We consider Prandtl’s 1933 model for calculating circulation distribution function \(\Gamma \) of a finite wing which minimizes induced drag, under the constraints of prescribed total lift and moment of inertia. We prove existence of a global minimizer of the problem without the restriction of nonnegativity \(\Gamma \ge 0\) in an appropriate function space. We also consider an improved model, where the prescribed moment of inertia takes into account the bending moment due to the weight of the wing itself, which leads to a more efficient solution than Prandtl’s 1933 result.

中文翻译：

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对 Prandtl 1933 模型的改进，用于最小化诱导阻力

我们考虑 Prandtl 的 1933 年模型，用于计算有限机翼的循环分布函数\(\Gamma \)，该函数在规定的总升力和惯性矩的约束下最小化诱导阻力。我们在适当的函数空间中证明了问题的全局最小化器的存在，而不受非负性\(\Gamma \ge 0\)的限制。我们还考虑了一种改进的模型，其中规定的惯性矩考虑了机翼本身重量引起的弯矩，这导致了比普朗特 1933 年结果更有效的解决方案。