Cone-Valued Lyapunov Functions

Abstract

**Please note that the full text is embargoed** ABSTRACT: It is very well known that employing a single

Lyapunov function and the theory of scalar

differential inequality offers a useful

mechanism to study a variety of qualitative

problems of differential equations in a unified

way [10]. Nevertheless, when using this

powerful technique for concrete problems, the

main difficulty we face is the lack of general

method of constructing a Lyapunov function.

This naturally beads to the development of the

method of vector Lyapunov functions which

utilizes several Lyapunov-like functions and

the theory of vector differential inequalities

in a fruitful manner [5,8-12]. This method

offers a more flexible mechanism to discuss

qualitative properties of nonlinear systems.

Also, it provides an effective tool to

investigate the properties of large scale

interconnected dynamical and control systems

whose multivariability, composite structure,

multi-connection and the variety of the nature

of subsystems make the construction of a single

Lyapunov function much more difficult.

Moreover, several Lyapunov functions result in

a natural way in the study of such systems by

the decomposition and aggregation method

[1,3,5,6,13-15].