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SUMMARY:Adrien Tendani-Soler: Local exact controllability to a constant tr
ajectory for a compressible Navier-Stokes-Korteweg system
DTSTART:20241001T083000Z
DTEND:20241001T093000Z
DTSTAMP:20241015T003800Z
UID:indico-event-12651@indico.math.cnrs.fr
DESCRIPTION:In this presentation\, we explore the local exact controllabil
ity to a constant trajectory for the compressible Navier-Stokes-Korteweg s
ystem on the torus in dimensions $d\\in{1\,2\,3}$\, where the control acts
on an open subset. The Navier-Stokes-Korteweg system is a variant of the
compressible Navier-Stokes system\, which considers both viscosity and cap
illarity effects\, relevant to describing the phase transition between liq
uid and vapor. In the literature\, the parabolic behavior of the system is
highlighted\, only\, in the framework of the Cauchy theory. The aim of my
presentation is to explain how capture the hidden parabolic behavior of t
his system from the perspective of control theory. Specifically\, we achie
ve local exact controllability to the constant state $(\\rho_\\star\,0)$ f
or arbitrarily small positive times\, without imposing any geometric condi
tion on the control region. To this end\, we study the control properties
of the linearized equation and conduct a thorough analysis of the observab
ility of the associated adjoint equations. Notably\, we identify the parab
olic (and potentially dispersive) structure of these adjoint equations. Us
ing this insight\, we derive observability for the adjoint system through
specific Carleman estimates.\n\nhttps://indico.math.cnrs.fr/event/12651/
LOCATION:Salle 318 (IMB)
URL:https://indico.math.cnrs.fr/event/12651/
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