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

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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

Abstract
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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 & 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.

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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.

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