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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/cocv/2012017/
[1] Dirichlet boundary control of semilinear parabolic equations, Part 1 : Problems with no state constraints. Appl. Math. Optim. 45 (2002) 125-143. | MR | Zbl
and ,[2] Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Syst. 9 (2003) 1549-1570. | MR | Zbl
and ,[3] Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston (1993). | MR | Zbl
,[4] The time optimal control of Navier-Stokes equations. Syst. Control Lett. 30 (1997) 93-100. | MR | Zbl
,[5] On the “bang-bang” control problem. Q. Appl. Math. 14 (1956) 11-18. | Zbl
, and ,[6] Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinburgh 125 (1995) 31-61. | MR | Zbl
, and ,[7] Time optimal control of solutions of operational differential equations. SIAM J. Control 2 (1964) 54-59. | MR | Zbl
,[8] Infinite Dimensional Linear Control Systems : The Time Optimal and Norm Optimal Problems. North-Holland Math. Stud. 201 (2005). | MR | Zbl
,[9] Sufficiency of the maximum principle for time optimality. Cubo : A. Math. J. 7 (2005) 27-37. | MR | Zbl
,[10] 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
and ,[11] Optimal Control of Distributed Systems, Theory and Applications. American Mathematical Society, Providence (2000). | MR | Zbl
,[12] 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
and ,[13] Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston (1995). | MR | Zbl
and ,[14] Remarques sur la contrôlabilité approchée, in Jornadas Hispano-Francesas Sobre Control de Sistemas Distribuidos. University of Málaga, Spain (1991) 77-87. | MR | Zbl
,[15] Remarks on approximate controllability. J. Anal. Math. 59 (1992) 103-116. | MR | Zbl
,[16] 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
and ,[17] An abstract bang-bang principle and time optimal boundary control of the heat equation. SIAM J. Control Optim. 35 (1997) 1204-1216. | MR | Zbl
and ,[18] Pontryagin's principle for time-optimal problems. J. Optim. Theory Appl. 101 (1999) 375-402. | MR | Zbl
and ,[19] 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] The bang-bang principle of time optimal controls for the heat equation with internal controls. Syst. Control Lett. 56 (2007) 709-713. | MR | Zbl
and ,[21] The optimal time control of a phase-field system. SIAM J. Control Optim. 42 (2003) 1483-1508. | MR | Zbl
and ,[22] On the equivalence of minimal time and minimal norm controls for heat equations. SIAM J. Control Optim. 50 (2012) 2938-2958. | MR | Zbl
and ,[23] Elliptic and Parabolic Equations. World Scientific Publishing Corporation, New Jersey (2006). | MR | Zbl
, and ,[24] Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control Cybern. 28 (1999) 665-683. | MR | Zbl
,Cité par Sources :