Mathematical Sciences: Volterra Integral Equations in Abstract Spaces

本课题所要做的工作涉及到Banach空间中Volterra积分微分方程的几个相关问题。计划应用于粘弹性介质和相关控制问题建模中出现的类型的偏微分积分方程。这项工作建立在早期的研究基础上，涉及半群、余弦族和相关理论。重点将放在寻找适当的设置和模型，这将产生可以方便地应用的定性信息。要分析的具体问题包括检查巴拿赫空间中的方程，该空间依赖于包含所有过去历史的数据。对于不依赖于过去全部历史的方程，我们将考虑其中一个代表极限方程的方程的作用。先前发展的某些无限历史问题的半群公式将继续受到关注。对对称双曲系统的研究也将继续。这一努力的连接主题是边界控制。

Work to be done on this project concerns several related problems concerning Volterra integrodifferential equations in Banach spaces. Applications are planned to partial differential integral equations of the type occurring in the modeling of viscoelastic media and associated control problems. The work builds on earlier investigations and involves semigroups, cosine families and the related theory. Emphasis will be placed on finding appropriate settings and models which will yield qualitative information which can conveniently be applied. Specific problems to be analyzed include the examination of equations in Banach spaces which depend on data that incorporate all past history. The role of one of these equations representing the limiting equation for equations which do not depend on the full past history will be considered. A semigroup formulation for certain infinite history problems developed earlier will continue to receive attention. Work on symmetric hyperbolic systems will also be continued. The connecting theme in this effort is that of boundary control.