Real-time computational optimal control of an MHD flow system with parameter uncertainty quantification

In this paper, we consider a magnetic control problem arising in a one-dimensional (1-D) MHD flow system governed by a set of coupled partial differential equations (PDEs) with parameter uncertainty quantification. We first formulate the control problem as a PDE-constrained optimization problem, which is then reduced to a semi-discrete optimal control problem governed by a set of ordinary differential equations (ODEs) through a finite-element approach. The control parameterization method is then utilized to convert the semi-discrete optimal control problem into an approximate parameter selection problem and the gradient formulas of the cost function corresponding to the decision variables are derived. In order to meet the requirement of real-time control design for the MHD flow system, we further propose a multi-fidelity probabilistic collocation method for the optimal control by introducing a high fidelity model and a low fidelity model to improve the computational optimization efficiency while maintaining the high computational accuracy. Numerical results are illustrated to validate the effectiveness of our proposed computational method.