

Author: Bhat Sanjay
Publisher: Springer Publishing Company
ISSN: 0932-4194
Source: MCSS Mathematics of Control, Signals and Systems, Vol.22, Iss.2, 2010-10, pp. : 155-184
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
Abstract
In this paper, fundamental relationships are established between convergence of solutions, stability of equilibria, and arc length of orbits. More specifically, it is shown that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium. Next, arc-length-based Lyapunov tests are derived for convergence and stability. These tests do not require the Lyapunov function to be positive definite. Instead, these results involve an inequality relating the right-hand side of the differential equation and the Lyapunov function derivative. This inequality makes it possible to deduce properties of the arc length function and thus leads to sufficient conditions for convergence and stability. Finally, it is shown that the converses of all the main results hold under additional assumptions. Examples are included to illustrate how our results are particularly suited for analyzing stability of systems having a continuum of equilibria.
Related content


Lyapunov Stability and Orbital Stability of Dynamical Systems
Differential Equations, Vol. 40, Iss. 8, 2004-08 ,pp. :


Lyapunov Stability of Measure Driven Impulsive Systems
Differential Equations, Vol. 40, Iss. 8, 2004-08 ,pp. :


On Stability of Boundary Equilibria in Systems with Cosymmetry
By Kurakin L.G.
Siberian Mathematical Journal, Vol. 42, Iss. 6, 2001-11 ,pp. :


l ~ -based stability criteria and its applications on FLC systems
Fuzzy Sets and Systems, Vol. 139, Iss. 1, 2003-10 ,pp. :