Interior local null controllability of one‐dimensional compressible flow near a constant steady state

Abstract

We consider the one‐dimensional compressible Navier–Stokes equations with periodic boundary conditions, with initial conditions in a small neighborhood of a state of uniform density and uniform nonzero velocity. We prove that, with a control given only by a body force localized in a subinterval, we can steer the system to uniform density and velocity. Copyright © 2016 John Wiley & Sons, Ltd.

Interior local null controllability for multi-dimensional compressible flow near a constant state

“…The purpose of the present paper is to extend the result of [15]. Our method of analysis rather closely follows that of [15], but there are some differences. Due to the different behavior of Sobolev embeddings, we need to work in spaces of higher regularity.…”

“…The result of [9] has recently been extended to higher dimensions [10]. The purpose of the present paper is to extend the result of [15]. Our method of analysis rather closely follows that of [15], but there are some differences.…”

“…A partial result on the null controllability of the nonlinear system in one space dimension is given in [1]. Null controllability of compressible one-dimensional flows is established in [9] and [15]. In [9], the authors establish a result on boundary controllability for Dirichlet boundary conditions, using a technique of proof which is actually based on using a distributed control located outside the domain of physical interest.…”

“…In [9], the authors establish a result on boundary controllability for Dirichlet boundary conditions, using a technique of proof which is actually based on using a distributed control located outside the domain of physical interest. In [15], periodic boundary conditions are considered, and a distributed control is added to the momentum equation only. Both approaches are based on combining Carleman estimates for the heat equation with the characteristics method for the transport equation.…”

“…They then take advantage of the Carleman weight to accomplish the decoupling of heat and transport equation, choosing one of the parameters sufficiently large. In contrast, the argument in [15] is, roughly speaking, based on the intuition that controllability should not be affected by compact perturbations as long as approximate controllability holds. A transformation of variables reduces the differential order of the coupling, basically making it weak for high Fourier modes.…”