The most widely used tool for the solution of optimal control problems is the Pontryagin
Maximum Principle. But the Maximum Principle is, in general, only a necessary
condition for optimality. It is therefore desirable to have supplementary conditions, for
example second order sufficient conditions, which confirm optimality (at least locally) of
an extremal arc, meaning one that satisfies the Maximum Principle.
Standard second order sufficient conditions for optimality, when they apply, yield the
information not only that the extremal is locally minimizing, but that it is also locally
unique. There are problems of interest, however, where minimizers are not locally unique,
owing to the fact that the cost is invariant under small perturbations of the extremal of
a particular structure (translations, rotations or time-shifting). For such problems the
standard second order conditions can never apply.
The first contribution of this thesis is to develop new second order conditions for
optimality of extremals which are applicable in some cases of interest when minimizers
are not locally unique. The new conditions can, for example, be applied to problems with
periodic boundary conditions when the cost is invariant under time translations.
The second order conditions investigated here apply to normal extremals. These
extremals satisfy the conditions of the Maximum Principle in normal form (with the
cost multiplier taken to be 1). It is, therefore, of interest to know when the Maximum
Principle applies in normal form. This issue is also addressed in this thesis, for optimal
control problems that can be expressed as calculus of variations problems. Normality of the
Maximum Principle follows from the fact that, under the regularity conditions developed,
the highest time derivative of an extremal arc is essentially bounded.
The thesis concludes with a brief account of possible future research directions.