Time-optimal linear problems with controls discontinuous on a set of positive measure

Linear problems of optimal time to the origin of coordinates with constant coefficients and constant convex set giving geometric constraints on the control are considered. It is proved that if the dimension of the phase space is greater than two, then arbitrarily small perturbations of such problems can lead to the situation that any optimal control is a function discontinuous on a set of positive measure in the “perturbed” problem for all initial states in some neighborhood of a given initial state $x_0$.
Figures: 1.
Bibliography: 14 titles.