For Stokes equations under divergence-free and mixed boundary conditions, the inverse problem of shape identification from boundary measurement is investigated. Taking the least-square misfit as an objective function, the state-constrained optimization is treated by using an adjoint state within the Lagrange approach. The directional differentiability of a Lagrangian function with respect to shape variations is proved within the velocity method, and a Hadamard representation of the shape derivative by boundary integrals is derived explicitly. The application to gradient descent methods of iterative optimization is discussed.

中文翻译：

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从Stokes方程的边界测量中识别形状的反问题：拉格朗日形状的可微性

对于无散度和混合边界条件下的斯托克斯方程，研究了通过边界测量识别形状的逆问题。以最小二乘不拟合为目标函数，通过使用拉格朗日方法中的伴随状态来处理状态约束的优化。在速度方法中证明了拉格朗日函数相对于形状变化的方向可微性，并通过边界积分明确导出了形状导数的Hadamard表示。讨论了迭代优化在梯度下降法中的应用。