Controllability of nonlinear fractional Langevin delay systems

Pitchaikkannu Suresh Suresh Kumar,Krishnan Balachandran,Natarajan Annapoorani

doi:10.15388/na.2018.3.3

Pitchaikkannu Suresh Suresh Kumar, Krishnan Balachandran + Show 1 more

Open Access

https://doi.org/10.15388/na.2018.3.3

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Abstract

In this paper, we discuss the controllability of fractional Langevin delay dynamical systems represented by the fractional delay differential equations of order 0 < α,β ≤ 1. Necessary and sufficient conditions for the controllability of linear fractional Langevin delay dynamical system are obtained by using the Grammian matrix. Sufficient conditions for the controllability of the nonlinear delay dynamical systems are established by using the Schauders fixed-point theorem. The problem of controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays and distributed delays in control are studied by using the same technique. Examples are provided to illustrate the theory.

Highlights

The concept of controllability plays a major role in both finite and infinite dimensional spaces for systems represented by ordinary differential equations and partial differential equations
It is natural to study this concept for dynamical systems represented by fractional differential equations and fractional delay differential equations
A sliding mode control for linear fractional systems with input and state delays is investigated by Si-Ammour et al [27]

Summary

Introduction

The concept of controllability plays a major role in both finite and infinite dimensional spaces for systems represented by ordinary differential equations and partial differential equations. Yi et al [34] discussed the controllability and observability of systems of linear delay differential equations via the matrix Lambert W function. Zhang et al [36] discussed the controllability criteria for linear fractional differential systems delay in state and impulse. Yu et al [35] discussed the existence and uniqueness of solutions of initial value problems for nonlinear Langevin equations involving two fractional orders This motivates us to study the fractional Langevin delay differential equations. Sufficient conditions for the controllability of nonlinear fractional Langevin delay dynamical systems are established by using the Schauders fixed-point theorem. Controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays in control and distributed delays are studied by using the same technique

Preliminaries

Linear delay systems

Nonlinear delay systems

Systems with multiple delays

Systems with distributed delays

Examples