Given a nonempty closed set D М Rn, a point x О D and direction y О Rn, find the intersection points x– and x+ of the line x + ly and the boundary of D or report on the absence of such intersections. This procedure is referred to as a boundary oracle for the set D it plays a key role in the modern randomized methods in optimization and control. We present the construction of boundary oracles for some matrix sets arising in typical control problems along with their use in the randomized methods.