Ever since we formulated the general optimal control problem in Section 3.3, we have allowed the state to take arbitrary values in . However, in many situations of interest is not an adequate state space (just like is not always an adequate control space). Recall, for example, the pendulum dynamics (2.55) that we derived on page . The angle variable naturally lives on a circle. Combining it with the angular velocity , we obtain the state that evolves on a cylinder. State spaces of this kind are quite typical for mechanical systems.
Surfaces such as a sphere, a cylinder, or a torus are all examples of (differentiable) manifolds. The purpose of this section is to reformulate our optimal control problem and the maximum principle in the geometric language of manifolds. Besides a higher level of generality, casting the maximum principle in the framework of manifolds has another important benefit: it clarifies the intrinsic meaning of the adjoint vector (costate), thereby greatly elucidating the essence of the maximum principle even in the familiar setting of control problems in . We begin this task in Section 7.1.1 where we describe manifolds more precisely as surfaces defined by equality constraints and discuss some fundamental objects associated with manifolds. It is worth noting that scenarios in which a state space is characterized by inequality constraints, although equally significant, are not captured by the present set-up.