Optimality conditions for fractional differential inclusions with nonsingular Mittag\u2013Leffler kernel

G M Bahaa,Adnane Hamiaz

doi:10.1186/s13662-018-1706-8

G M Bahaa, Adnane Hamiaz

Open Access

https://doi.org/10.1186/s13662-018-1706-8

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Abstract

In this paper, by using the Dubovitskii–Milyutin theorem, we consider a differential inclusions problem with fractional-time derivative with nonsingular Mittag–Leffler kernel in Hilbert spaces. The Atangana–Baleanu fractional derivative of order α in the sense of Caputo with respect to time t, is considered. Existence and uniqueness of solution are proved by means of the Lions–Stampacchia theorem. The existence of solution is obtained for all values of the fractional parameter alphain(0, 1). Moreover, by applying control theory to the fractional differential inclusions problem, we obtain an optimality system which has also a unique solution. The controllability of the fractional Dirichlet problem is studied. Some examples are analyzed in detail.

Highlights

Fractional calculus was proposed independently by Newton and Leibniz
The optimal control problems and calculus of variation for variational equality with fractional time derivative with nonlocal and nonsingular Mittag–Leffler kernel are studied in many papers
Proof According to the generalized Dubovitskii–Milyutin theorem [35], we approximate the set representing the inequality constraints by a regular admissible cone, the equality constraints by a regular tangent cone, and the performance functional by a regular improvement cone

Summary

Introduction

Fractional calculus was proposed independently by Newton and Leibniz. The theory of a non-integer order operators (integral and differential) was initiated in 1695. We shall mostly be considering a more recently developed definition for fractional differintegrals due to [10] This new type of calculus addresses the same underlying challenge as that of Caputo and Fabrizio, but it uses a kernel which is nonlocal as well as nonsingular, namely the Mittag–Leffler function: ABRDαa+f (t). The optimal control problems and calculus of variation for variational equality with fractional time derivative with nonlocal and nonsingular Mittag–Leffler kernel are studied in many papers (see, for example, [1, 2, 10, 11, 21] and the references therein). We study the optimal control problem for variational inequality with fractional time derivative with nonlocal and nonsingular Mittag–Leffler kernel. The spaces considered in this paper are assumed to be real

Definitions of cones and Lyusternik theorem

Conclusions