*Article* **Optimal Feedback Control Problem for the Fractional Voigt-***α* **Model**

#### **Victor Zvyagin 1,\*,†, Andrey Zvyagin 2,† and Anastasiia Ustiuzhaninova 3,†**


Received: 14 June 2020; Accepted: 20 July 2020; Published: 21 July 2020

**Abstract:** The study of the existence of an optimal feedback control problem for the initial-boundary value problem that describes the motion of the fractional Voigt-*α* model of a viscoelastic medium is investigated in this paper. In this model, the Voigt rheological relation is considered with the left-side fractional Riemann-Liouville derivative, which allows to take into account the memory of the medium. Also in this model, the memory is considered along the trajectory of the motion of fluid particles, determined by the velocity field. Due to the insufficient smoothness of the velocity field and, as a consequence, the impossibility of uniquely determining the trajectory for the velocity field for any initial value, a weak solution to the problem under study is introduced using regular Lagrangian flows. Based on the approximation-topological approach to the study of fluid dynamic problems, the existence of an optimal solution that gives a minimum to a given cost functional is proved.

**Keywords:** optimal feedback control; Voigt model; alpha-model; fractional derivative

**MSC:** 76D55; 49J20; 35Q35
