Global exact controllability of the viscous and resistive MHD system in a rectangle thanks to the lateral sides and to distributed phantom forces

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-11-01 DOI:10.1051/cocv/2023078

Jiajiang Liao

引用次数: 0

Abstract

We consider the 2-D incompressible viscous and resistive magnetohydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron’s return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra’s local exact controllability result.

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由于侧向和分布的幻力，粘阻MHD系统在矩形内具有全局精确的可控性

我们考虑矩形中的二维不可压缩粘阻磁流体动力学(MHD)系统，其两侧有控制。速度满足狄利克雷边界条件，而磁场在边界的其余不受控制部分遵循完全导电壁边界条件。我们将Coron等人在[Ann PDE 5(2):1-49, 2019]中的小时全局精确零可控性结果从Navier-Stokes方程扩展到MHD方程，并借助分布式幻力，可以在任何给定的Sobolev空间中选择任意小的幻力。我们的分析依赖于Coron的回归法、精心准备的耗散法、长时间非线性Cauchy-Kovalevskaya估计和Badra的局部精确可控性结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

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7.10%

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期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.