The optimal start control problem for 2D Boussinesq equations

Abstract

We consider the problem of the optimal start control for two-dimensional Boussinesq equations describing non-isothermal flows of a viscous fluid in a bounded domain. Using the study of the properties of admissible tuples and of the evolution operator, we prove the solubility of the optimization problem under natural assumptions about the model data. We derive a variational inequality which is satisfied by the optimal control provided that the objective functional is determined by the final observation. We also obtain sufficient conditions for the uniqueness of an optimal control.

Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity

“…Therefore, passing to the limit in (26) as m goes to +∞, and taking into account the strong convergence (24), we obtain…”

“…We also mention that there is a large number of mathematical works devoted to the study of optimal control problems for PDEs describing flows of a homogeneous fluid (see, for example, refs. [22][23][24][25][26] and the numerous references therein). Such problems are currently fairly well understood, while control and optimization problems for nonhomogeneous fluid flows remain a serious challenge.…”

Analysis and Computations of Optimal Control Problems for Boussinesq Equations

“…The solvability of the stationary boundary control problem for the Boussinesq equation is studied in [13,14], considering as boundary controls the velocity, the temperature, and the heat flux. Recently, new approaches to the study of the optimal control of Boussinesq equations have been proposed [15][16][17]. In [15], the solvability of an optimal control problem for steady non-isothermal incompressible creeping flows was proven.…”

“…In [16], the optimal Neumann control problem for non-isothermal steady flows in low-concentration aqueous polymer solutions was considered, and sufficient conditions for the existence of optimal solutions were established. The problem of the optimal start control for unsteady Boussinesq equations was investigated in [17] to prove their solvability.…”

“…We consider three different control mechanisms: distributed, Dirichlet, and Neumann boundary controls. The solvability of the stationary optimal control problem for the Boussinesq equations has already been widely investigated in previous studies, considering as controls the forces and heat sources acting on the domain, together with the velocity, pressure, heat flux, and temperature on a portion of the boundary [3,5,[9][10][11][12][13][14][15][16][17]. However, only a few studies show the numerical results of the optimal control problem for the Boussinesq equation and consider only a single control mechanism [2,11,14].…”