 
Summary: SYMPLECTIC METHODS FOR OPTIMIZATION AND CONTROL
A. Agrachev and R. Gamkrelidze
Contents
§1. Introduction
§2. Lagrange multipliers and Lagrangian submanifolds
§3. The problem of Optimal control
§4. Geometry of Lagrange Grassmannians
§5. The index of the second variation and the Maslov index
§6. Bangbang extremals
§7. Jacobi curves
§1. Introduction
1. The language of Symplectic geometry is successfully employed in many branches
of contemporary mathematics, but it is worth to remind that the original develop
ment of Symplectic geometry was greatly influenced by variational problems. In
Optimal control crucial role was plaid by the Hamiltonian system of Pontryagin's
Maximum principle, which itself is the object of Symplectic geometry. In further
development of Optimal control priorities were given to Convex analysis. Though
Convex and Functional analysis are very helpful in developing the general theory
of Extremal problems, they are not at all effective for investigating essentially non
linear problems in higher approximations, when the convex approximation fails.
