Oberseminar Partielle Differentialgleichungen Detail

Vortragende Person/Vortragende Personen:
Herr Dr. Buddhika Priyasad - Karls-Universität Prag

Zusammenfassung: The problem of uniform stabilization of the Navier-Stokes equations in the vicinity of an unstable equilibrium solution was introduced by Andrei Fursikov (2000), originally with an open-loop, infinite dimensional, boundary control. Despite the intense interest generated, whether it was possible to obtain such uniform stabilization for the 3D Navier-Stokes equations by means of localized, static, boundary-based, closed-loop, feedback controllers that moreover are finite dimensional and explicit has been an open problem till about 2021. Its solution required abandoning the traditional Hilbert setting (and use of Riccati equations) and replacing it with a suitable Besov space of tight indices (“close” to L 3 (Omega) for D = 3) (and use of spectral analysis). An analogous uniform stabilization problem with a localized, static, finite dimensional thermal boundary control was later shown for the Boussinesq system. Next, one seeks an optimal stabilization strategy for these dynamics under more practically relevant control constraints. This can be achieved by showing the differentiability of the corresponding value function of the system. Finally, this talk will present the latest results pertaining to the dynamical system governed by the Ladyzhenskaya model of Navier-Stokes equations subject to dynamic boundary conditions and possible stabilization approaches. These results are based on several projects with co-authors Lasiecka, Roberto Triggiani, Karl Kunisch, and Dalibor Prazak.