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.