Approximation of Distributions by Integrable Systems

Willem Sluis, Control and Dynamical Systems, California Institute of Technology

In the geometric theory of nonlinear control systems, the notion of a distribution and the dual notion of codistribution play a central role. Many results in nonlinear control theory require certain distributions to be integrable. Distributions (and codistributions) are not generically integrable and, moreover, the integrability property is not likely to persist under small perturbations of the system.

Therefore, it is natural to consider the problem of approximating a given codistribution by an integrable codistribution, and to determine to what extent such an approximation may be used for obtaining approximate solutions to various problems in control theory.

In this talk, I concentrate on the mathematical problem of approximating a given codistribution by an integrable codistribution and present an algorithm for approximating an $m$-dimensional nonintegrable codistribution on $\R^n$ by an integrable one using a homotopy approach. The method yields an approximating codistribution that agrees with the original codistribution on an $m$-dimensional submanifold of $\R^n$.