Morse families in optimal control problems

TL;DR

This paper introduces a geometric framework using Morse families to analyze optimal control problems, enabling the derivation of necessary conditions, integrability algorithms, and handling of discontinuous controls within the Hamiltonian and Lagrangian formalisms.

Contribution

It develops a novel geometric approach using Morse families for optimal control, extending classical conditions and algorithms to include discontinuous controls and singular cases.

Findings

Derived classical first order necessary conditions for optimality.

Adapted integrability algorithm to include discontinuous control trajectories.

Analyzed singular and overdetermined optimal control problems.

Abstract

We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained. From the Hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the Lagrangian formalism by means of a proper Lagrangian submanifold. Singular optimal control problems and overdetermined ones are also studied along the paper.