Abstract
In this paper, we consider a diffusion equation with fractional time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. We first prove the existence and uniqueness of solution by means of a spectral argument. Then, we consider a distributed controlled fractional diffusion problem. We show that there exists a unique optimal control, which can act on the system in order to approach the state of the system by a given state at minimal cost. Finally, using the Euler–Lagrange first-order optimality condition, we obtain an optimality system, which characterizes the optimal control.
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