Mathematical Sciences: Optimal Control Theory of Input-Output Systems - Control Theory of Partial Differential Equations

这项工作将集中于偏微分方程的控制理论。将进行两次主要推力。研究的第一行处理由偏微分方程、泛函微分方程和相关应用描述的输入-输出系统的最优控制。这项工作将在首席研究员最近将经典庞特里亚金极大原理推广到无限维一般非线性系统的工作的基础上进行扩展。到目前为止，所得的收敛原理的应用仅在经典案例中得到了利用。现在的工作将指向至少两个新的领域：泛函微分方程和形状优化，其中控制是一个时间相关的集合。由于状态方程是变量域的偏微分方程，不适合任何可用的极大值原理，所以这里特别适用一般极大值原理。还计划对热方程和波动方程的可达到的状态集进行表征，首先主要是在特殊的几何域上。在这个活跃的研究领域中，没有规定控制作用所通过的边界条件。这些问题往往归结为傅立叶分析中的问题，其中需要对序列的范数进行估计。基本问题集中于通过边界控制来确定可达到的状态，这种状态由空间变量中的函数表示，人们希望在规定的时间内从初始形状达到该状态。

This work will focus on the control theory of partial differential equations. Two primary thrusts will be undertaken. The first line of investigation deals with optimal control of input-output systems described by partial differential equations, functional differential equations and associated applications. This work will expand on recent work of the principal investigator which extended the classical Pontryagin maximum principle to general nonlinear systems in infinite dimensions. Applications of the resulting convergence principles have only been exploited in classical cases to date. Work will now be directed to at least two new areas: functional differential equations and to shape optimization where the control is a time-dependent set. Here the general maximum principle is especially suited since the state equation is a partial differential equation in the variable domain and does not fit any of the available maximum principles. Work is also planned on the problem of characterizing the set of attainable states for both the heat equation and the wave equation, primarily on special geometric domains at first. In this active area of research, boundary conditions through which the control action is exercised are not specified. These questions often reduce to problems in Fourier analysis in which estimates on the norms of sequences are required. Fundamental questions focus on determining attainable states by means of boundary control, a state being represented by a function in the space variables which one wishes to reach in prescribed time from an initial shape.