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.

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We consider the optimal control problem for a mathematical model describing steady flows of a nonlinear-viscous incompressible fluid in a bounded three-dimensional (or a two-dimensional) domain with impermeable solid walls. The control parameter is the surface force at a given part of the flow domain boundary. For a given bounded set of admissible controls, we construct generalized (weak) solutions that minimize a given cost functional.

We study the initial boundary value problem for the nonlinear system, which describes the dynamics of an incompressible viscoelastic fluid with the Jeffreys constitutive law under the Navier slip boundary condition. We construct a global (in time) weak solution to this problem.