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.
@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}, pages = {879--915}, publisher = {Elsevier}, volume = {30}, number = {5}, year = {2013}, doi = {10.1016/j.anihpc.2012.12.003}, mrnumber = {3103174}, zbl = {06295445}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2012.12.003/} }
TY - JOUR AU - Perrollaz, Vincent TI - Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law JO - Annales de l'I.H.P. Analyse non linéaire PY - 2013 SP - 879 EP - 915 VL - 30 IS - 5 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2012.12.003/ DO - 10.1016/j.anihpc.2012.12.003 LA - en ID - AIHPC_2013__30_5_879_0 ER -
%0 Journal Article %A Perrollaz, Vincent %T Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law %J Annales de l'I.H.P. Analyse non linéaire %D 2013 %P 879-915 %V 30 %N 5 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2012.12.003/ %R 10.1016/j.anihpc.2012.12.003 %G en %F 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, Tome 30 (2013) no. 5, pp. 879-915. doi : 10.1016/j.anihpc.2012.12.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2012.12.003/
[1] On the attainable set for Temple class systems with boundary controls, SIAM J. Control Optim. 43 no. 6 (2005), 2166-2190 | MR | Zbl
, ,[2] On the attainable set for scalar nonlinear conservation laws with boundary control, SIAM J. Control Optim. 36 no. 1 (1998), 290-312 | MR | Zbl
, ,[3] 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 | Zbl
, ,[4] On Lyapunov stability of linearized Saint-Venant equations for a sloping channel, Netw. Heterog. Media 4 no. 2 (2009), 177-187 | MR | Zbl
, , ,[5] Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim. 47 no. 3 (2008), 1460-1498 | MR | Zbl
, , ,[6] First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 9 (1979), 1017-1034 | MR | Zbl
, , ,[7] On the boundary control of systems of conservation laws, SIAM J. Control Optim. 41 no. 2 (2002), 607-622 | MR | Zbl
, ,[8] Global controllability of non-viscous and viscous Burgers type equations, SIAM J. Control Optim. 48 no. 3 (2009), 1567-1599 | MR | Zbl
,[9] Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems 5 no. 3 (1992), 295-312 | MR | Zbl
,[10] Control and Nonlinearity, Math. Surveys Monogr. vol. 136, Amer. Math. Soc., Providence, RI (2007) | MR
,[11] Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 no. 6 (1977), 1097-1119 | MR | Zbl
,[12] Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss. vol. 325, Springer-Verlag, Berlin (2005) | MR | Zbl
,[13] Differential equations with discontinuous right-hand side, Mat. Sb. (N.S.) 51 no. 93 (1960), 99-128 | MR | Zbl
,[14] On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc. 9 no. 3 (2007), 427-486 | EuDML | MR | Zbl
,[15] On the uniform controllability of the Burgers equation, SIAM J. Control Optim. 46 no. 4 (2007), 1211-1238 | MR | Zbl
, ,[16] Lectures on Nonlinear Hyperbolic Differential Equations, Math. Appl. vol. 26, Springer-Verlag, Berlin (1997) | Zbl
,[17] On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var. 3 (1998), 83-95 | EuDML | Numdam | Zbl
,[18] 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
,[19] Etude du problème mixte pour une équation quasi-linéaire du premier ordre, C. R. Acad. Sci. Paris Sér. A 285 (1977), 351-354 | Zbl
,[20] On the convergence of the Godounovʼs scheme for first order quasi linear equations, Proc. Japan Acad. 52 no. 9 (1976), 488-491
,[21] First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81 no. 123 (1970), 228-255
,[22] Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London (1996) | Zbl
, , , ,[23] Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. Ser. 2 26 (1963), 95-172 | Zbl
,[24] Initial–boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math. 322 no. 8 (1996), 729-734 | Zbl
,[25] 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
,[26] The global solution of the scalar nonconvex conservation law with boundary condition, J. Partial Differ. Equ. 11 no. 1 (1998), 1-8 | Zbl
, ,[27] Comparison properties for scalar conservation laws with boundary conditions, Nonlinear Anal. 28 no. 4 (1997), 633-653 | Zbl
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