A Short Survey on Measure-Driven Optimal Control Problems

In the middle of the last century, space exploration raised the need to solve rocket space navigation and control problems in which the jumps of state variables naturally arose. Then, alongside the development of control theory, the discontinuous arcs-solutions to calculus of variations problems began to attract increasing attention from researchers. Gradually, they started to be considered and investigated within the framework of a more general theory which is known today as the theory of optimal impulsive control. This article contains a short survey on the extension approach to the impulsive control theory by basing on the concept of measure-driven control differential system. The meaning of extension and of well-posed extension is defined and discussed. Various types of extensions are considered starting from the simplest linear case and finishing with a general nonlinear situation.