Computational bilinear optimal control for a class of one-dimensional MHD flow systems

Area of Science:

• Fluid Dynamics
• Plasma Physics
• Computational Science

Background:

• Magnetohydrodynamics (MHD) describes fluid motion influenced by magnetic fields.
• Controlling MHD flow is crucial for applications like fusion energy and astrophysics.
• Bilinear optimal control problems present unique challenges due to multiplicative control effects.

Purpose of the Study:

• To develop and validate an effective optimal control strategy for one-dimensional MHD flow.
• To steer the flow velocity towards a target value at a specific time.
• To address the challenges posed by multiplicative control inputs in MHD models.

Main Methods:

• Semi-discrete approximation using the Galerkin method with quadratic B-spline functions.
• Control parameterization combined with time-scaling transformation for optimization.
• Analytical computation of cost function gradients.
• Sequential Quadratic Programming (SQP) for solving the approximate optimal parameter selection problem.

Main Results:

• Convergence of the semi-discrete approximation problem was rigorously proved.
• An efficient method for computing exact gradients of the cost function was derived.
• Numerical simulations demonstrated the effectiveness of the proposed control strategy.
• The method successfully drove the flow velocity close to the desired target.

Conclusions:

• The presented optimal control method is effective for 1-D MHD flow problems.
• The combination of Galerkin approximation and control parameterization provides a robust solution.
• The findings have implications for controlling plasma behavior in various scientific and engineering applications.