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The bang-bang property of time optimal controls for the Burgers equation
1. | Institut für Mathematik, Karl-Franzens-Universität Graz, A-8010 Graz, Austria |
2. | School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China |
References:
[1] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. |
[2] |
V. Barbu, The time optimal control of Navier-Stokes equations, Systems Control Lett., 30 (1997), 93-100.
doi: 10.1016/S0167-6911(96)00083-7. |
[3] |
V. Barbu and M. Röckner, Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise, Arch. Ration. Mech. Anal., 209 (2013), 797-834.
doi: 10.1007/s00205-013-0632-x. |
[4] |
C. Bardos and L. Tartar, Sur l'unicité retrograde des équations paraboliques et quelques questions voisines, Arch. Ration. Mech. Anal., 50 (1973), 10-25. |
[5] |
J. M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Acad. Sci. Amsterdam, 43 (1940), 2-12. |
[6] |
A. B. Cruzeiro and E. Shamarova, On a forward-backward stochastic system associated to the Burgers equation, in Stochastic Analysis with Financial Applications, Springer Basel AG, (2011), 43-59.
doi: 10.1007/978-3-0348-0097-6_4. |
[7] |
H. O. Fattorini, Time optimal control of solutions of operational differential equations, SIAM J. Control, 2 (1964), 54-59.
doi: 10.1137/0302005. |
[8] |
H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies 201, ELSEVIER, 2005. |
[9] |
E. Fernández-Cara and S. Guerrero, Remarks on the null controllability of the Burgers equation, C. R. Acad. Sci. Paris, Ser. I, 341 (2005), 229-232.
doi: 10.1016/j.crma.2005.06.005. |
[10] |
E. Fernández-Cara and S. Guerrero, Null controllability of the Burgers system with distributed controls, Systems Control Lett., 56 (2007), 366-372.
doi: 10.1016/j.sysconle.2006.10.022. |
[11] |
A. V. Fursikov and O. Yu. Imanuvilov, On controllability of certain systems simulating a fluid flow, in Flow Control, Springer Verlag, New York, 68 (1995), 149-184.
doi: 10.1007/978-1-4612-2526-3_7. |
[12] |
T. Horsin, On the controllability of the Burgers equations, ESAIM: Control Optim. Calc. Var., 3 (1998), 83-95.
doi: 10.1051/cocv:1998103. |
[13] |
K. Kunisch and L. J. Wang, Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints, J. Math. Anal. Appl., 395 (2012), 114-130.
doi: 10.1016/j.jmaa.2012.05.028. |
[14] |
K. Kunisch and L. J. Wang, Time optimal control of the heat equation with pointwise control constraints, ESAIM: Control Optim. Calc. Var., 19 (2013), 460-485.
doi: 10.1051/cocv/2012017. |
[15] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of parabolic Type, American Mathematical Society, 1968. |
[16] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. |
[17] |
J. Lohéac and M. Tucsnak, Maximum principle and bang-bang property of time optimal control for Schrödinger type systems, Available from http://hal.archives-ouvertes.fr/docs/00/75/86/82/PDF/Loheac_Tucsnak.pdf. |
[18] |
K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.
doi: 10.1016/j.jfa.2010.04.015. |
[19] |
K. D. Phung and G. S. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[20] |
K. D. Phung, L. J. Wang and C. Zhang, Bang-bang Property for Time Optimal Control of Semilinear Heat Equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2013.
doi: 10.1016/j.anihpc.2013.04.005. |
[21] |
A. R. Bahadir, A fully implicit finite-difference scheme for two-dimensional Burgers' equation, Appl. Math. Comput., 137 (2003), 131-137.
doi: 10.1016/S0096-3003(02)00091-7. |
[22] |
F. Tröltzsch and S. Volkwein, The SQP method for control constrained optimal control of the Burgers equation, ESAIM: Control Optim. Calc. Var., 6 (2001), 649-674.
doi: 10.1051/cocv:2001127. |
[23] |
G. S. Wang, $L^\infty$-null controllability for the heat equtaion and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720.
doi: 10.1137/060678191. |
[24] |
G. S. Wang and L. J. Wang, The Bang-Bang principle of time optimal controls for the heat equation with internal controls, Systems Control Lett., 56 (2007), 709-713.
doi: 10.1016/j.sysconle.2007.06.001. |
[25] |
L. J. Wang and G. S. Wang, The optimal time control of a phase-field system, SIAM J. Control Optim., 42 (2003), 1483-1508.
