Optimal control for realizing target flow velocity in 1D MHD flow

In this paper, we consider an optimal control problem arising in a one-dimensional (1D) Magnetohydrodynamic (MHD) flow, which can be modelled by a coupled partial differential equations (PDEs) where the external control input (external induction of magnetic field) takes the multiplicative effect exerted on both state variables (momentum and magnetic components). The aim is to derive the flow velocity to within close proximity of a desired target flow velocity at the pre-indicated terminal time. We first use the Galerkin method to obtain a low dimensional dynamical ordinary differential equation (ODE) model based on the original coupled PDEs. Then, we combine the control parameterization method with the time-scaling transformation technique to obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques such as sequential quadratic programming (SQP). The exact gradients of the cost functional with respect to the decision parameters are computed based on the analytical equations. Finally, we conclude the paper with simulation results for an example of the 1D MHD flow.