scheutzow, M., Stabilization and destabilization by noise in the plane, Stochastic Anal. Appl., 11, 1, 97-113 (1993) · Zbl 0766.60072
[12] Wu, R.; Mao, X., Existence and uniqueness of the solutions of stochastics differential equations, Stochastics, 11, 19-32 (1983) · Zbl 0535.60054
[13] Zhabko, A. P.; Kharitonov, V. L., Problem of vibrational stabilization of linear systems, Avtomat. Telemekh., 2, 31-34 (1980), (English translation) · Zbl 0553.93047
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