Abstract
In this paper we study the long time behavior of solutions for an optimal control problem associated with the viscous incompressible electrically conducting fluid modeled by the magnetohydrodynamic (MHD) equations in a bounded two dimensional domain through the adjustment of distributed controls. We first construct a quasi-optimal solution for the MHD systems which possesses exponential decay in time. We then derive some preliminary estimates for the long-time behavior of all admissible solutions of the MHD systems. Next we prove the existence of a solution for the optimal control problem for both finite and infinite time intervals. Finally, we establish the long-time decay properties of the solutions for the optimal control problem.
Highlights
Magnetohydrodynamics (MHD) is the branch of continuum mechanics that studies the macroscopic interaction of electrically conducting fluids and electromagnetic fields
The main goal of this paper is to study the dynamics of solutions to an optimal control problem in magneto-hydrodynamics
The study of long-time behavior of solutions of optimal control problems associated with MHD systems is of great importance in many fluid dynamic applications such as stabilization and drag minimization
Summary
Magnetohydrodynamics (MHD) is the branch of continuum mechanics that studies the macroscopic interaction of electrically conducting fluids and electromagnetic fields. The study of long-time behavior of solutions of optimal control problems associated with MHD systems is of great importance in many fluid dynamic applications such as stabilization and drag minimization. In this article we study the long time behavior of solutions for optimal control problems associated with the magneto-hydrodynamic equations. On the Dynamics of Controlled Magnetohydrodynamic Systems that is, we assume the boundary is perfectly conducting (no tangential electric field and no normal magnetic field), see [5]. Our objective of matching the candidate flow field and magnetic field with the desired ones in ideal setting means matching the desired flow at each time instance This warrants minimizing a cost functional defined in terms of a pointwise norm in t. The physical objective of this minimization problem is to match a desired flow and magnetic field by adjusting the controls curl j and f.
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