Some properties of vector field systems that are not altered by small perturbations

Abstract

In the study of systems of vector fields from the point of view of Control Theory, the properties of accessibility and complete controllability (defined in Section 2) are of fundamental importance. The purpose of this paper is to prove (a) that both properties are stable under small perturbations and (b) that the first one is generic. More precisely, let ,.V(M)B denote the set of all K-tuples of C’ vector fields on the Cr+l n-dimensional separable manifold M (1 n2 + n, and he later improved this to r >, 2n (cf. [5]). Stefan [7, Theorem 2.61 proves the same result for M compact, r > 1. (It is clear that [5, 71 do not improve the density part of the result of [4], but only the openess part.) The work of Lobry and Stefan proves that the set ,L, of c.L.s. systems is dense and contains an open set. We prove that ?Ak is open and dense (Theorems 5.3 and 7.1) and this trivially implies that ,.Lk is open and dense (Theorem 8.1). Thus, the main novelty of our work is the result that ,L, (and ?Ak) are actually open in the fine Cl topology. Since ,A, is a proper subset of Ak , our density theorem is slightly better than those of [4, 5, 71. In order to prove the openness result, it becomes necessary to isolate a