Geometric characterization of singular differential algebraic equations

Singular differential algebraic equations (differential inclusions) are considered from a geometric point of view. The differential algebraic equations are defined as implicitly written ordinary differential equations. They appear in a natural way in the modelling of physical systems, as sets of differential algebraic relations describing the constitution of the systems. The main problem considered is that of finding the solution space and the infinitesimal generator (the dynamic equation) for a given differential algebraic equation. This question is far from being trivial, due to the fact that the solutions of a general differential algebraic equation are restricted to a lower dimensional manifold embedded in the configuration space. In that case, the differential algebraic equation is said to be singular, or algebraic incomplete. It is proved that the solution space of the equation is the set union of all (maximal) invariant submanifolds of the configuration space. A general iterative procedure for selecting invariant submanifolds is proposed, which is effective for a very broad class of differential algebraic equations. The theorems are proved, in which the trans-versality conditions ensuring that the iterates end in the following step are given. Examples illustrating the considerations are given.