This paper concerns control systems of the form   for which C. OLECR [2] showed that each point attainable from x₀, can also be attained by a piecewise extremal control function if the control regions U are compact. This yiels the following continuity properties for “Bang-Rang” control functions:

1) Let system (1) be linear, i.e. f(t, u) = B(t) u and let the control region U satisfy cortain additional hypotheses, then all extremal control functions u(t) are piecewise continuous and the values of u(t) are extrema! points of the convex hull of U at their points of continuity.

2) In the nonlinear case, all sets   excluding a finite set of times t_k may be strongiy convex. Then for ail extrernai functions u(t) the functions   (X(t)is the fundamental matrix of the homogeneous system xΔ = A (t) x) are piecewise continuous and they may be discontinuous only for t=t_k.