Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 5, p. 879-915

In this paper, we study the problem of asymptotic stabilization by closed loop feedback for a scalar conservation law with a convex flux and in the context of entropy solutions. Besides the boundary data, we use an additional control which is a source term acting uniformly in space.

DOI : https://doi.org/10.1016/j.anihpc.2012.12.003
Keywords: Asymptotic stabilization, Conservation law, Entropy solution, Generalized characteristics, PDE control
@article{AIHPC_2013__30_5_879_0,
author = {Perrollaz, Vincent},
title = {Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {30},
number = {5},
year = {2013},
pages = {879-915},
doi = {10.1016/j.anihpc.2012.12.003},
zbl = {06295445},
mrnumber = {3103174},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2013__30_5_879_0}
}

Perrollaz, Vincent. Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 5, pp. 879-915. doi : 10.1016/j.anihpc.2012.12.003. http://www.numdam.org/item/AIHPC_2013__30_5_879_0/

[1] F. Ancona, G.M. Coclite, On the attainable set for Temple class systems with boundary controls, SIAM J. Control Optim. 43 no. 6 (2005), 2166-2190 | MR 2179483 | Zbl 1087.93010

[2] F. Ancona, A. Marson, On the attainable set for scalar nonlinear conservation laws with boundary control, SIAM J. Control Optim. 36 no. 1 (1998), 290-312 | MR 1616586 | Zbl 0919.35082

[3] F. Ancona, A. Marson, Asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point, Control Methods in PDE-Dynamical Systems, Contemp. Math. vol. 426, Amer. Math. Soc., Providence, RI (2007), 1-43 | MR 2311519 | Zbl 1128.35069

[4] G. Bastin, J.-M. Coron, B. DʼAndréa-Novel, On Lyapunov stability of linearized Saint-Venant equations for a sloping channel, Netw. Heterog. Media 4 no. 2 (2009), 177-187 | MR 2525202 | Zbl 1185.37175

[5] J.-M. Coron, G. Bastin, B. DʼAndréa-Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim. 47 no. 3 (2008), 1460-1498 | MR 2407024 | Zbl 1172.35008

[6] C. Bardos, A.Y. Leroux, J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 9 (1979), 1017-1034 | MR 542510 | Zbl 0418.35024

[7] A. Bressan, G.M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim. 41 no. 2 (2002), 607-622 | MR 1920513 | Zbl 1026.35075

[8] M. Chapouly, Global controllability of non-viscous and viscous Burgers type equations, SIAM J. Control Optim. 48 no. 3 (2009), 1567-1599 | MR 2516179 | Zbl 1282.93050

[9] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems 5 no. 3 (1992), 295-312 | MR 1164379 | Zbl 0760.93067

[10] J.-M. Coron, Control and Nonlinearity, Math. Surveys Monogr. vol. 136, Amer. Math. Soc., Providence, RI (2007) | MR 2302744

[11] C.M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 no. 6 (1977), 1097-1119 | MR 457947 | Zbl 0377.35051

[12] C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss. vol. 325, Springer-Verlag, Berlin (2005) | MR 2169977 | Zbl 1078.35001

[13] A.F. Filippov, Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.) 51 no. 93 (1960), 99-128 | MR 114016 | Zbl 0138.32204

[14] O. Glass, On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. 9 no. 3 (2007), 427-486 | MR 2314104 | Zbl 1139.35014

[15] O. Glass, S. Guerrero, On the uniform controllability of the Burgers equation, SIAM J. Control Optim. 46 no. 4 (2007), 1211-1238 | MR 2346380 | Zbl 1140.93013

[16] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Math. Appl. vol. 26, Springer-Verlag, Berlin (1997) | Zbl 0881.35001

[17] T. Horsin, On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var. 3 (1998), 83-95 | Numdam | Zbl 0897.93034

[18] Tatsien Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math. vol. 3, American Institute of Mathematical Sciences (AIMS)/Higher Education Press, Springfield, MO/Beijing (2010) | Zbl 1198.93003

[19] A.Y. Leroux, Etude du problème mixte pour une équation quasi-linéaire du premier ordre, C. R. Acad. Sci. Paris Sér. A 285 (1977), 351-354 | Zbl 0366.35019

[20] A.Y. Leroux, On the convergence of the Godounovʼs scheme for first order quasi linear equations, Proc. Japan Acad. 52 no. 9 (1976), 488-491

[21] S.N. Kruzkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81 no. 123 (1970), 228-255

[22] J. Malek, J. Necas, M. Rokyta, M. Ruzicka, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London (1996) | Zbl 0851.35002

[23] O.A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. Ser. 2 26 (1963), 95-172 | Zbl 0131.31803

[24] F. Otto, Initial–boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math. 322 no. 8 (1996), 729-734 | Zbl 0852.35013

[25] V. Perrollaz, Exact controllability of scalar conservation laws with an additional control in the context of entropy solutions, SIAM J. Control Optim. 50 no. 4 (2012), 2025-2045 | Zbl 1252.93027

[26] T. Pan, L. Lin, The global solution of the scalar nonconvex conservation law with boundary condition, J. Partial Differ. Equ. 11 no. 1 (1998), 1-8 | Zbl 0905.35050

[27] A. Terracina, Comparison properties for scalar conservation laws with boundary conditions, Nonlinear Anal. 28 no. 4 (1997), 633-653 | Zbl 0873.35049