The Turnpike Property for Two-Dimensional Navier-Stokes Equations

TL;DR

This paper investigates the turnpike phenomenon in optimal control problems for 2D Navier-Stokes equations, demonstrating exponential turnpike properties for both time-dependent and stationary controls under smallness conditions.

Contribution

It establishes the exponential turnpike property for 2D Navier-Stokes control problems, extending understanding of long-term optimal control behavior in fluid dynamics.

Findings

Exponential turnpike property holds for time-dependent controls.

Turnpike property also valid for stationary controls.

Smallness conditions on stationary states are sufficient.

Abstract

This paper is devoted to the study of the turnpike phenomenon arising in the optimal distributed control tracking-type problem for the Navier-Stokes equations. We obtain a positive answer to this property in the case when the controls are time-dependent functions, and also when are independent of time. In both cases we prove an exponential turnpike property assuming that the stationary optimal state satisfy certain properties of smallness.