Optimal control of flows of viscoelastic semi-compressible fluids

TL;DR

This paper develops optimal control strategies for semi-compressible viscoelastic fluid flows, extending classical incompressible models to more physically realistic semi-compressible fluids, with analysis in 2D and 3D cases.

Contribution

It introduces a novel optimal control framework for semi-compressible fluid models, including boundary pressure control and analysis of control-to-state mapping.

Findings

Uniqueness of control-to-state mapping in 2D

First-order optimality conditions derived for 2D case

Outline for 3D case analysis

Abstract

Semilinear parabolic systems with bi-linear nonlinearities cover a lot of applications and their optimal control leads to relatively simple optimality conditions. An example is the incompressible Navier-Stokes system for homogeneous fluids, which is however here modified towards a physically reasonable model of slightly (so-called ``semi'') compressible liquids rather than fully compressible gases. An optimal control problem optimizing also pressure on the boundary is considered and, in the simple variant, analysed as far as uniqueness of the control-to-state mapping and 1st-order optimality conditions in the 2-dimensional case and outlined in a nonsimple variant for the 3-dimensional case. Some other bi-linear parabolic systems as Cahn-Hilliard diffusion or magneto-hydrodynamics can be treated analogously.