As in Section 3.3.2, we define an additional state variable, , to be the solution of
and arrive at the augmented system
and write the system (4.5) more compactly as
An optimal trajectory of the original system in corresponds in an obvious way to an optimal trajectory of the augmented system in . The first component of describes the evolution of the cost in the original problem, and is recovered from by projection onto parallel to the -axis. This situation is depicted in Figure 4.1 (note that is not necessarily positive, so need not actually be increasing along ). In this and all subsequent figures, the -axis will be vertical.
From now on, we let denote the terminal time of the optimal trajectory (or, what is the same, of ). The next exercise offers a geometric interpretation of optimality; it will not be directly used in the current proof, but we will see a related idea in Section 4.2.6.
In the particular case when , the claim in the exercise should be obvious: no other trajectory starting from some point on the optimal trajectory can hit the line at a point lower than . In other words, a final portion of the optimal trajectory must itself be optimal with respect to its starting point as the initial condition. This idea, known as the principle of optimality, is illustrated in Figure 4.2. (The reader will notice that we are using different axes in different figures.)
Another simple observation, which will be useful later, is that the Hamiltonian (4.2) can be equivalently represented as the following inner product in :