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In this paper, we study connections between the estimation entropy of a switched linear system and its Lyapunov exponents. We prove lower and upper bounds for the estimation entropy in terms of the Lyapunov exponents and show that, under the so-called regularity assumption, those bounds coincide. To do that, we use a geometric object called Oseledets' fltration of the system. Further, we show how to use the exponents and the Oseledets' fltration to design a quantization scheme for state estimation of switched linear systems. Then, we prove that we can make this algorithm work at an average data-rate arbitrarily close to the upper bound we provided for the estimation entropy of the given system. Furthermore, we can choose the average data-rate to be arbitrarily close to the estimation entropy whenever the switched linear system is regular. We show that, under the regularity assumption, the quantization scheme is completely causal in the sense that it depends only on information that is available up to the current time instant. We show that regularity is a natural property of many practical systems, such as Markov jump linear systems, and give sufcient conditions for it. [ABSTRACT FROM AUTHOR]

This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results, together with universal formulae for feedback stabilization of nonlinear systems, constitute our primary tools for control design. [ABSTRACT FROM AUTHOR]

This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple Lyapunov functions. For switched systems perturbed by white noise, we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones. In particular, we study stability of deterministic and stochastic switched systems under average dwell-time switching. [ABSTRACT FROM AUTHOR]

It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference. [ABSTRACT FROM AUTHOR]

We study spectral properties of certain families of linear second-order differential operators arising from linear stochastic differential equations. We construct a basis in the Hilbert space of square-integrable functions using modified Hermite polynomials, and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed. In particular, we completely describe the spectrum of the Fokker--Planck operator on an appropriate invariant subspace of rapidly decaying functions. The eigenvalues of the Fokker--Planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state, which is important for stochastic estimation and control applications. We show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator Lie algebra, which leads to a simple expression for the flow of their eigenfunctions. [ABSTRACT FROM AUTHOR]