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

Karl Kunisch; Lijuan Wang

doi:10.3934/dcds.2016.36.279

Article Contents

2016, Volume 36, Issue 1: 279-302. Doi: 10.3934/dcds.2016.36.279

This issue Previous Article Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation Next Article Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds

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

Karl Kunisch^1, and
Lijuan Wang^2,

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: May 2017

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Semilinear parabolic equation,
bang-bang property,
time optimal control,
observability estimate from measurable sets.

Mathematics Subject Classification: Primary: 35K10, 49J20; Secondary: 49J30, 93B07.

Citation:

Related Papers

Cited by

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]	H. O. Fattorini, Time optimal control of solutions of operational differential equations, J. SIAM Control, 2 (1964), 54-59.doi: 10.1137/0302005.
[4]	H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies 201, ELSEVIER, 2005.
[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), 114-130.doi: 10.1016/j.jmaa.2012.05.028.
[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), 460-485.doi: 10.1051/cocv/2012017.
[7]	K. Kunisch and L. J. Wang, Bang-bang property of time optimal controls of Burgers equation, Discrete Contin. Dyn. Syst., 34 (2014), 3611-3637.doi: 10.3934/dcds.2014.34.3611.
[8]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of parabolic Type, American Mathematical Society, 1968.
[9]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.
[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), 1204-1216.doi: 10.1137/S0363012996265470.
[11]	K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.doi: 10.1016/j.jfa.2010.04.015.
[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), 477-499.doi: 10.1016/j.anihpc.2013.04.005.
[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), 1701-1720.doi: 10.1137/060678191.
[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), 709-713.doi: 10.1016/j.sysconle.2007.06.001.
[15]	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.