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

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

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.
Citation: Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete & Continuous Dynamical Systems - A, 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, (1993).

[2]

V. Barbu, The time optimal control of Navier-Stokes equations,, Systems Control Lett., 30 (1997), 93. 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. 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.

[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.

[6]

A. B. Cruzeiro and E. Shamarova, On a forward-backward stochastic system associated to the Burgers equation,, in Stochastic Analysis with Financial Applications, (2011), 43. 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. 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, (2005).

[9]

E. Fernández-Cara and S. Guerrero, Remarks on the null controllability of the Burgers equation,, C. R. Acad. Sci. Paris, 341 (2005), 229. 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. 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, 68 (1995), 149. 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. 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. 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. 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, (1971).

[17]

J. Lohéac and M. Tucsnak, Maximum principle and bang-bang property of time optimal control for Schrödinger type systems,, Available from , ().

[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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0363012902405455.

show all references

References:
[1]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems,, Academic Press, (1993).

[2]

V. Barbu, The time optimal control of Navier-Stokes equations,, Systems Control Lett., 30 (1997), 93. 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. 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.

[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.

[6]

A. B. Cruzeiro and E. Shamarova, On a forward-backward stochastic system associated to the Burgers equation,, in Stochastic Analysis with Financial Applications, (2011), 43. 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. 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, (2005).

[9]

E. Fernández-Cara and S. Guerrero, Remarks on the null controllability of the Burgers equation,, C. R. Acad. Sci. Paris, 341 (2005), 229. 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. 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, 68 (1995), 149. 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. 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. 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. 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, (1971).

[17]

J. Lohéac and M. Tucsnak, Maximum principle and bang-bang property of time optimal control for Schrödinger type systems,, Available from , ().

[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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0363012902405455.

[1]

Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005

[4]

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 & Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547

[5]

Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1137-1155. doi: 10.3934/dcdsb.2011.16.1137

[6]

M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511

[7]

Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043

[8]

Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925

[9]

N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549

[10]

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 & Continuous Dynamical Systems - B, 2015, 20 (9) : 3131-3163. doi: 10.3934/dcdsb.2015.20.3131

[11]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15

[12]

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47

[13]

Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219

[14]

Vilmos Komornik, Paola Loreti. Observability of rectangular membranes and plates on small sets. Evolution Equations & Control Theory, 2014, 3 (2) : 287-304. doi: 10.3934/eect.2014.3.287

[15]

Fredi Tröltzsch, Daniel Wachsmuth. On the switching behavior of sparse optimal controls for the one-dimensional heat equation. Mathematical Control & Related Fields, 2018, 8 (1) : 135-153. doi: 10.3934/mcrf.2018006

[16]

Zhengchun Zhou, Xiaohu Tang. New nearly optimal codebooks from relative difference sets. Advances in Mathematics of Communications, 2011, 5 (3) : 521-527. doi: 10.3934/amc.2011.5.521

[17]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121

[18]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[19]

Y. Peng, X. Xiang. Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial & Management Optimization, 2008, 4 (1) : 17-32. doi: 10.3934/jimo.2008.4.17

[20]

Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]