On the uniform boundedness of solutions of contingent equations

It is proved that under standard conditions (in particular the convexity of the velocity set), the boundedness of the individual solutions of a contingent equation implies the uniform boundedness of the solutions and consequently the boundedness of the reachable sets. This result is used to obtain an existence theorem in optimal control theory. By means of an example it is shown that the convexity condition cannot be omitted. Keywords: Contingent equations, existence theorems, closure theorems