3.4.5 Critique of the variational approach and preview of the maximum principle

The variational approach presented in Sections 3.4.1-3.4.3 has led us, quite quickly, to the necessary conditions for optimality expressed by the canonical equations and the Hamiltonian maximization property (we did not actually prove the latter property, but we will see that it is indeed correct). While it helps us build intuition for what the correct statement of the maximum principle should look like, the variational approach has several limitations which, upon closer inspection, turn out to be quite severe.

CONTROL SET. Recall that our starting point was to consider perturbed controls of the form (3.24). Such perturbations make sense when the values of are interior points of the control set . This may not be the case, though, if has a boundary, and bounded (or even finite) control sets are common in control applications. As we will see in the next chapter, the statement that the function must have a maximum at is true even in such situations and, moreover, this maximum is in fact global. However, we cannot hope to establish this fact using the variational approach, because need not be 0 when the maximum is achieved at a boundary point of .

FINAL STATE. In the preceding we treated the case when the final state is free, but we know (see Section 3.3.3) that in general we may have some target set . Consider, for example, the case of a fixed endpoint: . Then the control perturbation is no longer arbitrary, since the resulting state perturbation must satisfy . In view of the fact that and are related by the system (3.28) with , it is easy to show that admissible perturbations must satisfy the constraint

where is the transition matrix for in (3.28). The second equation in (3.38) needs to hold only for admissible perturbations, and not for all . This condition is no longer strong enough to let us conclude that . We see that the prospects of extending the variational approach beyond free-endpoint problems do not look very promising.

DIFFERENTIABILITY. When developing the first
variation, we were tacitly
assuming that
is differentiable with respect to
(as well as
). Since the Hamiltonian
is defined via (3.29), both
and
must thus
be differentiable with respect to
. The reader can readily check that
differentiability of
with respect to
was *not*
one of the assumptions we made in Section 3.3.1 to ensure
existence
and uniqueness of solutions for our control system. In other words,
the variational approach requires extra regularity
assumptions to be imposed on the system.
Having to assume differentiability of
with respect to
is also undesirable, as
it rules out otherwise quite reasonable cost functionals like
Furthermore, the analysis based on the second variation--which
is needed to distinguish between a minimum and a
maximum--involves second-order partial derivatives
of
with respect to
. It is clear that the variational
approach would take us on a path of overly restrictive regularity assumptions.
Instead, we would like
to establish the Hamiltonian maximization property more directly, not by
working with derivatives.

CONTROL PERTURBATIONS. When considering the control and state perturbations as in (3.24) and (3.25) with near 0, we are allowing only small deviations in both and . For the system , this would correspond exactly to the notion of a weak minimum from calculus of variations. However, as we already discussed as early as Section 2.2.1 (see in particular Example 2.1), we would like to have a larger family of control perturbations. More precisely, we want to capture optimality with respect to control perturbations that may be large, as long as the corresponding state trajectories are close to the given one. For example, Figure 3.6 illustrates a particular control perturbation (for controls that switch between only two values) which is very reasonable but falls outside the scope of the variational approach. As we will soon see, working with a richer perturbation family is crucial for obtaining sharper necessary conditions for optimality.

In summary, while the basic form of the necessary conditions provided by the maximum principle will be similar to what we obtained using the variational approach, several shortcomings of the variational approach must be overcome in order to obtain a more satisfactory result. Specifically, we need to accommodate constraints on the control set, constraints on the final state, and weaker differentiability assumptions. A less restrictive notion of ``closeness" of controls will be the key to achieving these goals. Borrowing a colorful expression from [PB94], we can describe the task ahead of us as ``the cutting of the umbilical cord between the calculus of variations and optimal control theory." The maximum principle is a very nontrivial extension of the variational approach, and was developed many years later. The proof of the maximum principle is quite different from the argument given in this section; in particular, it is much more geometric in nature. We are now ready, in terms of both technical preparation and conceptual motivation, to tackle this proof in the next chapter.