An Extension of Lyapunov’s First Method to Nonlinear Systems With Non-Continuously Differentiable Vector Fields

Let$M$be an analytic connected 2-manifold with empty boundary, over the ground field$\mathbb{F}=\mathbb{R}$or$\mathbb{C}$. Let$Y$and$X$denote differentiable vector fields on$M$. We say that$Y$tracks$X$if$[Y,X]=fX$for some continuous function$f:\,M\rightarrow \mathbb{F}$. A subset$K$of the zero set$\mathsf{Z}(X)$is an essential block for$X$if it is non-empty, compact and open in$\mathsf{Z}(X)$, and the Poincaré–Hopf index$\mathsf{i}_{K}(X)$is non-zero. Let${\mathcal{G}}$be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field$X$. Let$K\subset \mathsf{Z}(X)$be an essential block. Assume that if$M$is complex and$\mathsf{i}_{K}(X)$is a positive even integer, no quotient of${\mathcal{G}}$is isomorphic to$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then${\mathcal{G}}$has a zero in$K$(main result). As a consequence, if$X$and$Y$are analytic,$X$is non-trivial, and$Y$tracks$X$, then every essential component of$\mathsf{Z}(X)$meets$\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

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A Theory of Index for Point Mapping Dynamical Systems

Journal of Applied Mechanics ◽

10.1115/1.3153601 ◽

1980 ◽

Vol 47 (1) ◽

pp. 185-190 ◽

Cited By ~ 8

Author(s):

C. S. Hsu

Keyword(s):

Dynamical Systems ◽

Nonlinear Systems ◽

Periodic Solutions ◽

Discrete Time ◽

Difference Equations ◽

Vector Fields ◽

Time Difference ◽

Point Mapping ◽

Strongly Nonlinear

Dynamical systems governed by discrete time-difference equations are referred to as point mapping dynamical systems in this paper. Based upon the Poincare´ theory of index for vector fields, a theory of index is established for point mapping dynamical systems. Besides its intrinsic theoretic value, the theory can be used to help search and locate periodic solutions of strongly nonlinear systems.

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Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields

Czechoslovak Mathematical Journal ◽

10.21136/cmj.1981.101777 ◽

1981 ◽

Vol 31 (4) ◽

pp. 614-632 ◽

Cited By ~ 1

Author(s):

Jean Mawhin

Keyword(s):

Vector Fields ◽

Differentiable Vector

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Asymptotic stability at infinity for differentiable vector fields of the plane

Journal of Differential Equations ◽

10.1016/j.jde.2006.07.025 ◽

2006 ◽

Vol 231 (1) ◽

pp. 165-181 ◽

Cited By ~ 5

Author(s):

Carlos Gutierrez ◽

Benito Pires ◽

Roland Rabanal

Keyword(s):

Asymptotic Stability ◽

Vector Fields ◽

Differentiable Vector

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Robust Observer Design for Lipschitz Nonlinear Systems With Parametric Uncertainty

Volume 3: Nonlinear Estimation and Control; Optimization and Optimal Control; Piezoelectric Actuation and Nanoscale Control; Robotics and Manipulators; Sensing; System Identification (Estimation for Automotive Applications, Modeling, Therapeutic Control in Bio-Systems); Variable Structure/Sliding-Mode Control; Vehicles and Human Robotics; Vehicle Dynamics and Control; Vehicle Path Planning and Collision Avoidance; Vibrational and Mechanical Systems; Wind Energy Systems and Control ◽

10.1115/dscc2013-4104 ◽

2013 ◽

Author(s):

Yan Wang ◽

David M. Bevly

Keyword(s):

Nonlinear Systems ◽

Vector Fields ◽

Observer Design ◽

Nonlinear Observer ◽

Performance Criteria ◽

Adaptation Algorithm ◽

Algebraic Set ◽

Observer Error ◽

Robust Observer ◽

Lipschitz Nonlinear Systems

This paper discusses optimal and robust observer design for the Lipschitz nonlinear systems. The stability analysis for the Lure problem is first reviewed. Then, a two-DOF nonlinear observer is proposed so that the observer error dynamic model can be transformed to an equivalent Lure system. In this framework, the difference of the nonlinear parts in the vector fields of the original system and observer is modeled as a nonlinear memoryless block that is covered by a multivariable sector condition or an equivalent semi-algebraic set defined by a quadratic polynomial inequality. Then, a sufficient condition for asymptotic stability of the observer error dynamics is formulated in terms of the feasibility of polynomial matrix inequalities (PMIs), which can be solved by Lasserre’s moment relaxation. Furthermore, various quadratic performance criteria, such as H2 and H∞, can be easily incorporated in this framework. Finally, a parameter adaptation algorithm is introduced to cope with the parameter uncertainty.

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