Mathematical Sciences: Optimal Design of Sensors and Actuators for Robust Control of Partial Differential Equation Systems

9409506 King计划研究的目标是为优化传感器和执行器设计开发严格的理论和计算方法。调查将集中在由偏微分方程（PDEs）控制的系统上。许多反馈控制设计方法产生的解决方案（即，最优控制）是写在一个反馈算子和最优状态。对于有限维系统，这种控制也可以写成增益和状态的乘积，并且可以使用增益的形式做出有关传感器设计的决策。在许多偏微分方程控制问题中，增益算子是希尔伯特空间之间的有界线性变换。表示定理通常可以用于获得增益算子的Rintegral表示，为数值近似和分析提供基础。最初，研究者将考虑在没有关于致动器类型的事先假设的情况下找到最佳增益算子的问题。鲁棒控制设计将用于制定相应的传感器设计问题。PDE系统的最佳传感器/致动器可能是空间分布的，尽管人们可能希望近似最佳设计并实现有限数量的传感器和致动器。这需要各种类型的控制算子的近似定理。＊＊＊

9409506 King The goal of the projected research is to develop rigorous theoretical and computational methods for optimal sensor and actuator design. The investigation will concentrate on systems governed by partial differential equations (PDEs). Many feedback control design methods yield a solution (i.e., an optimal control) which is written in terms of a feedback operator and the optimal state. For finite dimensional systems, this control can also be written as a product of gains and states and decisions regarding sensor design can be made using the form of the gains. In many PDE control problems, the gain operator is a bounded linear transformation between Hilbert spaces. Representation theorems often can be applied to obtain an Rintegral representationS of the gain operator that provides the basis for numerical approximations and analysis. Initially, the investigator will consider the problem of finding the optimal gain operator without prior assumptions regarding actuator type. Robust control designs will be used to formulate the corresponding sensor design problem. The optimal sensors/actuators for the PDE systems may be spatially distributed, though one may wish to approximate the optimal designs and implement a finite number of sensors and actuators. This requires approximation theorems for the various types of control operators. ***