Geometric Aspects of the Theory
of Necessary Conditions
for Optimal Control
Necessary conditions for an optimum
in the classical calculus of variations and optimal control theory include
the Euler-Lagrange equation and the Pontryagin Maximum Principle--both
of which have versions with various degrees of nonsmoothness in addition
to the classical smooth versions--as well as various additional conditions
involving "high-order variations." All these conditions turn out
to have geometric significance when expressed in a coordinate-free way.
This is well known for the Euler-Lagrange equation, which can be rewritten
in Hamiltonian form and related to symplectic geometry, but it is not so
widely known for the optimal control conditions, which turn out to be related
to connections along curves, and to have coordinate-free formulations in
terms of Lie brackets. These formulations have important applications,
some of which will be discussed in the course. In particular, we
will focus on using the geometrically invariant formulation to derive properties
of the optimal arcs, such as bang-bang and regularity results. The
general ideas will be illustrated with examples such as the recent results
by Agrachev and Gauthier on subanalyticity of
Carnot-Caratheorody distances.
In addition, we will present an introduction to the approach based on flows
and generalized differentials, which unifies a number of smooth, nonsmooth
and high-order conditions into a single theory.