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.