doi: 10.1137/S0363012902405455. |
show all references
References:
[1] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, 1993. |
[2] |
V. Barbu, The time optimal control of Navier-Stokes equations, Systems Control Lett., 30 (1997), 93-100.
doi: 10.1016/S0167-6911(96)00083-7. |
[3] |
V. Barbu and M. Röckner, Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise, Arch. Ration. Mech. Anal., 209 (2013), 797-834.
doi: 10.1007/s00205-013-0632-x. |
[4] |
C. Bardos and L. Tartar, Sur l'unicité retrograde des équations paraboliques et quelques questions voisines, Arch. Ration. Mech. Anal., 50 (1973), 10-25. |
[5] |
J. M. Burgers, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Acad. Sci. Amsterdam, 43 (1940), 2-12. |
[6] |
A. B. Cruzeiro and E. Shamarova, On a forward-backward stochastic system associated to the Burgers equation, in Stochastic Analysis with Financial Applications, Springer Basel AG, (2011), 43-59.
doi: 10.1007/978-3-0348-0097-6_4. |
[7] |
H. O. Fattorini, Time optimal control of solutions of operational differential equations, SIAM J. Control, 2 (1964), 54-59.
doi: 10.1137/0302005. |
[8] |
H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies 201, ELSEVIER, 2005. |
[9] |
E. Fernández-Cara and S. Guerrero, Remarks on the null controllability of the Burgers equation, C. R. Acad. Sci. Paris, Ser. I, 341 (2005), 229-232.
doi: 10.1016/j.crma.2005.06.005. |
[10] |
E. Fernández-Cara and S. Guerrero, Null controllability of the Burgers system with distributed controls, Systems Control Lett., 56 (2007), 366-372.
doi: 10.1016/j.sysconle.2006.10.022. |
[11] |
A. V. Fursikov and O. Yu. Imanuvilov, On controllability of certain systems simulating a fluid flow, in Flow Control, Springer Verlag, New York, 68 (1995), 149-184.
doi: 10.1007/978-1-4612-2526-3_7. |
[12] |
T. Horsin, On the controllability of the Burgers equations, ESAIM: Control Optim. Calc. Var., 3 (1998), 83-95.
doi: 10.1051/cocv:1998103. |
[13] |
K. Kunisch and L. J. Wang, Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints, J. Math. Anal. Appl., 395 (2012), 114-130.
doi: 10.1016/j.jmaa.2012.05.028. |
[14] |
K. Kunisch and L. J. Wang, Time optimal control of the heat equation with pointwise control constraints, ESAIM: Control Optim. Calc. Var., 19 (2013), 460-485.
doi: 10.1051/cocv/2012017. |
[15] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of parabolic Type, American Mathematical Society, 1968. |
[16] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. |
[17] |
J. Lohéac and M. Tucsnak, Maximum principle and bang-bang property of time optimal control for Schrödinger type systems, Available from http://hal.archives-ouvertes.fr/docs/00/75/86/82/PDF/Loheac_Tucsnak.pdf. |
[18] |
K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.
doi: 10.1016/j.jfa.2010.04.015. |
[19] |
K. D. Phung and G. S. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[20] |
K. D. Phung, L. J. Wang and C. Zhang, Bang-bang Property for Time Optimal Control of Semilinear Heat Equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2013.
doi: 10.1016/j.anihpc.2013.04.005. |
[21] |
A. R. Bahadir, A fully implicit finite-difference scheme for two-dimensional Burgers' equation, Appl. Math. Comput., 137 (2003), 131-137.
doi: 10.1016/S0096-3003(02)00091-7. |
[22] |
F. Tröltzsch and S. Volkwein, The SQP method for control constrained optimal control of the Burgers equation, ESAIM: Control Optim. Calc. Var., 6 (2001), 649-674.
doi: 10.1051/cocv:2001127. |
[23] |
G. S. Wang, $L^\infty$-null controllability for the heat equtaion and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720.
doi: 10.1137/060678191. |
[24] |
G. S. Wang and L. J. Wang, The Bang-Bang principle of time optimal controls for the heat equation with internal controls, Systems Control Lett., 56 (2007), 709-713.
doi: 10.1016/j.sysconle.2007.06.001. |
[25] |
L. J. Wang and G. S. Wang, The optimal time control of a phase-field system, SIAM J. Control Optim., 42 (2003), 1483-1508.
doi: 10.1137/S0363012902405455. |
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