Spectral Theory of Differential Operators Optimal Harvesting of Fish

鱼类最优捕捞微分算子谱理论

• 批准号: 246371334
• 负责人: Professor Dr. Horst Behncke
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 无数据
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/246371334?language=en
• 关键词: Spectral; Theory; Differential; Operators; Optimal

Projekt: I Spectral Theory of Differential Operators II Optimal Sustainable Harvesting of Fish I) In the last 10 years Professor D.B. Hinton and I have published 8 papers on higher order differential operators [e.g. B.H. 2011, 2011]. In these papers it is shown that the spectra are absolutely continuous, if the coefficients satisfy mild regularity assumptions. It is known from Sturm-Liouville operators that these assumptions are almost optional. The main techniques are asymptotic integration and the M-matrix. Recently Brown, Evans, and Plum have constructed the M-matrix for not necessarily selfadjoint operators, by using the method of Weyl circles. Applications to constant coefficient operators show that this technique is hardly useful for spectral theory. Thus Hinton and Ihave devised a new method, which will also permit to determine the spectral type. For Hamiltonian systems with almost constant coefficients this method seems to be promising. This, however, requires a detailed analysis of the characteristic polynomial P(lambda, z) and the algebraic curves P(lambda, z) = 0. The method of the M-matrix is typically based on one regular endpoint. Many problems of mathematical physiscs, however, lead to two sigular endpoints, infinity and a Bessel type singularity at 0. Generally this problem is treated by the decomposition method, leading to spectra of higher multiplicity [B.H. 2010]. If, however, the left partial operator has only discrete spectrum, this does not hold any more. In this case to total M-matrix is not any more of Nevanlinna type. Problems of this type will be studied separately. II) Some years ago I presented a talk on optimal harvesting of fish at the University of Tennessee, which takes into account the width of the fishing nets and age structure of the fish population. This led to a joint work with Prof. S. Lenhart and Dr. S. Ding. it is intended to continue this work in particular for cod and herring and to determine the optimal sustainable yield and analyze the corresponding optimal control problems.

项目： I 微分算子的谱理论 II 鱼类的最佳可持续收获 I) 在过去的 10 年里，D.B. 教授Hinton 和我发表了 8 篇关于高阶微分算子的论文 [例如B.H. 2011，2011]。这些论文表明，如果系数满足温和的规律性假设，则光谱是绝对连续的。从 Sturm-Liouville 算子得知，这些假设几乎是可选的。主要技术是渐近积分和M矩阵。最近，Brown、Evans 和 Plum 利用 Weyl 圆的方法构造了不一定自共轭算子的 M 矩阵。常数系数算子的应用表明该技术对于谱理论几乎没有用处。因此，辛顿和我设计了一种新方法，该方法也可以确定光谱类型。对于具有几乎恒定系数的哈密顿系统，这种方法似乎很有前途。然而，这需要对特征多项式 P(lambda, z) 和代数曲线 P(lambda, z) = 0 进行详细分析。M 矩阵的方法通常基于一个规则端点。然而，数学物理的许多问题都会导致两个奇异端点，无穷大和 0 处的贝塞尔型奇点。通常，这个问题通过分解方法来处理，导致更高的多重性谱[B.H. 2010]。然而，如果左部分算子仅具有离散谱，则这不再成立。在这种情况下，总 M 矩阵不再是 Nevanlinna 类型。此类问题将单独研究。 II）几年前，我在田纳西大学做了一次关于最佳鱼类收获的演讲，其中考虑了渔网的宽度和鱼类种群的年龄结构。这促成了与 S. Lenhart 教授和 S. Ding 博士的合作。计划继续开展这项工作，特别是针对鳕鱼和鲱鱼，并确定最佳可持续产量并分析相应的最佳控制问题。