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Abstract This paper provides a (rigorous) theoretical framework for the numerical approximation of Riccati-based feedback control problems of hyperbolic-like dynamics over a finite-time horizon, with emphasis on genuine unbounded control action. Both continuous and approximation theories are illustrated by specific canonical hyperbolic-like equations with boundary control, where the abstract assumptions are actually sharp regularity properties of the hyperbolic dynamics under discussion. Assumptions are divided in two groups. A first group of dynamical assumptions (actually dynamic properties) imply some preliminary critical properties of the control problem, including the definition of the would-be Riccati operator, in terms of the original data. However, in order to guarantee that such an operator is moreover the unique solution (within a specific class) of the corresponding Differential/Integral Riccati Equation, additional smoothing assumptions on the operators defining the performance index are required. The ultimate goal is to show that the the discrete finite dimensional Riccati based feedback operator, when inserted into the original PDE dynamics, provides near optimal performance. 

Abstract We consider thed-dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain,$$d = 2,3$$ $d=2,3$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$$\displaystyle \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega )\times \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega ), \, 1< p < \frac{2q}{2q-1}, \, q > d,$$ ${\tilde{B}}_{q,p}^{2{-}^2{/}_p}\left(\Omega \right)\times {\tilde{B}}_{q,p}^{2{-}^2{/}_p}\left(\Omega \right),1<p<\frac{2q}{2q-1},q>d,$of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to$$\textbf{L}^3(\Omega )$$ $L^3\left(\Omega \right)$for$$d = 3$$ $d=3$). Showing maximal$$L^p$$ $L^p$-regularity up to$$T = \infty $$ $T=\infty$for the feedback stabilized linear system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem.