Chapter 1: Optimization Foundations and Variational Methods
From finite-dimensional decisions to variations of entire trajectories
Optimal control begins with a change in what is being chosen: the decision variable grows from a number, to a vector, and finally to an entire trajectory. This chapter follows that progression deliberately. It starts with finite-dimensional optimization, develops functionals and first variations, and then shows how moving or free endpoints generate transversality and natural boundary conditions. These ideas supply the variational language used throughout the rest of the book.