Riemann integrability of an optimal control in linear time-optimal problems

A linear time-optimal problem is considered. It is shown that the set of initial states for which a three-dimensional time-optimal problem at the origin, with constant coefficients, has no solution in the class of Riemann integrable controls can fill up a given ball to an arbitrary extent. For a large class of multidimensional systems it is shown that this set does not contain isolated points. Sufficient conditions for the existence of a Riemann integrable control are studied for problems with variable coefficients.
Bibliography: 8 titles.