Invariant submanifolds for affine control systems

TL;DR

This paper introduces an algorithmic method for constructing invariant submanifolds in affine control systems by reducing Pfaffian systems and identifying integrals, assuming constant rank conditions.

Contribution

It provides a systematic approach to find invariant submanifolds using reduction of Pfaffian systems and integrals, advancing control theory techniques.

Findings

Algorithm for constructing invariant submanifolds under affine control systems.

Use of Pfaffian system reduction to identify integrable subsystems.

Method applicable when the span of system vector fields has constant rank.

Abstract

Given an affine control system $\dot{\mathbf x} = f({\mathbf x}) + \sum_{j=1}^m g_j({\mathbf x}) u_j$ we present an algorithmic process of construction of submanifolds that are invariant under controls assuming that the linear span of $f, g_1, \ldots, g_m$ has constant rank. We use the method of reduction of Pfaffian systems to a largest integrable subsystem and finding the first integrals and the generalized first integrals for the vector fields.