Controllability and constrained controllability for nonlocal Hilfer fractional differential systems with Clarke\u2019s subdifferential

Hamdy M Ahmed,Mahmoud M El-Borai,A S Okb El Bab,M Elsaid Ramadan

doi:10.1186/s13660-019-2184-6

Abstract

Sobolev-type nonlocal fractional differential systems with Clarke’s subdifferential are studied. Sufficient conditions for controllability and constrained controllability for Sobolev-type nonlocal fractional differential systems with Clarke’s subdifferential are established, where the time fractional derivative is the Hilfer derivative. An example is given to illustrate the obtained results.

Highlights

A Sobolev-type equation appears in several physical problems such as flow of fluids through fissured rocks, thermodynamics and propagation of long waves of small amplitude
Controllability means to steer a dynamical system from an arbitrary initial state to the desired final state in a given finite interval of time by using the admissible controls, and controllability results for linear and nonlinear integer order differential systems were studied by several authors
The constrained controllability is concerned with the existence of an admissible control that steers the state to a given target set from a specified initial state

Summary

Introduction

A Sobolev-type equation appears in several physical problems such as flow of fluids through fissured rocks, thermodynamics and propagation of long waves of small amplitude (see [1,2,3]). Nonlinear fractional differential equations can be observed in many areas such as population dynamics, heat conduction in materials with memory, seepage flow in porous media, autonomous mobile robots, fluid dynamics, traffic models, electro magnetic, aeronautics, economics (see [4,5,6,7,8,9,10,11,12,13]). Controllability means to steer a dynamical system from an arbitrary initial state to the desired final state in a given finite interval of time by using the admissible controls, and controllability results for linear and nonlinear integer order differential systems were studied by several authors (see [14,15,16,17,18,19,20,21,22,23,24,25,26,27]). The Clarke subdifferential has been applied in mechanics and engineering, especially in nonsmooth analysis and optimization [32, 33]

Objectives

Conclusion