Fractional Derivatives of Convex Lyapunov Functions and Control Problems in Fractional Order Systems

TL;DR

This paper develops control procedures for fractional order systems using convex Lyapunov functions and fractional derivatives, providing new estimates and an example demonstrating their effectiveness.

Contribution

It introduces a novel approach to control fractional systems via convex Lyapunov functions and fractional derivatives, extending existing estimates and including an illustrative example.

Findings

Proposed control procedures ensure system proximity under fractional dynamics.

Generalized estimates for fractional derivatives of convex Lyapunov functions.

Validated control method through a practical example.

Abstract

The paper is devoted to the development of control procedures with a guide for conflict-controlled dynamical systems described by ordinary fractional differential equations with the Caputo derivative of an order $\alpha \in (0, 1).$ For the case when the guide is in a certain sense a copy of the system, a mutual aiming procedure between the initial system and the guide is elaborated. The proof of proximity between motions of the systems is based on the estimate of the fractional derivative of the superposition of a convex Lyapunov function and a function represented by the fractional integral of an essentially bounded measurable function. This estimate can be considered as a generalization of the known estimates of such type. An example is considered which illustrates the workability of the proposed control procedures.