Abstract
We analyze an optimal boundary control problem for heat convection equations in a three-dimensional domain, with mixed boundary conditions. We prove the existence of optimal solutions, by considering boundary controls for the velocity vector and the temperature. The analyzed optimal control problem includes the minimization of a Lebesgue norm between the velocity and some desired field, as well as the temperature and some desired temperature. By using the Lagrange multipliers theorem we derive an optimality system. We also give a second-order sufficient condition.
Highlights
Let Ω ⊂ R3 be a connected bounded domain with Lipschitz boundary Γ
We prove the existence of optimal solutions, by considering boundary controls for the velocity vector and the temperature
We deal with the existence of weak solutions of a boundary value problem describing the motion of a viscous heat conduction fluid in Ω, with a Navier slip condition on the boundary for the velocity vector
Summary
Let Ω ⊂ R3 be a connected bounded domain with Lipschitz boundary Γ. In the context of the classic Navier-Stokes equations there is a good amount of work with Navier friction boundary condition, among which we can mention [3, 14,15,16,17,18,19], and references therein, where the existence, regularity, and uniqueness of weak and strong solutions are studied. For optimal control problems for the classical Boussinesq stationary model, see [22,23,24,25,26,27], and the references therein In such works, standard results have been obtained, such as the existence of an optimal solution and stability as well as first-order necessary optimality conditions, from which the authors derive an optimality system.
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