Toward a global theory of {f,g}-invariant distributions with singularities

Abstract

In this paper, I outline a program, initiated in independent and in joint efforts by A.J. Krener, A. Isidori, and myself, whose goal is to extend the linear (A,B)-invariant subspace theory to nonlinear problems beyond the existing local theory, including particularly those problems which are either globally formulated or which require an analysis of singular control systems. Perhaps most significant is that the analysis of certain of the local singular problems requires, in part, a global formulation of the local theory and for this reason I first present some global results. Next, it is shown how certain local singular problems may be recast as a nonsingular but now global problems by the method of resolution of singularities, as applied to real analytic foliations. The observation that this singular-global transition is applicable is perhaps the most novel point here, and its generality is currently under study ([3],[4],[5]), requiring both an analysis of the "resolution of singularities" of real analytic foliations, which would make significant contact with Hironaka's fundamental contribution [7], and an analysis of a categorical "leaf space" M/Δ for quite general foliations Δ of M [3]. As an illustration, in this paper, certain low-dimensional, singular problems are analyzed by a combination of Morse Theory, blowing-up, and global techniques from differential and algebraic topology. Indeed, as applications, new results on global disturbance decoupling in the presence of singularities are systematically presented for the first time. It is our expectation to have more to say about the general case in the future.

Research supported by U.S. Department of Energy under Contract No. DE-AC01-80RA50421 at Scientific Systems, Inc., Cambridge, Mass 02140. The author also wishes to acknowledge the support and hospitality of the Istituto d'Automatica, Universita Roma, Via Eudossiana, Roma, Italia.