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.