Bang-bang property for time optimal control of semilinear heat equation
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, p. 477-499
The full text of recent articles is available to journal subscribers only. See the article on the journal's website
This paper studies the bang-bang property for time optimal controls governed by semilinear heat equation in a bounded domain with control acting locally in a subset. Also, we present the null controllability cost for semilinear heat equation and an observability estimate from a positive measurable set in time for the linear heat equation with potential.
DOI : https://doi.org/10.1016/j.anihpc.2013.04.005
Keywords: Semilinear heat equation, Time optimal control, Bang-bang property, Observability estimate from measurable sets
@article{AIHPC_2014__31_3_477_0,
author = {Phung, Kim Dang and Wang, Lijuan and Zhang, Can},
title = {Bang-bang property for time optimal control of semilinear heat equation},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {3},
year = {2014},
pages = {477-499},
doi = {10.1016/j.anihpc.2013.04.005},
zbl = {1295.49005},
mrnumber = {3208451},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_3_477_0}
}

Phung, Kim Dang; Wang, Lijuan; Zhang, Can. Bang-bang property for time optimal control of semilinear heat equation. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 477-499. doi : 10.1016/j.anihpc.2013.04.005. http://www.numdam.org/item/AIHPC_2014__31_3_477_0/

[1] J. Apraiz, L. Escauriaza, G. Wang, C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (2013) | MR 3283402 | Zbl 1302.93040

[2] S. Anita, D. Tataru, Null controllability for the dissipative semilinear heat equation, Appl. Math. Optim. 46 (2002), 97 -105 | MR 1944754 | Zbl 1031.93018

[3] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston (1993) | MR 1195128

[4] T. Duyckaerts, X. Zhang, E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008), 1 -41 | Numdam | MR 2383077 | Zbl 1248.93031

[5] L. Escauriaza, F.J. Fernandez, S. Vessella, Doubling properties of caloric functions, Appl. Anal. 85 (2006), 205 -223 | MR 2198840 | Zbl 1090.35050

[6] H.O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, North-Holland Math. Stud. vol. 201 , Elsevier, Amsterdam (2005) | MR 2158806 | Zbl 1125.49020

[7] E. Fernández-Cara, S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim. 45 (2006), 1399 -1446 | MR 2257228 | Zbl 1121.35017

[8] E. Fernández-Cara, E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincare Anal. Non Lineaire 17 (2000), 583 -616 | Numdam | MR 1791879 | Zbl 0970.93023

[9] K. Kunisch, L. Wang, Time optimal controls of the linear Fitzhugh–Nagumo equation with pointwise control constraints, J. Math. Anal. Appl. 395 (2012), 114 -130 | MR 2943607 | Zbl 1251.35174

[10] K. Kunisch, L. Wang, Time optimal control of the heat equation with pointwise control constraints, ESAIM Control Optim. Calc. Var. 19 (2013), 460 -485 | Numdam | MR 3049719 | Zbl 1272.35109

[11] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin (1971) | MR 271512

[12] V. Mizel, T. Seidman, An abstract bang-bang principle and time optimal boundary control of the heat equation, SIAM J. Control Optim. 35 (1997), 1204 -1216 | MR 1453296 | Zbl 0891.49014

[13] K.D. Phung, G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal. 259 (2010), 1230 -1247 | MR 2652187 | Zbl 1215.35042

[14] K.D. Phung, G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc. 15 (2013), 681 -703 | MR 3017049 | Zbl 1258.93037

[15] G. Wang, The existence of time optimal control of semilinear parabolic equations, Systems Control Lett. 53 (2004), 171 -175 | MR 2092507 | Zbl 1157.49301

[16] G. Wang, ${L}^{\infty }$-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim. 47 (2008), 1701 -1720 | MR 2421326 | Zbl 1165.93016

[17] G. Wang, L. Wang, The bang-bang principle of time optimal controls for the heat equation with internal controls, Systems Control Lett. 56 (2007), 709 -713 | MR 2356456 | Zbl 1120.49002