Mathematical Sciences: Approximation of Singular Neutral Control Systems

本项目涉及中立型泛函积分-微分方程组的逼近和控制计算技术的研究。这类系统是在某些气动弹性控制问题的建模中出现的，其中包括许多奇异积分-微分方程组。该方法基于状态空间形式，并利用泛函解析半群理论中的逼近结果来分析特定数值格式的收敛。本课题的主要目的是：(1)为奇异中立型泛函微分方程(SNFDE)选择一个“合适的”状态空间，并研究与齐次SNFDE相关的解半群的无穷小生成元的性质；(2)建立了状态空间上SNFDE的一般近似框架，分析了SNFDE近似解的半离散化(即平均投影和各种样条型格式)和全离散数值格式的收敛性质；(3)发展了SNFDE的对偶理论，讨论了控制和观测问题。**//

This project involves the study of computational techniques for the approximation and control of systems governed by functional integro-differential equations of neutral type. Such systems arise in the modeling of certain aeroelastic control problems and include many singular integro-differential equations. The approach here is based upon a state-space formulation and makes use of approximation results in functional analytic semigroup theory to analyze the convergence of particular numerical schemes. The main objectives of this project are: (1) Select an "appropriate" state-space for singular neutral functional differential equations (SNFDE) and study the properties of the infinitesimal generator of the solution semigroup associated with the homogeneous SNFDE; (2) Develop a general approximation framework for SNFDEs on the state-space and analyze the convergence properties of semi- discretized (i.e., averaging projections and various spline schemes) and fully-discretized numerical schemes for the approximate solutions to the SNFDE; (3) Develop the duality theory for SNFDEs and discuss control and observation problems. **//