no. 3
Partial differential equations
Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions
[Contrôlabilité exacte aux trajectoires pour des lois de conservation scalaires multidimensionnelles]

Présenté par : Jean-Michel Coron

Donadello, Carlotta1 ; Perrollaz, Vincent2
1 Université de Bourgogne Franche-Comté, Laboratoire de mathématiques, CNRS UMR6623, 16, route de Gray, 25000 Besançon, France
2 Université de Tours, Institut Denis-Poisson, CNRS UMR 7013, Parc de Grandmont, 37000 Tours, France
Comptes Rendus. Mathématique, Tome 357 (2019) no. 3, pp. 263-271.

On décrit dans cet article une nouvelle méthode permettant d'obtenir un résultat de contrôlabilité exacte aux trajectoires pour des lois de conservation scalaires en plusieurs dimensions d'espace dans le cadre des solutions entropiques et sous une simple hypothèse de non-dégénérescence du flux et une hypothèse géométrique naturelle.

We describe a new method that allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a natural geometric condition.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.01.012
Donadello, Carlotta 1 ; Perrollaz, Vincent 2

1 Université de Bourgogne Franche-Comté, Laboratoire de mathématiques, CNRS UMR6623, 16, route de Gray, 25000 Besançon, France
2 Université de Tours, Institut Denis-Poisson, CNRS UMR 7013, Parc de Grandmont, 37000 Tours, France 
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Donadello, Carlotta; Perrollaz, Vincent. Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions. Comptes Rendus. Mathématique, Tome 357 (2019) no. 3, pp. 263-271. doi : 10.1016/j.crma.2019.01.012. http://www.numdam.org/articles/10.1016/j.crma.2019.01.012/

[1] Adimurthi; Ghoshal, S.S.; Gowda, V. Exact controllability of scalar conservation laws with strictly convex flux, Math. Control Relat. Fields, Volume 4 (2014), pp. 401-449

[2] Ammar, K.; Wittbold, P.; Carillo, J. Scalar conservation laws with general boundary conditions and continuous flux function, J. Differ. Equ., Volume 228 (2006), pp. 111-139

[3] Ancona, F.; Coclite, G.M. On the attainable set for Temple class systems with boundary controls, SIAM J. Control Optim., Volume 43 (2005) no. 6, pp. 2166-2190

[4] Ancona, F.; Marson, A. On the attainable set for scalar nonlinear conservation laws with boundary control, SIAM J. Control Optim., Volume 36 (1998) no. 1, pp. 290-312

[5] Ancona, F.; Marson, A. 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, American Mathematical Society, Providence, RI, USA, 2007, pp. 1-43

[6] Andreianov, B.; Donadello, C.; Ghoshal, S.S.; Razafison, U. On the attainable set for a class of triangular systems of conservation laws, J. Evol. Equ., Volume 15 (2015) no. 3, pp. 503-532

[7] Andreianov, B.; Donadello, C.; Marson, A. On the attainable set for a scalar nonconvex conservation law, SIAM J. Control Optim., Volume 55 (2017) no. 4, pp. 2235-2270

[8] Bardos, C.; Leroux, A.-Y.; Nédélec, J.-C. First order quasilinear equations with boundary conditions, Commun. Partial Differ. Equ., Volume 9 (1979), pp. 1017-1034

[9] Bastin, G.; Coron, J.-M. Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Springer PLNDE Subseries in Control, vol. 88, 2016

[10] Blandin, S.; Litrico, X.; Dalle Monache, M.L.; Piccoli, B.; Bayen, T. Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws, IEEE Trans. Autom. Control, Volume 62 (2017) no. 4, pp. 1620-1635

[11] Bressan, A. Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000

[12] Bressan, A.; Coclite, G.M. On the boundary control of systems of conservation laws, SIAM J. Control Optim., Volume 41 (2002) no. 2, pp. 607-622

[13] Chapouly, M. Global controllability of non-viscous and viscous Burgers type equations, SIAM J. Control Optim., Volume 48 (2009) no. 3, pp. 1567-1599

[14] Corghi, M.; Marson, A. On the attainable set for scalar balance laws with distributed control, ESAIM Control Optim. Calc. Var., Volume 22 (2016), pp. 236-266

[15] Coron, J.-M. On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim., Volume 37 (1999) no. 6, pp. 1874-1896

[16] Coron, J.-M. Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, USA, 2007

[17] Coron, J.-M.; Bastin, G.; d'Andréa-Novel, B. Dissipative boundary conditions for one dimensional nonlinear hyperbolic systems, SIAM J. Control Optim., Volume 47 (2008) no. 3, pp. 1460-1498

[18] Coron, J.-M.; Ervedoza, S.; Ghoshal, S.S.; Glass, O.; Perrollaz, V. Dissipative boundary conditions for 2×2 hyperbolic systems of conservation laws for entropy solutions in BV, J. Differ. Equ., Volume 262 (2017), pp. 1-30

[19] Dafermos, C.M. Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., Volume 26 (1977), pp. 1097-1119

[20] Dafermos, C.M. Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, vol. 325, Springer-Verlag, Berlin, 2005

[21] Glass, O. On the controllability of the 1-D isentropic Euler equation, J. Eur. Math. Soc., Volume 9 (2007) no. 3, pp. 427-486

[22] Glass, O. On the controllability of the non-isentropic 1-D Euler equation, J. Differ. Equ., Volume 257 (2014), pp. 638-719

[23] Glass, O.; Guerrero, S. On the uniform controllability of the Burgers equation, SIAM J. Control Optim., Volume 46 (2007) no. 4, pp. 1211-1238

[24] Horsin, T. On the controllability of the Burger equation, ESAIM Control Optim. Calc. Var., Volume 3 (1998), pp. 83-95

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

[26] Léautaud, M. Uniform controllability of scalar conservation laws in the vanishing viscosity limit, SIAM J. Control Optim., Volume 50 (2012), pp. 1661-1699

[27] Leroux, A.Y. Étude du problème mixte pour une équation quasi linéaire du premier ordre, C. R. Acad. Sci. Paris, Ser. A, Volume 285 (1977), pp. 351-354

[28] Li, T. Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, vol. 3, American Institute of Mathematical Sciences (AIMS), 2010

[29] Li, T.; Yu, L. One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws, J. Math. Pures Appl. (9), Volume 107 (2017) no. 1, pp. 1-40

[30] Malek, J.; Necas, J.; Rokyta, M.; Ruzicka, M. Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman & Hall, London, 1996

[31] Otto, F. Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris, Ser. I, Volume 322 (1996) no. 8, pp. 729-734

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

[33] Perrollaz, V. Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 5, pp. 879-915

[34] Perrollaz, V. Asymptotic stabilization of stationary shock waves using boundary feedback law (preprint) | arXiv

[35] Vasseur, A. Strong traces for solutions to multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., Volume 170 (2001) no. 3, pp. 181-193

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