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January  2016, 36(1): 279-302. doi: 10.3934/dcds.2016.36.279

Bang-bang property of time optimal controls of semilinear parabolic equation

1. 

Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz

2. 

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072

Received  June 2014 Revised  March 2015 Published  June 2015

The bang-bang property of time optimal controls for a semilinear parabolic equation, with homogeneous Dirichlet boundary condition 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 a linear parabolic equation, with potential 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. 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
References:
[1]

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

[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.  Google Scholar

[3]

H. O. Fattorini, Time optimal control of solutions of operational differential equations,, J. SIAM Control, 2 (1964), 54.  doi: 10.1137/0302005.  Google Scholar

[4]

H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems,, North-Holland Mathematics Studies 201, (2005).   Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

K. Kunisch and L. J. Wang, Bang-bang property of time optimal controls of Burgers equation,, Discrete Contin. Dyn. Syst., 34 (2014), 3611.  doi: 10.3934/dcds.2014.34.3611.  Google Scholar

[8]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[9]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer, (1971).   Google Scholar

[10]

V. J. Mizel and T. I. Seidman, An abstract bang-bang principle and time optimal boundary control of the heat equation,, SIAM J. Control Optim., 35 (1997), 1204.  doi: 10.1137/S0363012996265470.  Google Scholar

[11]

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.  Google Scholar

[12]

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, 31 (2014), 477.  doi: 10.1016/j.anihpc.2013.04.005.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

H. O. Fattorini, Time optimal control of solutions of operational differential equations,, J. SIAM Control, 2 (1964), 54.  doi: 10.1137/0302005.  Google Scholar

[4]

H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems,, North-Holland Mathematics Studies 201, (2005).   Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

K. Kunisch and L. J. Wang, Bang-bang property of time optimal controls of Burgers equation,, Discrete Contin. Dyn. Syst., 34 (2014), 3611.  doi: 10.3934/dcds.2014.34.3611.  Google Scholar

[8]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of parabolic Type,, American Mathematical Society, (1968).   Google Scholar

[9]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer, (1971).   Google Scholar

[10]

V. J. Mizel and T. I. Seidman, An abstract bang-bang principle and time optimal boundary control of the heat equation,, SIAM J. Control Optim., 35 (1997), 1204.  doi: 10.1137/S0363012996265470.  Google Scholar

[11]

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.  Google Scholar

[12]

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, 31 (2014), 477.  doi: 10.1016/j.anihpc.2013.04.005.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

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