Optimality conditions for fractional differential inclusions with nonsingular Mittag–Leffler kernel

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 α ∈ (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. MSC: 46C05; 49J20; 93C20

A complex analysis approach to Atangana–Baleanu fractional calculus

“…This can be proved in the same way as Lemma 4. Indeed, the series (42) is just a modification of the series (28) by a factor of…”

“…where E ( , ) denotes the modified double Mittag-Leffler function defined by (28) and H denotes the Hankel contour defined by ( 4) and (6).…”

“…In particular, we shall focus on the Atangana-Baleanu model, which we define in Section 2 below using Mittag-Leffler functions. Note that the Mittag-Leffler function has a strong and well-established connection with fractional calculus [24,31], and the Atangana-Baleanu model in particular has been proven useful by its many applications [3,9,10,11,13,27]. However, so far the Atangana-Baleanu model has been defined only for real orders of differintegration, so a complex formulation is something lacking in the field.…”

“…Note that the Mittag-Leffler function has a strong and well-established connection with fractional calculus, 23,24 and the Atangana-Baleanu model in particular has been proven useful by its many applications. 7,[25][26][27][28][29] However, so far the Atangana-Baleanu model has been defined only for real orders of differintegration, so a complex formulation is something lacking in the field. Thus, all existing literature on this type of fractional calculus has been restricted to the real domain only, unable to take advantage of the wealth of methods that can be found in complex analysis.…”

Optimal control problem and maximum principle for fractional order cooperative systems

“…Both fractional calculus of variations and fractional optimal control problems were developed by many authors it is enough to see, for example, works in (see [1] - [3], [11] - [14,21,35,36] and the papers and references therein) similar to a differential equations with integer time derivatives (see [10], [17] - [20], [24] - [26], [38] and the papers and references therein).…”