Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T15:59:41.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Time optimal control of the heat equation with pointwisecontrol constraints***

Published online by Cambridge University Press:  15 February 2013

Karl Kunisch  and
Lijuan Wang
Karl Kunisch
Affiliation:
Institut für Mathematik, Karl-Franzens-Universität Graz, A-8010 Graz, Austria. [email protected]
Lijuan Wang*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China; [email protected]
*
∗Corresponding author.
Get access

Abstract

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

Keywords

Bang-bang propertytime optimal controlnorm optimal control
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 460 - 485
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Supported in part by theFonds zur Förderung der wissenschaftlichen Forschung under SFB 32, “MathematicalOptimization and Applications in Biomedical Sciences”.

**

This work was carried out, inpart, while the author was guest-researcher at the Radon Institute, Linz, supported bythe Austrian Academy of Sciences. It was also partially supported by the NationalNatural Science Foundation of China under Grants Nos. 10971158 and11161130003.

References

Arada, N. and Raymond, J.P., Dirichlet boundary control of semilinear parabolic equations, Part 1 : Problems with no state constraints. Appl. Math. Optim. 45 (2002) 125–143. Google Scholar
Arada, N. and Raymond, J.P., Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Syst. 9 (2003) 1549–1570. Google Scholar
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston (1993).
Barbu, V., The time optimal control of Navier-Stokes equations. Syst. Control Lett. 30 (1997) 93–100. Google Scholar
Bellman, R.E., Glicksberg, I. and Gross, O.A., On the “bang-bang” control problem. Q. Appl. Math. 14 (1956) 11–18. Google Scholar
Fabre, C., Puel, J.P. and Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinburgh 125 (1995) 31–61. Google Scholar
Fattorini, H.O., Time optimal control of solutions of operational differential equations. SIAM J. Control 2 (1964) 54–59. Google Scholar
Fattorini, H.O., Infinite Dimensional Linear Control Systems : The Time Optimal and Norm Optimal Problems. North-Holland Math. Stud. 201 (2005). Google Scholar
Fattorini, H.O., Sufficiency of the maximum principle for time optimality. Cubo : A. Math. J. 7 (2005) 27–37. Google Scholar
Fernández-Cara, E. and Zuazua, E., Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17 (2000) 583–616. Google Scholar
A.V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications. American Mathematical Society, Providence (2000).
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. CrossRef
X.J. Li and J.M. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston (1995).
J.L. Lions, Remarques sur la contrôlabilité approchée, in Jornadas Hispano-Francesas Sobre Control de Sistemas Distribuidos. University of Málaga, Spain (1991) 77–87.
Lions, J.L., Remarks on approximate controllability. J. Anal. Math. 59 (1992) 103–116. Google Scholar
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.
Mizel, V.J. and Seidman, T.I., An abstract bang-bang principle and time optimal boundary control of the heat equation. SIAM J. Control Optim. 35 (1997) 1204–1216. Google Scholar
Raymond, J.P. and Zidani, H., Pontryagin’s principle for time-optimal problems. J. Optim. Theory Appl. 101 (1999) 375–402. Google Scholar
Schmidt, E.J.P.G., The “bang-bang” principle for the time-optimal problem in boundary control of the heat equation. SIAM J. Control Optim. 18 (1980) 101–107. Google Scholar
Wang, G.S. and Wang, L.J., The bang-bang principle of time optimal controls for the heat equation with internal controls. Syst. Control Lett. 56 (2007) 709–713. Google Scholar
Wang, L.J. and Wang, G.S., The optimal time control of a phase-field system. SIAM J. Control Optim. 42 (2003) 1483–1508. Google Scholar
Wang, G.S. and Zuazua, E., On the equivalence of minimal time and minimal norm controls for heat equations. SIAM J. Control Optim. 50 (2012) 2938–2958. Google Scholar
Z.Q. Wu, J.X. Yin and C.P. Wang, Elliptic and Parabolic Equations. World Scientific Publishing Corporation, New Jersey (2006).
Zuazua, E., Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control Cybern. 28 (1999) 665–683. Google Scholar