A weak regularity theorem for real analytic optimal control problems (Q1099402)

The following question is addressed: Given a control system of the form \(x'=f(x)+ug(x)\), -1\(\leq u\leq 1\) on a real analytic manifold M, let us assume the existence of a measurable control function u(t) defined on the interval [0,T] which steers an initial point \(x_ 1\) to a final point \(x_ 2\). Does there exist another control v(t) defined on the interval [0,T] which steers \(x_ 1\) to \(x_ 2\) and belongs to some prescribed subclass \({\mathcal R}\) of control functions? It is proved by an example that the answer is no if f and g are \(C^{\infty}\) vector fields. However, if the system is real analytic and \({\mathcal R}\) consists of functions which are analytic on an open dense subset of [0,T], the answer is yes.