JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, Vol. I, No. 3, 1995, 319-324 ON REGULARITY PROPERTIES OF EXTREMAL CONTROLS A.A. AGRACHEV ABSTRACT. We prove some regularity properties of the optimal con- trois for the smooth bracket generating systems with scalar control parameters, and show that the Ca~ltor sets cannot be the sets of switching points. Owing to papers by H. Sussmann we know some regularity properties of optimal controls for general real-analytic systems; see [2], [3]. The same author demonstrates in [2] that optimal controls for general C~176 do not possess any regularity properties. In this note, we show that the situation is not so hopeless for bracket generating systems and establish a curious property of the sets of switching points, which is also new for real-anMytic systems. We consider the control system = f(x) + ug(x), x E M, I~l _< 1, where M is a Coo-manifold and f, g are Coo vector fields on M. Let Lie{f, g) be a Lie sub-algebra of the vector fields generated by f, g, and L~ g) be an ideal in Lie{f, g} generated by g. Suppose that {v(z) :veL~ =T~M, VzeM. (1) Let u(t), t E ~, be a measurable bounded function. The point to E ~ is called a density point for u if there exists a derivative to for t = to. 1991 Mathematics Subject Classification. 49J30. Key words ar phrases. Optimal control, brarket generating system, switching point. Partially supported by the Russian Fund for Fundamental Research and grant No. 95- 01-00310a. 319 1079-2724/95/0700-0319507.50]0 ~ 1995 Plenum Publishing Corporation