Dynamic equations in descriptor form

This paper studies a general form of sets of equations that is often the product of problem formulation in large-scale systems, especially when the equations are expressed in terms of the natural describing variables of the system. Such equations represent a broad class of time-evolutionary phenomena, and include as special cases ordinary static equations of arbitrary dimension, ordinary state-space equations, combinations of static and dynamic equations, and noncausal systems. The main thrust of the paper is to show (for sets of linear equations) that familiar concepts of dynamic system theory can be extended to this more general class, although sometimes with significant modification. Two new (and essentially dual) concepts, that of solvable and conditionable sets of equations, are found to be fundamental to the study of equations of this form. The notion of initial conditions, although not directly related to a state, is used as a general solution method for equations of this type. In addition a set of necessary and sufficient conditions for a set of dynamic equations to contain an embedded state-space representation is derived.