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Observers for compressible Navier-Stokes equation. (English) Zbl 1383.93017

Summary: We consider a multidimensional model of a compressible fluid in a bounded domain. We want to estimate the density and velocity of the fluid, based on the observations for only velocity. We build an observer exploiting the symmetries of the fluid dynamics laws. Our main result is that for the linearized system with full observations of the velocity field, we can find an observer which converges to the true state of the system at any desired convergence rate for finitely many but arbitrarily large number of Fourier modes. Our one-dimensional numerical results corroborate the results for the linearized, fully observed system, and also show similar convergence for the full nonlinear system and also for the case when the velocity field is observed only over a subdomain.

MSC:

93B07 Observability
93C20 Control/observation systems governed by partial differential equations
93C95 Application models in control theory
93B40 Computational methods in systems theory (MSC2010)
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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