Some Controllability Results for Linearized Compressible Navier-Stokes System

In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around $(\bar \rho,0,\bar \theta)$, with $\bar \rho > 0,$ $\bar \theta > 0$ is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove that the the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state $(\bar \rho,\bar v ,\bar \theta)$, with $\bar \rho > 0,$ $\bar v > 0,$ $\bar \theta > 0$ is not null controllable by localized interior control or by boundary control for small time $T.$ Next we consider two-dimensional compressible Navier-Stokes system for barotropic fluid linearized around a constant steady state $(\bar \rho, {\bf 0}).$ We prove that this system is also not null controllable by localized interior control

Some controllability results for linear viscoelastic fluids

We analyze the controllability properties of systems which provide a description, at first approximation, of a kind of viscoelastic fluid. We consider linear Maxwell fluids. First, we establish the large time approximate-finite dimensional controllability of the system, with distributed or boundary controls supported by arbitrary small sets. Then, we prove the large time exact controllability of fluids of the same kind with controls supported by suitable large sets. The proofs of these results rely on classical arguments. In particular, the approximate controllability result is implied by appropriate unique continuation properties, while exact controllability is a consequence of observability (inverse) inequalities. We also discuss questions concerning the controllability of viscoelastic fluids and some related open problems.Ministerio de Ciencia e Innovació