Calculus of variations and optimal control for generalized functions

TL;DR

This paper extends calculus of variations and optimal control to generalized functions within the category of generalized smooth functions, including distributions, and proves key theorems like Euler-Lagrange, Noether's theorem, and Pontryagin's principle.

Contribution

It introduces a framework for higher order calculus of variations and optimal control for generalized functions, including new proofs of classical theorems in this broader context.

Findings

Derived higher order Euler-Lagrange equations for generalized functions

Proved a form of the Pontryagin maximum principle for generalized control systems

Analyzed specific systems like the singular pendulum and Pais-Uhlenbeck oscillator

Abstract

We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler-Lagrange equations, the D'Alembert principle in differential form, the du Bois-Reymond optimality condition and the Noether's theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noether's theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais-Uhlenbeck oscillator with singular frequencies.