Approximate Controllability of Linearized Shape-Dependent Operators for Flow Problems

TL;DR

This paper investigates the approximate controllability of linearized shape-dependent operators in flow problems, revealing how infinitesimal shape changes can influence flow characteristics like wall velocity and shear stress.

Contribution

It introduces the linearizations of shape-dependent operators in flow problems, proves their well-posedness, and demonstrates their approximate controllability, advancing shape optimization methods.

Findings

Linearized operators are well-posed.

Approximate controllability is established.

Shape deformations influence flow properties.

Abstract

We study the controllability of linearized shape-dependent operators for flow problems. The first operator is a mapping from the shape of the computational domain to the tangential wall velocity of the potential flow problem and the second operator maps to the wall shear stress of the Stokes problem. We derive linearizations of these operators, provide their well-posedness and finally show approximate controllability. The controllability of the linearization shows in what directions the observable can be changed by applying infinitesimal shape deformations.