Optimal Control Problems with Time Delays: Constancy of the Hamiltonian

Abstract

This paper concerns necessary conditions of optimality for optimal control problems with time delays in the state variable. It is well known that, when there are no time delays and the dynamics are autonomous, the standard necessary conditions in the form of a maximum principle can be supplemented by an extra condition, namely “constancy of the Hamiltonian" along optimal trajectories (and associated costate trajectories). This property, possibly supplemented by other invariance principles, has been used to investigate properties of optimal trajectories, such as solution regularity, without the need to solve the underlying extremal equations. In classical mechanics, for example, the constancy of the Hamiltonian condition can be used to derive a conservation of energy principle from Hamilton's principle of least action. While the maximum principle has been generalized to cover time delays, the validity of constancy of the Hamiltonian-type conditions has not been previously investigated. We provide the first “extra" optimality condition of this nature for autonomous, time delay optimal control problems. The new “constancy of the Hamiltonian" condition involves a correction term, without which the condition is not valid. We illustrate the significance of this condition by applications to minimizer regularity and conservation laws in nonclassical Hamiltonian mechanics.