Controllability of the 3D Compressible Euler System

Abstract

Abstract. The paper is devoted to the controllability problem for 3D compressible Euler system. The control is a finite-dimensional external force acting only on the velocity equation. We show that the velocity and density of the fluid are simultaneously controllable. In particular, the system is approximately controllable and exactly controllable in projections.

Local Exact Controllability for the One-Dimensional Compressible Navier–Stokes Equation

“…where we have used in (4.18) that the function u constructed in Section 3 vanishes at x = L. Note that, due to the fact that ρ f (t, 0) = ρ b (t, L) = 0, we have the following identities 19) which will be used in the sequel.…”

“…A local exact controllability result for the one dimensional isentropic Euler equation in the context of weak entropy solutions was established by Glass in [10]. Let us also mention a result of approximate controllability in the 3-dimensional case by means of a finite number of modes, see Nersisyan [19].…”

Maximal Dissipation and Well-posedness for the Compressible Euler System

“…For a given initial distribution of the density ̺ 0 , find a velocity field u 0 such that the corresponding solution of the compressible Euler system reaches the equilibrium state [̺, 0] at the time T , see Nersisyan [16], and Ervedoza et al [12] for related results in the viscous case. [10] for the incompressible Euler system, we may construct a different extension (as a matter of fact inifinitely many), namely ̺ =̺, u(T, ·) = 0, |u(t, ·)| = α > 0 for all t > T, where α is chosen small enough so that…”

Local exact controllability for the 1-d compressible Navier-Stokes equations

“…Let η 0 be a smooth cut-off function taking value 1 on {x ∈ T L , with d(x, Ω) ≤ 5ε + |u|T 0 } and vanishing on {x ∈ T L , d(x, Ω) ≥ 6ε + |u|T 0 }. We then introduce η the solution of 23) and the solutions r f and r b (here 'f ' stands for forward, 'b' for backward) of…”