Liapunov Theory Research Articles

Stability of large-scale systems with stable and unstable subsystems.

The purpose of this paper is to develop new methods for constructing vector Liapunov functions and broaden the application of Liapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. With minor technical adjustments, the same criterion can be used to determine connective asymptotic stability of large-scale systems subject to structural perturbations. By redefining the constraints imposed on the interconnections among the subsystems, the considered class of systems is broadened in an essential way to include composite systems with unstable subsystems. In this way, the theory is brought substantially closer to reality since stability of all subsystems is no longer a necessary assumption in establishing stability of the overall composite system.

A remark on a stability theorem of M. Marachkoff

By placing certain conditions on $f(t,x)$ for the system of ordinary differential equations (1) \[ x’ = f(t,x),\;f(t,0) \equiv 0,\] Marachkoff weakened the conditions on the Liapunov function of the classical asymptotic stability theorem of Liapunov theory and obtained asymptotic stability of the zero solution of (1). Later, Massera gave a shorter proof of Marachkoff’s result. In this note we show that Marachkoff’s theorem can be proven without the use of one of the conditions.

Application of Liapunov’s direct method to fixed point theorems

The direct method of Liapunov is applied to the existence of fixed points for multivalued functions. Many recent fixed point theorems are shown to be special cases of the Liapunov theory.

Application of Liapunov theory to boundary value problems

The theory of Liapunov's direct method is developed for bouindary value problems occurring in ordinary difierential equations. Conditions are given in terms of a Liaptunov function which are sufficient to insure uiniqueness and existence of solutions of boundary value problems. A suitable Liapunov f unction is ob- tained to give conditions obtainied by Hat tman as special cases. 1. IntroducUion. Many techniques and theories developed for boundary value problems of ordinary differential equations originated

Stochastic stability in nonlinear oscillating systems

This paper gives an extension of the Liapunov theory to the stability study of third- and higher-order nonlinear systems with stochastic coefficients. A Liapunov function V( x) is first obtained, similar to the deterministic case. Instead of V  ( x) = d V d t , introduce the stochastic analog ÃV( x), where à denotes the differential generator. Two third-order and four fourth-order examples are given. One fifth-order and one sixth-order nonlinear system are also included. Stability conditions are stated in each of the examples.

Frequency domain stability criteria--Part II

The objective of this paper is to illustrate the limitations of the generalized Popov Theorem in establishing the stability of a loop containing a single nonlinearity, and to use the Liapunov Theory to give a new frequency domain stability criterion for such systems. The new criterion differs from Popov's in that less restrictive assumptions are made on the linear part, and stronger assumptions are made on the nonlinearity. In this paper, it is assumed that the nonlinearity is monotone increasing. The approach used here is quite general, however, and in a companion paper various other restrictions are considered.

New theorems and examples in the Liapunov theory of stochastic stability

Liapunov theory of stochastic stability, discussing new theorems and examples for noise of white Gaussian type