1. Context

A control system is a dynamic system that can be manipulated by means of a command. Optimal control theory analyzes the properties of the system in order to bring it from an initial state to a desired final state, possibly respecting certain criteria. The systems studied cover a wide scientific spectrum, from mechanics and electronics to biology, chemistry and economics. In general, there are a multitude of ways to bring a system from its initial state to its final state. It can sometimes be interesting to choose among all these dynamics the one that will minimize or maximize a certain optimization criterion. Consider, for example, a car whose controls are the brake and gas pedal pedals and the steering wheel. To get from city A to city B, we may, for this controlled system, want to minimize travel time, fuel consumption or travel distance. These three cases are examples of costs that can...