Hilfer fractional evolution hemivariational inequalities with nonlocal initial conditions and optimal controls

Yatian Pei,Yong-Kui Chang

doi:10.15388/na.2019.2.3

Yatian Pei, Yong-Kui Chang

Open Access

https://doi.org/10.15388/na.2019.2.3

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Abstract

In this paper, we mainly consider a control system governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. We first establish sufficient conditions for the existence of mild solutions to the addressed control system via properties of generalized Clarke subdifferential and a fixed point theorem for condensing multivalued maps. Then we present the existence of optimal state-control pairs of the limited Lagrange optimal systems governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. The optimal control results are derived without uniqueness of solutions for the control system.

Highlights

As a generalization of the ordinary differentiation and integration to arbitrary noninteger order, fractional calculus has been recognized as one of the most powerful tools to describe long-memory processes in the last decades
Kumar considered the existence of optimal control for the system governed by semilinear Caputo fractional evolution equation of order (0, 1) with fixed delay in [16]
Liu and Wang in [18] dealt with optimal controls of systems governed by semilinear Caputo fractional differential equations of order (0, 1) with not instantaneous impulses

Summary

Introduction

As a generalization of the ordinary differentiation and integration to arbitrary noninteger order, fractional calculus has been recognized as one of the most powerful tools to describe long-memory processes in the last decades. In [27], Yang and Wang considered existence of mild solutions for a Hilfer fractional differential equation with nonlocal initial conditions. Kumar considered the existence of optimal control for the system governed by semilinear Caputo fractional evolution equation of order (0, 1) with fixed delay in [16]. Liu and Wang in [18] dealt with optimal controls of systems governed by semilinear Caputo fractional differential equations of order (0, 1) with not instantaneous impulses. Lu and Liu [19] studied the existence and controllability for a stochastic evolution hemivariational inequality in Caputo fractional derivative of order (0, 1). Liu et al [20] investigated solvability and optimal controls for a semilinear fractional evolution hemivariational inequality in Caputo sense of order (0, 1).

Preliminaries

Existence results

Optimal controls