Distributed-Order Non-Local Optimal Control

TL;DR

This paper extends fractional optimal control theory to systems with distributed-order derivatives, establishing a version of Pontryagin's maximum principle and optimality conditions for such complex non-local models.

Contribution

It introduces a novel framework for distributed-order fractional control systems and proves a weak Pontryagin maximum principle along with sufficient optimality conditions.

Findings

Established a weak Pontryagin maximum principle for distributed-order systems.

Derived sufficient optimality conditions under convexity assumptions.

Extended classical fractional control theory to more general distributed-order derivatives.

Abstract

Distributed-order fractional non-local operators have been introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical systems constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin's maximum principle and a sufficient optimality condition under appropriate convexity assumptions.