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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 and 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, Oxford, 2003.

[2]

J.-J. Alibert and J.-P. Raymond, A Lagrange multiplier theorem for control problems with state constraints, Numer. Funct. Anal. Optim., 19 (1998), 697-704. doi: 10.1080/01630569808816854.

[3]

J. F. Bonnans and E. Casas, Optimal control of semilinear multistate systems with state constraints, SIAM J. Contr. Opt., 27 (1989), 446-455. doi: 10.1137/0327023.

[4]

Eduardo Casas and Fredi Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem, Control and Cybernetics, 31 (2002), 695-712.

[5]

C. Clason and B. Kaltenbacher, Optimal control of a singular PDE modeling transient MEMS with control or state constraints, in "Mathematical Optimization and Applications in Biomedical Sciences,'' Technical Report, 2012-017, SFB, 2012.

[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), 738-751. doi: 10.1016/j.jmaa.2009.03.043.

[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), 503-523. doi: 10.3934/dcdss.2009.2.503.

[8]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537.

[9]

M. Wilke and S. Meyer, Optimal regularity and long time behavior of solutions for the Westervelt equation, Applied Mathematics and Optimization, 64 (2011), 257-273. doi: 10.1007/s00245-011-9138-9.

show all references

References:
[1]

R. A. Adams and J. F. Fournier, "Sobolev Spaces,'' Elsevier, Oxford, 2003.

[2]

J.-J. Alibert and J.-P. Raymond, A Lagrange multiplier theorem for control problems with state constraints, Numer. Funct. Anal. Optim., 19 (1998), 697-704. doi: 10.1080/01630569808816854.

[3]

J. F. Bonnans and E. Casas, Optimal control of semilinear multistate systems with state constraints, SIAM J. Contr. Opt., 27 (1989), 446-455. doi: 10.1137/0327023.

[4]

Eduardo Casas and Fredi Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem, Control and Cybernetics, 31 (2002), 695-712.

[5]

C. Clason and B. Kaltenbacher, Optimal control of a singular PDE modeling transient MEMS with control or state constraints, in "Mathematical Optimization and Applications in Biomedical Sciences,'' Technical Report, 2012-017, SFB, 2012.

[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), 738-751. doi: 10.1016/j.jmaa.2009.03.043.

[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), 503-523. doi: 10.3934/dcdss.2009.2.503.

[8]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537.

[9]

M. Wilke and S. Meyer, Optimal regularity and long time behavior of solutions for the Westervelt equation, Applied Mathematics and Optimization, 64 (2011), 257-273. doi: 10.1007/s00245-011-9138-9.

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