# A.A. Agrachev, R.V. Gamkrelidze, Symplectic geometry and necessary conditions for optimality

Title: Symplectic geometry and necessary conditions for optimality
Authors: A.A. Agrachev, R.V. Gamkrelidze
Journal title: Mathematics of the USSR-Sbornik
Year: 1992
Issue: 1
Volume: 182
Pages: 29-45
Citation:

A. A. Agrachev, R. V. Gamkrelidze, “Symplectic geometry and conditions necessary conditions for optimality”, Mathematics of the USSR-Sbornik, 1992, 72:1, 29–45

Abstract:

With the help of a sympletic technique the concept of a field of extremals in the classical calculus of variations is generalized to optimal control problems. This enables usto get new optimality conditions that are equally suitable for regular, bang-bang, and singular extremals. Special attention is given to systems of the form $\dot{x} = f_0 (x) + u f_1 (x)$ with a scalar control. New pointwise conditions for optimality and sufficient conditions for local controllability are obtained as a consequence of the general theory.