On the controllability of the non-isentropic 1-D Euler equation

TL;DR

This paper investigates the boundary controllability of the 1-D non-isentropic Euler equation for compressible gas, demonstrating controllability toward constant states under certain conditions and maintaining small total variation.

Contribution

It provides new results on boundary controllability for both Eulerian and Lagrangian forms of the 1-D non-isentropic Euler system, including limitations on the adiabatic constant.

Findings

Controllability toward constant states achieved for both systems.

Solutions preserve small total variation if initially small.

Limitations on the adiabatic constant for Lagrangian system.

Abstract

In this paper, we examine the question of the boundary controllability of the one-dimensional non-isentropic Euler equation for compressible polytropic gas, in the context of weak entropy solutions. We consider the system in Eulerian coordinates and the one in Lagrangian coordinates. We obtain for both systems a result of controllability toward constant states (with the limitation $\gamma < 5/3$ on the adiabatic constant for the Lagrangian system). The solutions that we obtain remain of small total variation in space if the initial condition is itself of sufficiently small total variation.