Spectral theory and optimal harvesting

光谱理论和最佳收获

• 批准号: 264898126
• 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: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 无数据
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/264898126?language=en
• 关键词: Spectral; theory; optimal; harvesting

Spectral TheoryIn 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. Application to constant coefficient operators show that this technique is hardly useful for spectral theory. Thus Hinton and I have 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 characteristic polynomial P(lambda, z) and the algebraic curves P(lambda, z) = 0. Moreover we need to determine a good representation of the resolvent, which allows, precise estimates in order to derive the connection with the M-matrix and derive properties of the spectral type.Optimal harvestingSome years ago I presented a talk on optimal havesting 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.

在过去的10年里，我和D.B. Hinton教授发表了8篇关于高阶微分算子的论文[例如B.H. 2011, 2011]。本文证明了谱是绝对连续的，如果系数满足温和正则性假设。从Sturm-Liouville算符可知，这些假设几乎是可选的。主要的技术是渐近积分和m矩阵。最近，Brown、Evans和Plum利用Weyl圆的方法构造了不一定自伴随算子的m矩阵。对常系数算符的应用表明，这种方法对谱理论几乎没有用处。因此，欣顿和我设计了一种新的方法，也可以确定光谱类型。对于几乎常系数的哈密顿系统，这种方法似乎很有前途。然而，这需要详细分析特征多项式P（lambda, z）和代数曲线P(lambda, z) = 0。此外，我们需要确定一个很好的解析表达式，它允许精确的估计，以便推导与m矩阵的联系，并推导谱型的性质。最佳收获几年前，我在田纳西大学做了一个关于鱼类最佳收获的演讲，其中考虑了渔网的宽度和鱼类种群的年龄结构。这导致了与S. Lenhart教授和S. Ding博士的联合工作。打算继续这项工作，特别是对鳕鱼和鲱鱼，并确定最佳可持续产量和分析相应的最优控制问题。