Three Spaces Problem for Lyapunov Theorem on Vector Measure

It is proved that a Banach space X has the Lyapunov property if its subspace Y and the quotient space X/Y have it.

Comments on "finite-time stability theorem of stochastic nonlinear systems" [Automatica 46 (2010) 2105-2108]

In this comment, we will point out some errors existing in Chen and Jiao (2010) from definitions to the proof of the main result, where the authors discussed the finite-time stability of stochastic nonlinear systems and proved a Lyapunov theorem on the finitetime stability.

A finite-time stability theorem of stochastic nonlinear systems and stabilization designs

This paper focuses on the finite-time stability and stabilization designs of stochastic nonlinear systems. We first present and discuss a definition on the finite-time stability in probability of stochastic nonlinear systems, then we introduce a stochastic Lyapunov theorem on the finite-time stability, which has been established by Yin et al. We also employ this theorem to design a continuous state feedback controller that makes a class of stochastic nonlinear systems to be stable in finite time. An example and a simulation are given to illustrate the theoretical analysis.

Wide-area control of aggregated power systems

This paper proposes a new method for stabilizing disturbed power systems using wide area measurement and FACTS devices. The approach focuses on both first swing and damping stability of power systems following large disturbances. A two step control algorithm based on Lyapunov Theorem is proposed to be applied on the controllers to improve the power systems stability. The proposed approach is simulated on two test systems and the results show significant improvement in the first swing and damping stability of the test systems.

Continuous finite-time state feedback stabilizers for some nonlinear stochastic systems

This paper is concerned with the problem of finite-time stabilization for some nonlinear stochastic systems. Based on the stochastic Lyapunov theorem on finite-time stability that has been established by the authors in the paper, it is proven that Euler-type stochastic nonlinear systems can be finite-time stabilized via a family of continuous feedback controllers. Using the technique of adding a power integrator, a continuous, global state feedback controller is constructed to stabilize in finite time a large class of two-dimensional lower-triangular stochastic nonlinear systems. Also, for a class of three-dimensional lower-triangular stochastic nonlinear systems, a recursive design scheme of finite-time stabilization is given by developing the technique of adding a power integrator and constructing a continuous feedback controller. Finally, a simulation example is given to illustrate the theoretical results. © 2014 John Wiley & Sons, Ltd.

Some properties of finite-time stable stochastic nonlinear systems

In this paper we study some properties of finite-time stable stochastic nonlinear systems. We begin by showing several continuous dependence theorems of solutions on initial values under some conditions on the coefficients of stochastic systems. We then derive some regular properties of its stochastic settling time for a finite-time stable stochastic nonlinear system. We show continuity, positive definiteness and boundedness of the expected stochastic settling time under appropriate conditions. Finally, a Lyapunov function is constructed by making use of the expectation of the stochastic settling time, and the infinitesimal generator of the stochastic system defined on the Lyapunov function is also given, and hence resulting in a converse Lyapunov theorem of finite-time stochastic stability.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

On the Lyapunov transformation for stable matrices

The matrices studied here are positive stable (or briefly stable). These are matrices, real or complex, whose eigenvalues have positive real parts. A theorem of Lyapunov states that A is stable if and only if there exists H ˃ 0 such that AH + HA* = I. Let A be a stable matrix. Three aspects of the Lyapunov transformation L_A :H → AH + HA* are discussed.

1. Let C₁ (A) = {AH + HA* :H ≥ 0} and C₂ (A) = {H: AH+HA* ≥ 0}. The problems of determining the cones C₁(A) and C₂(A) are still unsolved. Using solvability theory for linear equations over cones it is proved that C₁(A) is the polar of C₂(A*), and it is also shown that C₁ (A) = C₁(A^-1). The inertia assumed by matrices in C₁(A) is characterized.

2. The index of dissipation of A was defined to be the maximum number of equal eigenvalues of H, where H runs through all matrices in the interior of C₂(A). Upper and lower bounds, as well as some properties of this index, are given.

3. We consider the minimal eigenvalue of the Lyapunov transform AH+HA*, where H varies over the set of all positive semi-definite matrices whose largest eigenvalue is less than or equal to one. Denote it by ψ(A). It is proved that if A is Hermitian and has eigenvalues μ₁ ≥ μ₂…≥ μ_n ˃ 0, then ψ(A) = -(μ₁-μ_n)²/(4(μ₁ + μ_n)). The value of ψ(A) is also determined in case A is a normal, stable matrix. Then ψ(A) can be expressed in terms of at most three of the eigenvalues of A. If A is an arbitrary stable matrix, then upper and lower bounds for ψ(A) are obtained.

A differential lyapunov framework for contraction analysis

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves. © 2013 IEEE.

Application of a new discreet form of Gauss’ theorem for measuring volume

Volume measurements are useful in many branches of science and medicine. They are usually accomplished by acquiring a sequence of cross sectional images through the object using an appropriate scanning modality, for example x-ray computed tomography (CT), magnetic resonance (MR) or ultrasound (US). In the cases of CT and MR, a dividing cubes algorithm can be used to describe the surface as a triangle mesh. However, such algorithms are not suitable for US data, especially when the image sequence is multiplanar (as it usually is). This problem may be overcome by manually tracing regions of interest (ROIs) on the registered multiplanar images and connecting the points into a triangular mesh. In this paper we describe and evaluate a new discreet form of Gauss’ theorem which enables the calculation of the volume of any enclosed surface described by a triangular mesh. The volume is calculated by summing the vector product of the centroid, area and normal of each surface triangle. The algorithm was tested on computer-generated objects, US-scanned balloons, livers and kidneys and CT-scanned clay rocks. The results, expressed as the mean percentage difference ± one standard deviation were 1.2 ± 2.3, 5.5 ± 4.7, 3.0 ± 3.2 and −1.2 ± 3.2% for balloons, livers, kidneys and rocks respectively. The results compare favourably with other volume estimation methods such as planimetry and tetrahedral decomposition.

Stabilization of parametric roll resonance with active u-tanks via Lyapunov control design

Parametric ship roll resonance is a phenomenon where a ship can rapidly develop high roll motion while sailing in longitudinal waves. This effect can be described mathematically by periodic changes of the parameters of the equations of motion, which lead to a bifurcation. In this paper, the control design of an active u-tank stabilizer is carried out using Lyapunov theory. A nonlinear backstepping controller is developed to provide global exponential stability of roll. An extension of commonly used u-tank models is presented to account for large roll angles, and the control design is tested via simulation on a high-fidelity model of a vessel under parametric roll resonance.

Exploring the fundamental theorem of arithmetic in excel 2007

This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

The validity of the Adler-Bardeen theorem in the supersymmetric U(1) gauge model

Using the dimensional reduction regularization scheme, we show that radiative corrections to the anomaly of the axial current, which is coupled to the gauge field, are absent in a supersymmetric U(1) gauge model for both 't Hooft-Veltman and Bardeen prescriptions for γ5. We also discuss the results with reference to conventional dimensional regularization. This result has significant implications with respect to the renormalizability of supersymmetric models.