Weak Pontryagin's Maximum Principle for Optimal Control Problems Involving a General Analytic Kernel

TL;DR

This paper develops a weak Pontryagin's maximum principle for optimal control problems involving fractional operators with a general analytic kernel, extending classical results to more general fractional dynamics.

Contribution

It introduces a duality relation, an integration by parts formula, and a weak maximum principle for fractional control problems with general kernels, including mixed integer and fractional orders.

Findings

Established a duality relation for fractional operators with general kernels

Proved a new Gronwall's inequality for fractional differential equations

Derived necessary optimality conditions for mixed integer and fractional order problems

Abstract

We prove a duality relation and an integration by parts formula for fractional operators with a general analytical kernel. Based on these basic results, we are able to prove a new Gronwall's inequality and continuity and differentiability of solutions of control differential equations. This allow us to obtain a weak version of Pontryagin's maximum principle. Moreover, our approach also allow us to consider mixed problems with both integer and fractional order operators and derive necessary optimality conditions for isoperimetric variational problems and other problems of the calculus of variations.