Fágner D. Araruna | Searcharxiv

arXiv2024

Uniform local null control of the Leray-$α$ model

This paper deals with the distributed and boundary controllability of the so called Leray-$α$ model. This is a regularized variant of the Navier-Stokes system ($α$ is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-$α$ equations are locally null controllable, with controls bounded independently of $α$. We also prove that, if the initial data are sufficiently small, the controls converge as $α\to 0^+$ to a null control of the Navier-Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.

math.OC↗

arXiv2024

On the control of the Burgers-alpha model

This work is devoted to prove the local null controllability of the Burgers-$α$ model. The state is the solution to a regularized Burgers equation, where the transport term is of the form $zy_x$, $z=(Id-α^2\frac{\partial^2}{\partial x^2})^{-1}y$ and $α>0$ is a small parameter. We also prove some results concerning the behavior of the null controls and associated states as $α\to 0^+$.

math.OC↗