Fractional Optimal Control of Navier-Stokes Equations

In this paper, the non-stationary incompressible fluid flows governed by the Navier-Stokes equations are studied in a bounded domain. This study focuses on the timefractional Navier-Stokes equations in the optimal control subject, where the control is distributed within the domain and the time-fractional derivative is proposed as RiemannLiouville sort. In addition, the control object is to minimize the quadratic cost functional. By using the Lax-Milgram lemma with the assistance of the fixed-point theorem, we demonstrate the existence and uniqueness of the weak solution to this system. Moreover, for a quadratic cost functional subject to the time-fractional Navier-Stokes equations, we prove the existence and uniqueness of optimal control. Also, via the variational inequality upon introducing the adjoint linearized system, some inequalities and identities are given to guarantee the first-order necessary optimality conditions. A direct consequence of the results obtained here is that when α → 1, the obtained results are valid for the classical optimal control of systems governed by the Navier-Stokes equations.