Abstract
We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form.
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References
D. G. Korenevskii, “The effect of random parametric perturbations of the white-noise type in linear discrete dynamical systems is solely destabilizing,” Dokl. Ross. Akad. Nauk, 378, No. 3, 310–313 (2001).
D. H. Korenivs’kyi, “On the impossibility of stabilization of solutions of a system of linear deterministic difference equations by perturbations of its coefficients by stochastic processes of “ white-noise” type,” Ukr. Mat. Zh., 54, No. 2, 285–288 (2002).
D. G. Korenevskii, Stability of Dynamical Systems under Random Perturbations of Their Parameters. Algebraic Criteria [in Russian], Naukova Dumka, Kiev (1989).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1719–1724, December, 2005.
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Korenivs’kyi, D.H. Destabilizing effect of random parametric perturbations of the white-noise type in some quasilinear continuous and discrete dynamical systems. Ukr Math J 57, 2021–2026 (2005). https://doi.org/10.1007/s11253-006-0046-0
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DOI: https://doi.org/10.1007/s11253-006-0046-0