The bang-bang property of time optimal controls for the Burgers equation

Previous Article
Invariant foliations for random dynamical systems
DCDS Home
This Issue
Next Article
Shadowing is generic---a continuous map case

September 2014, 34(9): 3611-3637. doi: 10.3934/dcds.2014.34.3611

The bang-bang property of time optimal controls for the Burgers equation

Karl Kunisch ^1, and Lijuan Wang ^2,

1.	Institut für Mathematik, Karl-Franzens-Universität Graz, A-8010 Graz, Austria
2.	School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

Received June 2013 Revised January 2014 Published March 2014

Abstract
Related Papers

The bang-bang property of time optimal controls for the Burgers equations in dimension up to three, with homogeneous Dirichlet boundary conditions and distributed controls acting on an open subset of the domain is established. This relies on an observability estimate from a measurable set in time for linear parabolic equations, with potentials depending on both space and time variables. The proof of the bang-bang property relies on a Kakutani fixed point argument.

Keywords: time optimal controls, bang-bang property, observability estimate from measurable sets., Burgers equation.

Mathematics Subject Classification: Primary: 49J20, 49J30; Secondary: 93B0.

Citation: Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3611-3637. doi: 10.3934/dcds.2014.34.3611

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.

[1]	Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279
[2]	Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial and Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443
[3]	Gengsheng Wang, Yubiao Zhang. Decompositions and bang-bang properties. Mathematical Control and Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005
[4]	M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014
[5]	Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547
[6]	M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511
[7]	Nicolas Hegoburu. Time optimal internal controls for the Lotka-McKendrick equation with spatial diffusion. Mathematical Control and Related Fields, 2019, 9 (4) : 697-718. doi: 10.3934/mcrf.2019047
[8]	Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1137-1155. doi: 10.3934/dcdsb.2011.16.1137
[9]	Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043
[10]	Xenia Kerkhoff, Sandra May. Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021054
[11]	Sebastian Engel, Karl Kunisch. Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach. Mathematical Control and Related Fields, 2020, 10 (3) : 591-622. doi: 10.3934/mcrf.2020012
[12]	Shulin Qin, Gengsheng Wang, Huaiqiang Yu. On switching properties of time optimal controls for linear ODEs. Mathematical Control and Related Fields, 2021, 11 (2) : 329-351. doi: 10.3934/mcrf.2020039
[13]	Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925
[14]	N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549
[15]	J. Leonel Rocha, Abdel-Kaddous Taha, Danièle Fournier-Prunaret. Explosion birth and extinction: Double big bang bifurcations and Allee effect in Tsoularis-Wallace's growth models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3131-3163. doi: 10.3934/dcdsb.2015.20.3131
[16]	Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[17]	Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160
[18]	Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47
[19]	Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219
[20]	Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control and Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Karl Kunisch Lijuan Wang
on Google Scholar
Karl Kunisch Lijuan Wang

[Back to Top]