Feedback Optimal Control Problem for a Network Model of Viscous Fluid Flows

We study a feedback optimal control problem for a three-dimensional model of a stationary flow of a non-Newtonian fluid (with variable viscosity) in a pipeline network with complex geometry. The control parameter is the dynamic pressure on connection surfaces of pipes to nodes. The flow model is a mixed boundary-value problem for a system of strongly nonlinear partial differential equations in a netlike domain with Kirchhoff-type transmission conditions at interior nodes of the network. The solvability of the optimization problem in the weak formulation is proved; namely, we establish sufficient conditions for the existence of a weak solution which minimizes a lower semicontinuous cost functional.