The stability of a motion with respect to a set
Abstract
Consideration of the problem of the stability and asymptotic stability of a motion with respect to a set (generally arbitrary) of points in a real ndimensional Euclidean space. Following the introduction of the concept of definiteness of a function with respect to a set, the applicability of Liapunov's (1950) second method to the solution of this problem is demonstrated. Theorems are then proven which correspond to the classical theorems of Liapunov and Chetaev (1955) concerning the stability, asymptotic stability, and instability of a motion with respect to an isolated point, but, as generalizations of these theorems, are valid with respect to an entire set of points. The use of the proposed theorem concerning the stability of a motion with respect to a set of points is illustrated by an example dealing with the motion of a heavy gyrostat about a fixed point. Two theorems concerning instability of a motion with respect to a set of points are applied to an example involving the motion of a heavy sphere rolling without skidding on a rough horizontal plane.
 Publication:

Zeitschrift Angewandte Mathematik und Mechanik
 Pub Date:
 December 1974
 DOI:
 10.1002/zamm.19740541106
 Bibcode:
 1974ZaMM...54..789L
 Keywords:

 Equilibrium Equations;
 Fixed Points (Mathematics);
 Gyroscopes;
 Motion Stability;
 Nonholonomic Equations;
 Set Theory;
 Euclidean Geometry;
 Hyperspaces;
 Liapunov Functions;
 Skidding;
 Spheres;
 Physics (General)