On the Nature and Stability of Differential-Algebraic Systems

Nonlinear dynamical systems described by a class of higher index differential-algebraic equations (DAE) are considered. A quantitative and qualitative analysis of their nature and of the stability properties of their solution is presented. Using tools from geometric control theory, higher index differential-algebraic systems are shown to be inherently unstable about their solution manifold. A qualitative geometric interpretation is given, and the consequences of this instability are discussed, in particular as they relate to the numerical solution of these systems. The paper concludes with ideas and directions for future research.