Time optimal control of the heat equation with pointwise control constraints

Karl Kunisch; Lijuan Wang

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 2, page 460-485
  • ISSN: 1292-8119

Abstract

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Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained.

How to cite

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Kunisch, Karl, and Wang, Lijuan. "Time optimal control of the heat equation with pointwise control constraints." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 460-485. <http://eudml.org/doc/272753>.

@article{Kunisch2013,
abstract = {Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained.},
author = {Kunisch, Karl, Wang, Lijuan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {bang-bang property; time optimal control; norm optimal control},
language = {eng},
number = {2},
pages = {460-485},
publisher = {EDP-Sciences},
title = {Time optimal control of the heat equation with pointwise control constraints},
url = {http://eudml.org/doc/272753},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Kunisch, Karl
AU - Wang, Lijuan
TI - Time optimal control of the heat equation with pointwise control constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 2
SP - 460
EP - 485
AB - Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained.
LA - eng
KW - bang-bang property; time optimal control; norm optimal control
UR - http://eudml.org/doc/272753
ER -

References

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  1. [1] N. Arada and J.P. Raymond, Dirichlet boundary control of semilinear parabolic equations, Part 1 : Problems with no state constraints. Appl. Math. Optim. 45 (2002) 125–143. Zbl1005.49016MR1874072
  2. [2] N. Arada and J.P. Raymond, Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Syst.9 (2003) 1549–1570. Zbl1076.49012MR2017681
  3. [3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston (1993). Zbl0776.49005MR1195128
  4. [4] V. Barbu, The time optimal control of Navier-Stokes equations. Syst. Control Lett.30 (1997) 93–100. Zbl0898.49011MR1449630
  5. [5] R.E. Bellman, I. Glicksberg and O.A. Gross, On the “bang-bang” control problem. Q. Appl. Math.14 (1956) 11–18. Zbl0073.11501
  6. [6] C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinburgh125 (1995) 31–61. Zbl0818.93032MR1318622
  7. [7] H.O. Fattorini, Time optimal control of solutions of operational differential equations. SIAM J. Control2 (1964) 54–59. Zbl0143.16803MR169738
  8. [8] H.O. Fattorini, Infinite Dimensional Linear Control Systems : The Time Optimal and Norm Optimal Problems. North-Holland Math. Stud. 201 (2005). Zbl1135.93001MR2158806
  9. [9] H.O. Fattorini, Sufficiency of the maximum principle for time optimality. Cubo : A. Math. J. 7 (2005) 27–37. Zbl1125.49020MR2191045
  10. [10] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17 (2000) 583–616. Zbl0970.93023MR1791879
  11. [11] A.V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications. American Mathematical Society, Providence (2000). Zbl1027.93500MR1726442
  12. [12] K. Kunisch and L.J. Wang, Time optimal controls of the linear Fitzhugh–Nagumo equation with pointwise control constraints. J. Math. Anal. Appl. (2012), doi: 10.1016/j.jmaa.2012.05.028. Zbl1251.35174MR2943607
  13. [13] X.J. Li and J.M. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston (1995). Zbl0817.49001MR1312364
  14. [14] J.L. Lions, Remarques sur la contrôlabilité approchée, in Jornadas Hispano-Francesas Sobre Control de Sistemas Distribuidos. University of Málaga, Spain (1991) 77–87. Zbl0752.93037MR1108876
  15. [15] J.L. Lions, Remarks on approximate controllability. J. Anal. Math.59 (1992) 103–116. Zbl0806.35101MR1226954
  16. [16] S. Micu and E. Zuazua, An introduction to the controllability of partial differential equations, in Quelques questions de théorie du contròle, edited by T. Sari. Collection Travaux en Cours Hermann (2004) 69–157. Zbl1231.93042
  17. [17] 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. Zbl0891.49014MR1453296
  18. [18] J.P. Raymond and H. Zidani, Pontryagin’s principle for time-optimal problems. J. Optim. Theory Appl.101 (1999) 375–402. Zbl0952.49020MR1684676
  19. [19] E.J.P.G. Schmidt, The “bang-bang” principle for the time-optimal problem in boundary control of the heat equation. SIAM J. Control Optim.18 (1980) 101–107. Zbl0441.93022MR560041
  20. [20] G.S. Wang and L.J. Wang, The bang-bang principle of time optimal controls for the heat equation with internal controls. Syst. Control Lett.56 (2007) 709–713. Zbl1120.49002MR2356456
  21. [21] L.J. Wang and G.S. Wang, The optimal time control of a phase-field system. SIAM J. Control Optim.42 (2003) 1483–1508. Zbl1048.93034MR2044806
  22. [22] G.S. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for heat equations. SIAM J. Control Optim.50 (2012) 2938–2958. Zbl1257.49005MR3022093
  23. [23] Z.Q. Wu, J.X. Yin and C.P. Wang, Elliptic and Parabolic Equations. World Scientific Publishing Corporation, New Jersey (2006). Zbl1108.35001MR2309679
  24. [24] E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control Cybern.28 (1999) 665–683. Zbl0959.93025MR1782020

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