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Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping
Avoiding degeneracy in the Westervelt equation by state constrained optimal control
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 |
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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