Uniform stabilization in Besov spaces with arbitrary decay rates of the magnetohydrodynamic system by finite-dimensional interior localized static feedback controllers

• Published in: Research in the mathematical sciences Vol. 12; no. 1
• Main Authors: Lasiecka, Irena, Priyasad, Buddhika, Triggiani, Roberto
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2025; Springer Nature B.V
• Subjects: Applications of Mathematics; Boundary conditions; Computational Mathematics and Numerical Analysis; Control systems; Decay rate; Feedback control; Function space; Initial conditions; Linear systems; Magnetohydrodynamics; Mathematics; Mathematics and Statistics; Regularity; Stabilization; Well posed problems
• ISSN: 2522-0144; 2197-9847
• DOI: 10.1007/s40687-024-00490-7

We consider the d -dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain, d = 2 , 3 with homogeneous boundary conditions, and subject to external sources assumed to cause instability. The initial conditions for both fluid and magnetic equations are taken of low regularity. We then seek to uniformly stabilize such MHD system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, static, feedback controls, which are localized on an arbitrarily small interior subdomain. In additional, they will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space B ~ q , p 2 - 2 / p ( Ω ) × B ~ q , p 2 - 2 / p ( Ω ) , 1 < p < 2 q 2 q - 1 , q > d , of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to L 3 ( Ω ) for d = 3 ). Showing maximal L p -regularity up to T = ∞ for the feedback stabilized linear system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem.