Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions

Abstract

We study the feedback stabilization of the Boussinesq system in a two dimensional domain, with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green's formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This provides a rigorous mathematical framework for models studied in the engineering literature. We prove the stabilizability by extending to the linearized Boussinesq system a local Carleman estimate already established for the Oseen system. Then we determine a feedback control law able to stabilize the linearized system around the stationary solution, with any prescribed exponential decay rate, and able to stabilize locally the nonlinear system.

Uniform stabilization of Boussinesq systems in critical <inline-formula><tex-math id="M1">\begin{document}$ \mathbf{L}^q $\end{document}</tex-math></inline-formula>-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

“…In effect, such reference [88] provides two additional UCPs, say for the original problem (1.34 a-b-c) with over determination this time on a small subdomain ω subtended by an arbitrary portion of the boundary. A proof yielding, say Theorem 1.3 with d = 2 and with limited regularity of the solution was given in [74]. Theorem 1.4 is in line with an "observability inequality" for the time dependent problem (1.1) needed in the study of local controllability to the origin or to a trajectory given in [30].…”

“…Indeed, the main obstacle in this 3 − d case is due to compatibility conditions between boundary terms and initial conditions, resulting from the necessity of using sufficiently regular [differentiable] solutions for d = 3. As it was originally the case with the Navier-Stokes equations, the Boussinesq case in the literature considers until now [7], [65], [74], [92] the problem within the Hilbert framework. Due to the Stokes non-linearity in 3 − d, this necessitates in the Navier-Stokes equations -whether alone or as a component of the Boussinesq system -the need of working with sufficiently high order Sobolev spaces which recognize boundary conditions, and thus impose compatibility conditions.…”

“…These results are in stark contrast with the present work, where -as noted in point 3 above -first, we establish that the stabilizing control is finite dimensional; and, second, we show that its fluid component is of reduced dimension, as expressed only by means of (d − 1)-components. Finally paper [74] studies the feedback stabilization problem with, this time, mixed boundary and claims to provide a rigorous treatment of problems studied in [15].…”

A penalty approach to the infinite horizon LQR optimal control problem for the linearized Boussinesq system

“…The optimal control of the Boussinesq system and of its linearization around a stationary state are matters of great interest in various applications elds, such as designing and exploiting energy ecient buildings, (see, for instance, [9], [33] and [10], [23]). A diculty which has to be handled in solving this type of problems is that, due to the free divergence condition for the velocity eld and to the presence of the pressure, the governing equations cannot be written as a well-posed control system in the sense of Salomon-Weiss (as described, for instance, in Curtain and Weiss [12]), but merely as an innite dimensional descriptor system (see, for instance, Reis [26]).…”

Robust output regulation of the linearized Boussinesq equations with boundary control and observation

“…[23,33]. Most of the previous work regarding control of the Boussinesq equations focuses on stabilization [12,26,37,42]. Examples of output tracking for both nonlinear and linear thermal fluid flows based on state feedback have been considered in [1] and references therein, and solution methods for the related regulator equations have been further developed in [2,3].…”