Bilinear Optimal Control Problem for the Stationary Navier–Stokes Equations with Variable Density and Slip Boundary Condition

Abstract

An optimal control problem for the stationary Navier–Stokes equations with variable density is studied. A bilinear control is applied on the flow domain, while Dirichlet and Navier boundary conditions for the velocity are assumed on the boundary. As a first step, we enunciate a result on the existence of weak solutions of the dynamical equation; this is done by firstly expressing the fluid density in terms of the stream-function. Then, the bilinear optimal control problem is analyzed, and the existence of optimal solutions are proved; their corresponding characterization regarding the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.