A Remarkable Property of the Dynamic Optimization Extremals

TL;DR

This paper reveals a generalized property of extremals in optimal control theory, extending the classical conservation of the Hamiltonian, and introduces methods to find conserved quantities and characterize problems with specific constants of motion.

Contribution

It generalizes the known property of Hamiltonian constancy in optimal control, providing new tools for identifying conserved quantities and characterizing problems with particular constants of motion.

Findings

Generalized the property of Hamiltonian derivatives along extremals.

Developed methods to find conserved quantities in optimal control.

Characterized problems with specified constants of motion.

Abstract

At the core of optimal control theory is the Pontryagin maximum principle - the celebrated first order necessary optimality condition - whose solutions are called extremals and which are obtained through a function called Hamiltonian, akin to the Lagrangian function used in ordinary calculus optimization problems. A remarkable property of the extremals is that the total derivative with respect to time of the corresponding Hamiltonian equals the partial derivative of the Hamiltonian with respect to time. In particular, when the Hamiltonian does not depend explicitly on time, the value of the Hamiltonian evaluated along the extremals turns out to be constant (a property that corresponds to energy conservation in classical mechanics). We present a generalization of the above property. As applications of the new relation, methods for obtaining conserved quantities along the Pontryagin extremals and for characterizing problems possessing given constants of the motion are obtained.