Non-linear functions and stability of motions of implicit differential systems

Abstract

The paper deals with a refined qualitative analysis of the motion of a broad class of continuous, time-varying, non-linear, implicit differential systems. These systems consist of a finite number of first-order implicit differential equations that cannot be set into a normal form. Some novel concepts of stability, boundedness and asymptotic stability, convenient for descriptions of the motion behaviour of the systems, are introduced, discussed and analysed. These concepts employ some nonlinear functions which are not usually in use in stability theory. For the concepts introduced, the general sufficient conditions are derived in terms of the existence of suitable Lyapunov functions. Also, for a subclass of the implicit systems considered, the results developed are specialized to use directly some inherent properties of these systems. The results obtained generalize a number of known results in stability theory, particularly those on the stability of singular (semistate) systems.