Control theory for nonlinear fractional dispersive systems

TL;DR

This paper develops control theory methods for nonlinear systems governed by fractional differential equations, extending classical control concepts to fractional calculus and establishing controllability results.

Contribution

It introduces a fixed point theorem approach for nonlinear fractional systems and reestablishes classical control theorems in the fractional setting.

Findings

Existence of controls for nonlinear fractional systems proven.

Representation of solutions via Gramian matrices extended to fractional systems.

Controllability and observability equivalence established in fractional context.

Abstract

We consider a terminal control problem for processes governed by a nonlinear system of fractional ODEs. In order to show existence of the control, we first consider the linear counterpart of the system and reprove a number of classical theorems in the fractional setting (representation of the solution through the Gramian type matrix, Kalman's principle, equivalence of the controllability and observability). We are then in the position to use a fixed point theorem approach and various techniques from the fractional calculus theory to get the desired result.