General formulation of an optimization problem. Calculation of variations

It means finding a permissible control that forces the system to follow a permissible trajectory that optimizes the performance index J.

In view of its close connection with mechanics, we're going to deal with the problem of optimization by calculating variations.

To do this, we'll need a number of results, not all of which we'll be able to establish.

Static optimization theory uses function theory: this is the classic theory of extremums.

The calculus of variations, which enables us to minimize (or maximize) J where an integral appears, uses the theory of functionals.