Pontryagin Maximum Principle for Distributed-Order Fractional Systems

TL;DR

This paper extends the Pontryagin maximum principle to distributed-order fractional optimal control problems, including controls with pointwise constraints, and demonstrates its applicability through an example.

Contribution

It introduces a novel Pontryagin maximum principle for distributed-order fractional systems with constrained controls, advancing optimal control theory in this area.

Findings

Established continuity of solutions under control perturbations

Derived differentiability of state solutions with respect to controls

Proved the Pontryagin maximum principle for distributed-order fractional systems

Abstract

We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.