On properness of impulsive extension

This article investigates properness of an impulsive extension of the conventional optimal control problem. This includes both the correctness of the solution to impulsive control systems arising as result of an impulsive extension of ordinary differential systems, and existence theorems. Correctness in the classic Cauchy sense is proved. New approximation lemmas are introduced proved that guarantee the sensitivity to small perturbations in control variables. Filippov-type existence theorems are established.