Skinner-Rusk Unified Formalism for Optimal Control Systems and Applications

TL;DR

This paper introduces a geometric Skinner-Rusk formalism for time-dependent optimal control problems, unifying Lagrangian and Hamiltonian approaches and extending to implicit and descriptor systems.

Contribution

It develops a unified geometric framework for optimal control, incorporating Pontryagin's Maximum Principle and handling implicit and algebraic systems.

Findings

Provides a geometric formulation of optimal control conditions

Extends the formalism to implicit and descriptor systems

Validates the approach for a broad class of control problems

Abstract

A geometric approach to time-dependent optimal control problems is proposed. This formulation is based on the Skinner and Rusk formalism for Lagrangian and Hamiltonian systems. The corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by Pontryagin's Maximum Principle, provided that the differentiability with respect to controls is assumed and the space of controls is open. Furthermore, our method is also valid for implicit optimal control systems and, in particular, for the so-called descriptor systems (optimal control problems including both differential and algebraic equations).