Avoiding degeneracy in the Westervelt equation by state constrained optimal control
Avoiding degeneracy in the Westervelt equation by state constrained optimal control
复制标题
通过状态约束最优控制避免 Westervelt 方程的简并性
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
B. Kaltenbacher
中科院分区：
文献类型：
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作者：
Christian Clason;B. Kaltenbacher
描述高强度超声应用中的非线性声波传播的Westvelt方程，在较大声压值时表现出潜在的简并性。虽然到目前为止，关于这个偏微分方程组的适定性结果都是基于高阶空间范数的解的小，但非简并性可以通过最小化问题中的逐点状态约束来显式地强制，从而允许具有大的梯度和高阶导数的压力，正如在上述应用中所要求的那样。利用线性化状态方程的正则性结果，利用ALibert和Raymond[2]的松弛方法，证明了偏微分方程约束优化问题的适定性和最优性必要条件。
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].