Avoiding degeneracy in the Westervelt equation by state constrained optimal control

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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

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

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References:

[1]	R. A. Adams and J. F. Fournier, "Sobolev Spaces,'', Elsevier, (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. 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. 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.
[5]	C. Clason and B. Kaltenbacher, Optimal control of a singular PDE modeling transient MEMS with control or state constraints,, in, (2012), 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. 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. 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.
[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.

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