Local Solvability of Degenerate, Overdetermined First-Order Partial-Differential Equations - A Control Theoretic Perspective

Abstract

In this paper we address the question of the existence of classical solutions to an overdetermined system of degenerate, first order partial differential equations. The adjective, degenerate, means that the vector fields, corresponding to the differential operators in these equations, are allowed to become linearly dependent at certain points of the ambient manifold. Thus these equations are, in a restricted sense, analogues of differential equations with singularities. The method of investigation is a combination of tools from nonlinear control theory involving invariant manifolds and foliations (with singularities) and ideas akin to the method of characteristics. The motivation for the study of this question arises from problems in nonlinear control theory. We provide some sufficient conditions for an affirmative answer to the the aforementioned question using techniques which are themselves inspired from the methods of control theory.