American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Higher differentiability in the context of Besov spaces for a class of nonlocal functionals
  • EECT Home
  • This Issue
  • Next Article
    Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
June  2013, 2(2): 281-300. doi: 10.3934/eect.2013.2.281

Avoiding degeneracy in the Westervelt equation by state constrained optimal control

Christian Clason 1, and  Barbara Kaltenbacher 2,

1. 

Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010 Graz, Austria
2. 

Institute for Mathematics, Alpen-Adria Universität Klagenfurt, Universitätsstr. 65-67, 9020 Klagenfurt

Received  September 2012 Revised  November 2012 Published  March 2013

The Westervelt equation, which describes nonlinear acoustic wave propagation in high intensity ultrasound applications, exhibits potential degeneracy for large acoustic pressure values. While well-posedness results on this PDE have so far been based on smallness of the solution in a higher order spatial norm, non-degeneracy can be enforced explicitly by a pointwise state constraint in a minimization problem, thus allowing for pressures with large gradients and higher-order derivatives, as is required in the mentioned applications. Using regularity results on the linearized state equation, well-posedness and necessary optimality conditions for the PDE constrained optimization problem can be shown via a relaxation approach by Alibert and Raymond [2].
Keywords: singular PDEs, nonlinear acoustics., Westervelt equation, Optimal control of PDEs, state constraints.
Mathematics Subject Classification: Primary: 49K20, 35L72; Secondary: 35L6.
Citation: Christian Clason, Barbara Kaltenbacher. Avoiding degeneracy in the Westervelt equation by state constrained optimal control. Evolution Equations & Control Theory, 2013, 2 (2) : 281-300. doi: 10.3934/eect.2013.2.281
References:
[1]

R. A. Adams and J. F. Fournier, "Sobolev Spaces,'', Elsevier, (2003). Google Scholar

[2]

J.-J. Alibert and J.-P. Raymond, A Lagrange multiplier theorem for control problems with state constraints,, Numer. Funct. Anal. Optim., 19 (1998), 697. doi: 10.1080/01630569808816854. Google Scholar

[3]

J. F. Bonnans and E. Casas, Optimal control of semilinear multistate systems with state constraints,, SIAM J. Contr. Opt., 27 (1989), 446. doi: 10.1137/0327023. Google Scholar

[4]

Eduardo Casas and Fredi Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem,, Control and Cybernetics, 31 (2002), 695. Google Scholar

[5]

C. Clason and B. Kaltenbacher, Optimal control of a singular PDE modeling transient MEMS with control or state constraints,, in, (2012), 2012. Google Scholar

[6]

C. Clason, B. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equation,, Journal of Mathematical Analysis and Applications, 356 (2009), 738. doi: 10.1016/j.jmaa.2009.03.043. Google Scholar

[7]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation,, Discrete and Continuous Dynamical Systems Ser. S, 2 (2009), 503. doi: 10.3934/dcdss.2009.2.503. Google Scholar

[8]

P. J. Westervelt, Parametric acoustic array,, The Journal of the Acoustic Society of America, 35 (1963), 535. Google Scholar

[9]

M. Wilke and S. Meyer, Optimal regularity and long time behavior of solutions for the Westervelt equation,, Applied Mathematics and Optimization, 64 (2011), 257. doi: 10.1007/s00245-011-9138-9. Google Scholar

show all references

References:
[1]

R. A. Adams and J. F. Fournier, "Sobolev Spaces,'', Elsevier, (2003). Google Scholar

[2]

J.-J. Alibert and J.-P. Raymond, A Lagrange multiplier theorem for control problems with state constraints,, Numer. Funct. Anal. Optim., 19 (1998), 697. doi: 10.1080/01630569808816854. Google Scholar

[3]

J. F. Bonnans and E. Casas, Optimal control of semilinear multistate systems with state constraints,, SIAM J. Contr. Opt., 27 (1989), 446. doi: 10.1137/0327023. Google Scholar

[4]

Eduardo Casas and Fredi Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem,, Control and Cybernetics, 31 (2002), 695. Google Scholar

[5]

C. Clason and B. Kaltenbacher, Optimal control of a singular PDE modeling transient MEMS with control or state constraints,, in, (2012), 2012. Google Scholar

[6]

C. Clason, B. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equation,, Journal of Mathematical Analysis and Applications, 356 (2009), 738. doi: 10.1016/j.jmaa.2009.03.043. Google Scholar

[7]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation,, Discrete and Continuous Dynamical Systems Ser. S, 2 (2009), 503. doi: 10.3934/dcdss.2009.2.503. Google Scholar

[8]

P. J. Westervelt, Parametric acoustic array,, The Journal of the Acoustic Society of America, 35 (1963), 535. Google Scholar

[9]

M. Wilke and S. Meyer, Optimal regularity and long time behavior of solutions for the Westervelt equation,, Applied Mathematics and Optimization, 64 (2011), 257. doi: 10.1007/s00245-011-9138-9. Google Scholar

[1]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183
[2]

Alexander Komech. Attractors of Hamilton nonlinear PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6201-6256. doi: 10.3934/dcds.2016071
[3]

Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305
[4]

Lorena Bociu, Barbara Kaltenbacher, Petronela Radu. Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications. Evolution Equations & Control Theory, 2013, 2 (2) : i-ii. doi: 10.3934/eect.2013.2.2i
[5]

Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579
[6]

Marianne Beringhier, Adrien Leygue, Francisco Chinesta. Parametric nonlinear PDEs with multiple solutions: A PGD approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 383-392. doi: 10.3934/dcdss.2016002
[7]

Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081
[8]

Swann Marx, Tillmann Weisser, Didier Henrion, Jean Bernard Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019032
[9]

Nikolaos Kazantzis, Vasiliki Kazantzi. Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors. Mathematical Biosciences & Engineering, 2010, 7 (2) : 401-419. doi: 10.3934/mbe.2010.7.401
[10]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553
[11]

Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[12]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022
[13]

J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341
[14]

Stephen W. Taylor. Locally smooth unitary groups and applications to boundary control of PDEs. Evolution Equations & Control Theory, 2013, 2 (4) : 733-740. doi: 10.3934/eect.2013.2.733
[15]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032
[16]

H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119
[17]

Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial & Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247
[18]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819
[19]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505
[20]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences