Optimal control of the unsteady, laminar, fully developed flow of a viscous, incompressible and electrically conducting fluid is considered under the effect of a time varying magnetic field $B_0\left(t\right)$ applied in the direction making an angle with the $y$–axis. Thus, the coefficients of convection terms in the Magnetohydrodynamics (MHD) equations are also time-dependent. The coupled time-dependent MHD equations are solved by using the mixed finite element method (FEM) in space and the implicit Euler scheme in time. FEM solutions are obtained for various values of the Hartmann number, Reynolds number, magnetic Reynolds number and for several types of time dependence of applied magnetic field at transient level and steady-state.

In this study, we aim to control the unsteady MHD flow by using the time varying coefficient function $f\left(t\right)$ in the applied magnetic field $B_0\left(t\right)=B_{0}f\left(t\right)$ as a control function. In addition, control problem is designed to involve the determination of the optimal parameters of the system (Reynolds number, magnetic Reynolds number and the angle $\theta$) regarded as control variables.

In the optimization, a discretize-then-optimize approach with a gradient based algorithm is followed. Cost function is designed to regain the prescribed velocity and induced magnetic field profiles as well as the smooth control function with respect to time. Controls are investigated for the regularization parameters included in the cost function. Optimal solutions are achieved for several states of the flow considering Hartmann number and at the time level where the flow stabilizes.

在时变磁场的影响下，考虑对粘性、不可压缩和导电流体的不稳定、层流、充分发展的流动进行最佳控制 ${乙}_0\left(吨\right)$ 应用在与形成一个角度的方向上 $是$-轴。因此，磁流体动力学 (MHD) 方程中的对流项系数也与时间有关。耦合的瞬态 MHD 方程在空间上使用混合有限元法 (FEM) 和隐式欧拉格式在时间上求解。FEM 解决方案是针对哈特曼数、雷诺数、磁雷诺数的各种值以及在瞬态水平和稳态下施加的磁场的几种类型的时间依赖性而获得的。

在本研究中，我们旨在通过使用时变系数函数来控制非定常 MHD 流 $F\left(吨\right)$ 在外加磁场中 ${乙}_0\left(吨\right)={乙}_{0}F\left(吨\right)$作为控制功能。此外，控制问题的设计涉及系统最优参数（雷诺数、磁雷诺数和角度）的确定。$\theta$) 视为控制变量。

在优化中，遵循基于梯度算法的离散化然后优化方法。成本函数旨在重新获得规定的速度和感应磁场分布以及相对于时间的平滑控制函数。对包含在成本函数中的正则化参数进行控制研究。考虑到哈特曼数和在流动稳定的时间水平，为流动的几种状态实现了最佳解决方案。