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Fast uniform stabilization of the linearized magnetohydrodynamics system by finite-dimensional localized feedback controllers
Lasiecka, Irena
Priyasad, Buddhika
Triggiani, Roberto
2025
Abstract
This research project considers 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 addition, the actuators will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space eB2−2/p q,p (Ω) × eB2−2/p q,p (Ω), 1 < p < 2q 2q−1 , q > d, of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to L3(Ω) for d = 3). It is known that such Besov space does not recognize compatibility conditions at the boundary, yet it provides a “minimal” level of regularity necessary to handle the nonlinear terms. In this paper we provide a solution of the first step: uniform stabilization of the linearized MHD. Showing maximal Lp-regularity up to T = ∞ for the feedback stabilized linearized system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem. The solution of the nonlinear stabilization problem is to be given in a successive paper [29].
Source
Irena Lasiecka, Buddhika Priyasad, and Roberto Triggiani, "Fast uniform stabilization of the linearized magnetohydrodynamics system by finite-dimensional localized feedback controllers" in: "Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.57 (2025)", EUT Edizioni Università di Trieste, Trieste, 2025, pp.
Languages
en
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International
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