Time optimal control of the heat equation with pointwise control constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 460-485.

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.

DOI : 10.1051/cocv/2012017
Classification : 35K05, 49J20, 49J30
Mots-clés : bang-bang property, time optimal control, norm optimal control 
@article{COCV_2013__19_2_460_0,
     author = {Kunisch, Karl and Wang, Lijuan},
     title = {Time optimal control of the heat equation with pointwise control constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {460--485},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {2},
     year = {2013},
     doi = {10.1051/cocv/2012017},
     mrnumber = {3049719},
     zbl = {1272.35109},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2012017/}
}
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
SP  - 460
EP  - 485
VL  - 19
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2012017/
DO  - 10.1051/cocv/2012017
LA  - en
ID  - COCV_2013__19_2_460_0
ER  - 
%0 Journal Article
%A Kunisch, Karl
%A Wang, Lijuan
%T Time optimal control of the heat equation with pointwise control constraints
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 460-485
%V 19
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2012017/
%R 10.1051/cocv/2012017
%G en
%F COCV_2013__19_2_460_0
Kunisch, Karl; Wang, Lijuan. Time optimal control of the heat equation with pointwise control constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 460-485. doi : 10.1051/cocv/2012017. http://archive.numdam.org/articles/10.1051/cocv/2012017/

[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. | MR | Zbl

[2] N. Arada and J.P. Raymond, Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Syst. 9 (2003) 1549-1570. | MR | Zbl

[3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston (1993). | MR | Zbl

[4] V. Barbu, The time optimal control of Navier-Stokes equations. Syst. Control Lett. 30 (1997) 93-100. | MR | Zbl

[5] R.E. Bellman, I. Glicksberg and O.A. Gross, On the “bang-bang” control problem. Q. Appl. Math. 14 (1956) 11-18. | Zbl

[6] C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinburgh 125 (1995) 31-61. | MR | Zbl

[7] H.O. Fattorini, Time optimal control of solutions of operational differential equations. SIAM J. Control 2 (1964) 54-59. | MR | Zbl

[8] H.O. Fattorini, Infinite Dimensional Linear Control Systems : The Time Optimal and Norm Optimal Problems. North-Holland Math. Stud. 201 (2005). | MR | Zbl

[9] H.O. Fattorini, Sufficiency of the maximum principle for time optimality. Cubo : A. Math. J. 7 (2005) 27-37. | MR | Zbl

[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. | Numdam | MR | Zbl

[11] A.V. Fursikov, Optimal Control of Distributed Systems, Theory and Applications. American Mathematical Society, Providence (2000). | MR | Zbl

[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. | MR | Zbl

[13] X.J. Li and J.M. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston (1995). | MR | Zbl

[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. | MR | Zbl

[15] J.L. Lions, Remarks on approximate controllability. J. Anal. Math. 59 (1992) 103-116. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[18] J.P. Raymond and H. Zidani, Pontryagin's principle for time-optimal problems. J. Optim. Theory Appl. 101 (1999) 375-402. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[23] Z.Q. Wu, J.X. Yin and C.P. Wang, Elliptic and Parabolic Equations. World Scientific Publishing Corporation, New Jersey (2006). | MR | Zbl

[24] E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control Cybern. 28 (1999) 665-683. | MR | Zbl

Cité par Sources :