The computation of Lyapunov quantities is closely connected with the important in engineering mechanics question of dynamical system behavior near to boundary of the stability domain. Followed by the work of Bautin [1], one differs "safe" or "dangerous" boundaries, a slight shift of which implies a small (invertible) or noninvertible changes of system status, respectively. Such changes correspo...

In this paper, we describe in detail a method of computing Lyapunov exponents for a continuous-time dynamical system and extend the method to discrete maps. Using this method, a partial Lyapunov spectrum can be computed using fewer equations as compared to the computation of the full spectrum, there is no difficulty in evaluating degenerate Lyapunov spectra, the equations are straightforward to...

In this article, we consider chaotic behavior happened in nonsmooth dynamical systems. To quantify such a behavior, a computation of Lyapunov exponents for chaotic orbits of a given nonsmooth dynamical system is focused. The Lyapunov exponent is a very important concept in chaotic theory, because this quantity measures the sensitive dependence on initial conditions in dynamical systems. Therefo...

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton-Jacobi-Bellman partial differential inequality of 2nd order. We give local and global versions of the First and Second Lyapunov Theorems assuming the e...

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jac...

We consider linear matrix equations where the linear mapping is the sum of a standard Lyapunov operator and a positive operator. These equations play a role in the context of stochastic or bilinear control systems. To solve them efficiently one can fall back on known efficient methods developed for standard Lyapunov equations. In the present paper we describe a direct and an iterative method ba...

In this paper, Gain-Scheduling (GS) control via parameter dependent Lyapunov function for crane is synthesized. A purpose of cranes system is to lift and carry a load to assigned positions. Cranes are required to transport the load fast, accurately and to reduce oscillations. In previous research, a method to robust control of cranes to suppress effects by variation of the rope length have been...

Title of dissertation: LINEAR STABILITY ANALYSIS USING LYAPUNOV INVERSE ITERATION Minghao Wu, Doctor of Philosophy, 2012 Dissertation directed by: Professor Howard Elman Department of Computer Science Institute for Advanced Computer Studies In this dissertation, we develop robust and efficient methods for linear stability analysis of large-scale dynamical systems, with emphasis on the incompres...

The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems ẋ(t) = A0x(t) +A1x(t− τ) +B0u(t). We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unkn...