Necessary and sufficient conditions for the existence of a Lyapunov function with ‘a quadratic form plus an integral term’

1996 ◽  
Vol 64 (4) ◽  
pp. 707-719 ◽  
Author(s):  
KAIQI XIONG
Keyword(s):  
Lyapunov Function ◽  
Quadratic Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Integral Term ◽  
Necessary And Sufficient

The quadratic form positive definite on a linear manifold

1951 ◽  
Vol 47 (1) ◽  
pp. 1-6 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Linear Manifold ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Real Vector Space ◽  
Real Vector ◽  
Necessary And Sufficient ◽  
Real Quadratic

1. Introduction. Necessary and sufficient conditions are established for a real quadratic form to be positive definite on a linear manifold, in a real vector space, explicit in terms of the dual Grassmann coordinates for the manifold.

Quadratic forms positive definite on a linear manifold

1952 ◽  
Vol 48 (1) ◽  
pp. 70-71
Author(s):  
L. S. Goddard
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Linear Manifold ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Real Quadratic

1. In a recent paper(1), Afriat has given necessary and sufficient conditions for a real quadratic form to be positive definite on a linear manifold, in terms of the dual Grassmannian coordinates of the manifold. Considerable matrix manipulations were used in Afriat's method, but most of these may be avoided by the method of the present paper, which depends on some well-known properties of the Grassmannian coordinates. We first show that the conditions may be expressed as a set of inequalities which are quadratic in the Grassmannian coordinates of the manifold. Then, by a standard theorem, these may be transformed into Afriat's conditions on the dual coordinates.

scholarly journals Solutions to Lyapunov stability problems: nonlinear systems with continuous motions

1994 ◽  
Vol 17 (3) ◽  
pp. 587-596 ◽  
Author(s):  
Ljubomir T. Grujic
Keyword(s):  
Lyapunov Function ◽  
Nonlinear Dynamical System ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Necessary And Sufficient Conditions ◽  
Nonlinear Dynamical ◽  
Bounded Set ◽  
Necessary And Sufficient ◽  
Bounded Sets ◽  
Selection Of

The necessary and sufficient conditions for accurate construction of a Lyapunov function and the necessary and sufficient conditions for a set to be the asymptotic stability domain are algorithmically solved for a nonlinear dynamical system with continuous motions. The conditions are established by utilizing properties of o-uniquely bounded sets, which are explained in the paper. They allow arbitrary selection of an o-uniquely bounded set to generate a Lyapunov function.Simple examples illustrate the theory and its applications.

Sign in / Sign up

Export Citation Format

Share Document