Structural Analysis for Differential-Algebraic Systems: Complexity, Formulation and Facets

In this paper we consider the structural analysis problem for differential-algebraic systems with conditional equations. This consists, given a conditional differential algebraic system, in verifying if the system is well-constrained for every state and if not in finding a state for which the system is bad-constrained. We first show that the problem reduces to the perfect matching free subgraph problem in a bipartite graph. We then show the NP-completeness of this problem and give a formulation as an integer linear program. We also discuss the polytope of the solutions of this problem and propose a Branch-and-Cut algorithm.