June  2011, 1(2): 231-250. doi: 10.3934/mcrf.2011.1.231

Strict Lyapunov functions for semilinear parabolic partial differential equations

1. 

Team INRIA DISCO, CNRS-Supelec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France

2. 

Department of Automatic Control, Gipsa-lab, 961 rue de la Houille Blanche, BP 46, 38402 Grenoble Cedex

Received  November 2010 Revised  March 2011 Published  June 2011

For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.
Citation: Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
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show all references

References:
[1]

Applied Mathematical Sciences, 83, Springer-Verlag, New York, 1989.  Google Scholar

[2]

Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[3]

Journal of Differential Equations, 78 (1989), 160-190.  Google Scholar

[4]

SIAM Journal on Control and Optimization, 47 (2008), 1460-1498. doi: 10.1137/070706847.  Google Scholar

[5]

IEEE Transactions on Automatic Control, 43 (1998), 608-618. doi: 10.1109/9.668828.  Google Scholar

[6]

IEEE Transactions on Automatic Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903.  Google Scholar

[7]

SIAM Journal on Control and Optimization, 43 (2004), 549-569. doi: 10.1137/S036301290342471X.  Google Scholar

[8]

Commun. Contemp. Math., 8 (2006), 535-567. doi: 10.1142/S0219199706002209.  Google Scholar

[9]

IEEE Transactions on Automatic Control, 53 (2008), 412-416. doi: 10.1109/TAC.2007.914948.  Google Scholar

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Automatica J. IFAC, 45 (2009), 312-323. doi: 10.1016/j.automatica.2008.07.012.  Google Scholar

[11]

European Journal of Control, 14 (2008), 539-555. doi: 10.3166/ejc.14.539-555.  Google Scholar

[12]

Advances in Design and Control, 16, SIAM, Philadelphia, PA, 2008.  Google Scholar

[13]

IEEE Transactions on Automatic Control, 53 (2008), 1575-1591. doi: 10.1109/TAC.2008.927798.  Google Scholar

[14]

Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1999.  Google Scholar

[15]

Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 2009.  Google Scholar

[16]

ESAIM Control Optim. Cal. Var., 11 (2005), 487-507. doi: 10.1051/cocv:2005016.  Google Scholar

[17]

IEEE Transactions on Automatic Control, 54 (2009), 177-183. doi: 10.1109/TAC.2008.2008353.  Google Scholar

[18]

Automatica J. IFAC, 44 (2008), 2415-2421. doi: 10.1016/j.automatica.2008.01.024.  Google Scholar

[19]

Mathematics of Control, Signals, and Systems, 19 (2007), 151-182. doi: 10.1007/s00498-007-0015-7.  Google Scholar

[20]

Duke Math. J., 86 (1997), 143-195. doi: 10.1215/S0012-7094-97-08605-1.  Google Scholar

[21]

Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[22]

IEEE Transactions on Automatic Control, 54 (2009), 1688-1693. doi: 10.1109/TAC.2009.2020642.  Google Scholar

[23]

in "Proc. of American Control Conference (ACC'10)," (2010), 190-195. Google Scholar

[24]

submitted for publication, (2011). Google Scholar

[25]

SIAM Journal on Control, 12 (1974), 500-508. doi: 10.1137/0312038.  Google Scholar

[26]

Automatica J. IFAC, 43 (2007), 1543-1556. doi: 10.1016/j.automatica.2007.02.014.  Google Scholar

[27]

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[28]

Nonlinear and Optimal Control Theory, Springer-Verlag, Berlin, (2007), 163-220. Google Scholar

[29]

Springer-Verlag, New York, 1997.  Google Scholar

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