Optimal Regional Tracking Control of Time-Fractional Diffusion Systems

TL;DR

This paper develops an optimal regional tracking control method for time-fractional diffusion systems, providing explicit control expressions and addressing systems with partial controllability, with validation through a numerical example.

Contribution

It introduces a novel approach using Hilbert uniqueness method for optimal control of time-fractional diffusion systems, including cases with fractional order one.

Findings

Explicit control signal expressions derived

Method applicable to partially controllable systems

Numerical example confirms theoretical results

Abstract

In this paper, we aim to explore optimal regional trajectory tracking control problems of the anomalous subdiffusion processes governed by time-fractional diffusion systems under the Neumann boundary conditions. Using eigenvalue theory of the system operator and the semigroup theory, we explore the existence and some estimates of the mild solution to the considered system. An approach on finding solution to the optimal problem that minimizes the regional trajectory tracking error and the corresponding control cost over a finite space and time domain is then explored via the Hilbert uniqueness method (HUM). The obtained results not only can be directly used to investigate the systems that are not controllable on the whole domain, but also yield an explicit expression of the control signal in terms of the desired trajectory. Most importantly, it is worth noting that our results in this paper are still novel even for the special case when the order of fractional derivative is equal to one. Finally, we provide a numerical example to illustrate our theoretical results.