Super-duality and necessary optimality conditions of order "infinity" in optimal control theory

TL;DR

This paper introduces a novel approach to analyze and solve nonlinear optimal control problems using increment representations inspired by classical calculus, leading to necessary conditions of arbitrary and infinite order, with applications to nonlocal continuity equations.

Contribution

The paper develops a new framework for deriving high-order and infinite-order necessary optimality conditions in optimal control, extending classical methods to nonlinear and metric space models.

Findings

Derived necessary optimality conditions of arbitrary order.

Formulated infinite-order optimality conditions with feedback mechanisms.

Applied the framework to control problems involving nonlocal continuity equations.

Abstract

We systematically introduce an approach to the analysis and (numerical) solution of a broad class of nonlinear unconstrained optimal control problems, involving ordinary and distributed systems. Our approach relies on exact representations of the increments of the objective functional, drawing inspiration from the classical Weierstrass formula in Calculus of Variations. While such representations are straightforward to devise for state-linear problems (in vector spaces), they can also be extended to nonlinear models (in metric spaces) by immersing them into suitable linear "super-structures". We demonstrate that these increment formulas lead to necessary optimality conditions of an arbitrary order. Moreover, they enable to formulate optimality conditions of "infinite order", incorporating a kind of feedback mechanism. As a central result, we rigorously apply this general technique to the optimal control of nonlocal continuity equations in the space of probability measures.