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Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions

Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions


Titill: Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions
Höfundur: Guðmundsson, Skúli
Hafstein, Sigurdur   orcid.org/0000-0003-0073-2765
Útgáfa: 2018
Tungumál: Enska
Umfang: 2895658
Háskóli/Stofnun: Háskóli Íslands
University of Iceland
Svið: Verkfræði- og náttúruvísindasvið (HÍ)
School of Engineering and Natural Sciences (UI)
Deild: Raunvísindadeild (HÍ)
Raunvísindastofnun (HÍ)
Birtist í: Complexity;
ISSN: 1076-2787
1099-0526 (eISSN)
DOI: 10.1155/2018/2895658
Efnisorð: Stærðfræði; Líkindareikningur
URI: https://hdl.handle.net/20.500.11815/845

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Tilvitnun:

Gudmundsson, S., & Hafstein, S. (2018). Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions. Complexity, 2018, 9. doi:10.1155/2018/2895658

Útdráttur:

We study stability for dynamical systems specifed by autonomous stochastic diferential equations of the form dX(t) = f(X(t))dt + g(X(t))dW(t), with (X(t))t≥0 an Rd -valued Ito process and ˆ (W(t))t≥0 an RQ-valued Wiener process, and the functions f : Rd → Rd and g : Rd → Rd×Q are Lipschitz and vanish at the origin, making it an equilibrium for the system. Te concept of asymptotic stability in probability of the null solution is well known and implies that solutions started arbitrarily close to the origin remain close and converge to it. Te concept therefore pertains exclusively to system properties local to the origin. We wish to address the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can they then be started? To this end we defne a probabilistic version of the basin of attraction, the y-BOA, with the property that any solution started within it stays close and converges to the origin with probability at least y. We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on y-BOA.

Leyfi:

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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