$k$-symplectic Pontryagin's Maximum Principle for some families of PDEs

TL;DR

This paper introduces a novel $k$-symplectic geometric framework for optimal control problems involving certain PDEs, extending Pontryagin's Maximum Principle to this setting and connecting it with classical control theory.

Contribution

It is the first to apply $k$-symplectic formalism to PDE-based control problems and establishes a geometric formulation of Pontryagin's Maximum Principle in this context.

Findings

Developed a $k$-symplectic formalism for PDE control problems

Proved Pontryagin's Maximum Principle within this framework

Connected classical and PDE control theories through geometric methods

Abstract

An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature $k$-symplectic formalism is used to provide the optimal control problems associated to some families of partial differential equations with a geometric formulation. A parallel between the classic formalism of optimal control theory with ordinary differential equations and the one with particular families of partial differential equations is established. This description allows us to state and prove Pontryagin's Maximum Principle on $k$-symplectic formalism. We also consider the unified Skinner-Rusk formalism for optimal control problems governed by an implicit partial differential equation.