Local boundary controllability to trajectories for the 1d compressible Navier Stokes equations

Ervedoza, Sylvain; Savel, Marc

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 211-235.

In this article, we show a local exact boundary controllability result for the 1d isentropic compressible Navier Stokes equations around a smooth target trajectory. Our controllability result requires a geometric condition on the flow of the target trajectory, which comes naturally when dealing with the linearized equations. The proof of our result is based on a fixed point argument in weighted spaces and follows the strategy already developed in [S. Ervedoza, O. Glass, S. Guerrero, J.-P. Puel, Arch. Ration. Mech. Anal. 206 (2012) 189–238] in the case of a non-zero constant velocity field. The main novelty of this article is in the construction of the controlled density in the case of possible oscillations of the characteristics of the target flow on the boundary.

P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system. Electron. J. Differ. Equ. 22 (2000) 15. | MR | Zbl

E.V. Amosova, Exact local controllability for the equations of viscous gas dynamics. Differ. Uravneniya 47 (2011) 1754–1772. | MR | Zbl

M. Badra, S. Ervedoza and S. Guerrero, Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations. Ann. Inst. Henri Poincaré (C) Non Lin. Anal. (2014). | MR

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238 (2003) 211–223. | DOI | MR | Zbl

D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Commun. Partial Differ. Equ. 28 (2003) 843–868. | DOI | MR | Zbl

F.W. Chaves-Silva, L. Rosier and E. Zuazua, Null controllability of a system of viscoelasticity with a moving control. J. Math. Pures Appl. 101 (2014) 198–222. | DOI | MR | Zbl

S. Chowdhury, M. Debanjana, M. Ramaswamy and M. Renardy, Null controllability of the linearized compressible Navier Stokes system in one dimension. J. Differ. Equ. 257 (2014) 3813–3849. | DOI | MR | Zbl

S. Chowdhury, M. Ramaswamy and J.-P. Raymond, Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension. SIAM J. Control Optim. 50 (2012) 2959–2987. | DOI | MR | Zbl

J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1996) 35–75. | Numdam | MR | Zbl

J.-M. Coron, On the controllability of $2$-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155–188. | MR | Zbl

J.-M. Coron, Control and nonlinearity, Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). | MR | Zbl

J.-M. Coron and A.V. Fursikov, Global exact controllability of the $2$D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429–448. | MR | Zbl

S. Ervedoza, O. Glass, S. Guerrero and J.-P. Puel, Local exact controllability for the one-dimensional compressible Navier-Stokes equation. Arch. Ration. Mech. Anal. 206 (2012) 189–238. | DOI | MR | Zbl

E. Fernández-Cara, S. Guerrero, O.Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. | DOI | MR | Zbl

A.V. Fursikov and O. Yu. Èmanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Uspekhi Mat. Nauk 54 (1999) 93–146. | MR | Zbl

A.V. Fursikov and O.Y. Imanuvilov, Controllability of evolution equations, Vol. 34 of Lecture Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996). | MR | Zbl

O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1–44. | Numdam | MR | Zbl

O. Glass, On the controllability of the 1-D isentropic Euler equation. J. Eur. Math. Soc. (JEMS) 9 (2007) 427–486. | DOI | MR | Zbl

O. Glass, On the controllability of the non-isentropic 1-D Euler equation. J. Differ. Equ. 257 (2014) 638–719. | DOI | MR | Zbl

O.Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39–72. | Numdam | MR | Zbl

T.-T. Li and B.-P. Rao, Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Control Optim. 41 (2003) 1748–1755. | DOI | MR | Zbl

D. Maity, Some controllability results for linearized compressible Navier-Stokes system. ESAIM: COCV 21 (2015) 1002–1028. | Numdam | MR | Zbl

P. Martin, L. Rosier and P. Rouchon, Null controllability of the structurally damped wave equation with moving control. SIAM J. Control Optim. 51 (2013) 660–684. | DOI | MR | Zbl

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20 (1980) 67–104. | MR | Zbl

L. Rosier and P. Rouchon, On the controllability of a wave equation with structural damping. Int. J. Tomogr. Stat. 5 (2007) 79–84. | MR

J. Simon, Compact sets in the space $L^p(0,T;B)$. Ann. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl

H. Triebel, Interpolation theory, function spaces, differential operators. North-Holland Publishing Co. Amsterdam New York (1978). | MR | Zbl