Dynamic Optimal Control of a One-Dimensional Magnetohydrodynamic System With Bilinear Actuation

The manipulation of the magnetic field has been a proven effective method to change the flow velocity in magnetohydrodynamics (MHD) flow systems. In this paper, we consider a novel bilinear magnetic control problem arising in a 1-D MHD flow system modeled by a set of coupled partial differential equations (PDEs). We formulate the control of the magnetic field as a finite-time PDE-constrained dynamic optimal control problem and our aim is to realize the desired stationary state of the flow velocity at a specific terminal time. A model order reduction technique, based on proper orthogonal decomposition and Galerkin projection procedure, is first adopted to approximate the original complex optimization problem governed by PDEs into a semi-discrete approximation problem governed by a low-dimensional reduced-order model, and therefore can efficiently reduce the computational burden of the dynamic system. Then, the piecewise-linear control parameterization method is used to obtain an approximate optimal parameter selection problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming. The exact formulas for the gradients of the defined cost functional with respect to the decision parameters are analytically derived via state sensitivity method. Numerical simulation results verify the effectiveness of our proposed computational method. The methodology proposed in this paper is a potential implementation of a real-time control strategy in a number of MHD flow systems.