Optimal Boundary Control of Non-Isothermal Viscous Fluid Flow

Abstract

We study an optimal control problem for the mathematical model that describes steady non-isothermal creeping flows of an incompressible fluid through a locally Lipschitz bounded domain. The control parameters are the pressure and the temperature on the in-flow and out-flow parts of the boundary of the flow domain. We propose the weak formulation of the problem and prove the existence of weak solutions that minimize a given cost functional. It is also shown that the marginal function of this control system is lower semi-continuous. D(u)def = (∇u + (∇u) T )/2, the function p = p(x) represents the pressure field, µ(θ) > 0 is the viscosity, κ(θ) > 0 is the thermal conductivity, α > 0 is a coefficient characterizing the heat transfer on solid walls of the flow domain, ω(x, θ) stands for the heat source intensity, n = n(x) is the unit outward normal to the surface ∂Ω, S is a flat (straight for d = 2) portion of ∂Ω or the union of several such portions. Functions ζ : S → R and π : S → R play the role of controls, U 1 × U 2 is the set of admissible controls, while J = J(u, θ, π, ζ) is a given cost functional. By the symbol τ we denote the tangential component of a vector, i.e., u τ def = u − (u · n)n. writing-review and editing, M.A.A.Funding: This research received no external funding. Conflicts of Interest:The authors declare no conflict of interest.

The Well Posedness for Nonhomogeneous Boussinesq Equations

“…) Now, we shall estimate the term δθ. By Proposition 3,s 2 = 1, p 1 = p 2 = p = N in(15), we obtain:) δu L1 T ( Ḃ1 N,∞ ) . (59)…”

“…In [14], they studied an optimum control problem of mathematical model describing steady non-isothermal creep of incompressible fluid through local Lipschitz bounded region. In [15], they studied an optimal control problem for the mathematical model that describes steady non-isothermal creeping flows of an incompressible fluid through a locally Lipschitz bounded domain. In [16], the initial-boundary value problem of completely incompressible Navier-Stokes equations with viscosity coefficient ν and heat conductivity κ varying with temperature by the power law of Chapman-Enskog are studied.…”

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

“…In the studying of optimal solutions, it is important to investigate the case when the collection of all admissible controls (in our problem, the set F × G u 0 ) can be expanded/reduced. Following the ideas developed in [37][38][39], we introduce the concept of the marginal function, which shows how the minimal value of the cost functional J changes under a variation of the set F × G u 0 .…”

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.…”

“…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. The temperature and the pressure in a flat portion of the local Lipschitz boundary played the role of controls.…”