Remarks on the Preservation of Various Controllability Properties under Sampling

This note studies the preservation of controllability (and other properties) under sampling of a nonlinear system. More detailed results are obtained in the cases of analytic systems and of systems with finite dimensional Lie algebras. 1. Preliminaries. When a system is regulated by a digital computer, control decisions are often restricted to be taken at fixed times 0,λ,2λ,...; one calls λ>0 the sampling time. For a (continuous time) plant, the resulting situation can be modelled through the constraint that the inputs applied be constant on intervals of length λ. It is thus of interest to characterize the various controllability properties when the controls are so restricted. This problem motivated the results in [KHN], which studied the case of linear systems; more recent references are [BL], [GH]. For nonlinear systems, it appears that the problem has not been studied systematically. The recent paper [SS] began such a study, using tools of geometric control theory. We continue that study here. Reasons of space prevent us from repeating the material in [SS], which will be needed in a few places. The definitions and statements of results, however, will be self-contained. The systems σ to be considered are those described by differential equations (1.1) (dx/dt)(t) = f(x(t),u(t)), x(t)∈M, u(t)∈U, where M is a smooth (Hausdorff, second countable) n-dimensional manifold, U is a subset (see below) of a smooth manifold P, f:M×P→TM is smooth, and Xu:=f(⋅,u) is a complete vector field on M for each u. "Smooth" means either infinitely differentiable or analytic; in the latter case, σ is an analytic system. The control set U can be very general; we shall only assume that V:= int(U) is connected and that the following (local) condition is satisfied at each u∈U: there exists a smooth path g:[0,1]→P with g(0)=u, g([0,1])⊆U, and g(t)∈V for almost all t. ("Smooth" meaning: defined and smooth in a ngbd of [0,1].) In particular, then, U must be path connected and it must satisfy U⊆clos(V); the former is the essential property for most results. *Research suported in part by US Air Force Grant 80-0196