In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space ℝ$$. As an application, we then show the exact null-controllability for this semilinear heat equation in ℝ$$. The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in ℝ$$. This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.
Keywords: Semilinear heat equation, null-controllability, uniform cost, equidistributed set
@article{COCV_2022__28_1_A8_0,
author = {Wang, Lijuan and Zhang, Can},
title = {A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022001},
mrnumber = {4368391},
zbl = {1515.35116},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022001/}
}
TY - JOUR AU - Wang, Lijuan AU - Zhang, Can TI - A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022001/ DO - 10.1051/cocv/2022001 LA - en ID - COCV_2022__28_1_A8_0 ER -
%0 Journal Article %A Wang, Lijuan %A Zhang, Can %T A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022001/ %R 10.1051/cocv/2022001 %G en %F COCV_2022__28_1_A8_0
Wang, Lijuan; Zhang, Can. A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 8. doi: 10.1051/cocv/2022001
[1] , , and , Observability inequalities and measurable sets. J. Eur. Math. Soc. 16 (2014) 2433–2475. | MR | Zbl | DOI
[2] , Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42 (2000) 73–89. | MR | Zbl | DOI
[3] , Exact null internal controllability for the heat equation on unbounded convex domains. ESAIM: COCV 20 (2014) 222–235. | MR | Zbl | Numdam
[4] , and , Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms. J. Optim. Theory Appl. 110 (2001) 245–264. | MR | Zbl | DOI
[5] , Control and Nonlinearity. American Mathematical Society, Providence, RI (2007). | MR | Zbl
[6] , Approximate controllability of a semilinear heat equation in . SIAM J. Control Optim. 36 (1998) 2128–2147. | MR | Zbl | DOI
[7] and , Approximate controllability of a semilinear heat equation in unbounded domains. Nonlinear Anal. 37 (1999) 1059–1090. | MR | Zbl | DOI
[8] , and , Observability inequalities for the heat equation with bounded potentials on the whole space. SIAM J. Control Optim. 58 (2020) 1939–1960. | MR | Zbl | DOI
[9] and Sharp geometric condition for null-controllability of the heat equation on and consistent estimates on the control cost. Arch. Math. (Basel) 111 (2018) 85–99. | MR | Zbl | DOI
[10] , , , , and , Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domains. Edited by , , , in Vol. 277 of Control Theory of Infinite-Dimensional Systems. Operator Theory: Advances and Applications. Birkhäuser, Cham (2020). | MR | Zbl
[11] , and , Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31–61. | MR | Zbl | DOI
[12] and , Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 583–616. | MR | Zbl | Numdam | DOI
[13] and , Controllability of Evolution Equations. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR
[14] and , Some results on controllability for linear and nonlinear heat equations in unbounded domains. Adv. Differ. Equ. 12 (2007) 1201–1240. | MR | Zbl
[15] , Global null-controllability and nonnegative-controllability of slightly superlinear heat equations. J. Math. Pures Appl. 135 (2020) 103–139. | MR | Zbl | DOI
[16] and , Null-controllability of the Kolmogorov equation in the whole phase space. J. Differ. Equ. 260 (2016) 3193–3233. | MR | Zbl | DOI
[17] , , and , Sharp estimates and homogenization of the control cost of the heat equation on large domains. ESAIM: COCV 26 (2020) 54. | MR | Zbl | Numdam
[18] , Carleman commutator approach in logarithmic convexity for parabolic equations. Math. Control Relat. Fields 8 (2018) 899–933. | MR | Zbl | DOI
[19] and , Quantitative unique continuation for the semilinear heat equation in a convex domain. J. Funct. Anal. 259 (2010) 1230–1247. | MR | Zbl | DOI
[20] and , An observability estimate for parabolic equations from a measurable set in time and its applications. J. Eur. Math. Soc. 15 (2013) 681–703. | MR | Zbl | DOI
[21] , and , Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 31 (2014) 477–499. | MR | Zbl | Numdam | DOI
[22] , and , The Three-Dimensional Navier-Stokes Equations. Classical Theory. Cambridge Studies in Advanced Mathematics, 157. Cambridge University Press, Cambridge (2016). | MR | Zbl
[23] and , Exhaustion approximation for the control problem of the heat or Schrödinger semigroup on unbounded domains. Arch. Math. (Basel) 115 (2020) 195–213. | MR | Zbl | DOI
[24] , , and , Time Optimal Control of Evolution Equations. Progress in Nonlinear Differential Equations and Their Applications, 92. Subseries in Control. Birkhäuser, Cham (2018). | MR | Zbl | DOI
[25] , , and , Observable set, observability, interpolation inequality and spectral inequality for the heat equation in . J. Math. Pures Appl. 126 (2019) 144–194. | MR | Zbl | DOI
[26] and , Observability inequalities from measurable sets for some abstract evolution equations. SIAM J. Control Optim. 55 (2017) 1862–1886. | MR | Zbl | DOI
[27] , Quantitative unique continuation for the heat equation with Coulomb potentials. Math. Control Relat. Fields 8 (2018) 1097–1116. | MR | Zbl | DOI
[28] and , On the optimality of the observability inequalities for Kirchhoff plate systems with potentials in unbounded domains. Hyperbolic problems: theory, numerics, applications. Springer, Berlin (2008) 233–243. | MR | Zbl
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