Mathematical Sciences: Algebraic Methods in Systems Theory

9610389 Rosenthal这个项目解决了系统理论和编码理论中的一些问题。该提案的主要目标是：1。一般矩阵扩展问题的统一代数理论，在特殊情况下，一边是极点放置和插值文献的结果，另一边是矩阵扩展的结果。2. 将一维系统中著名的Hermann-Martin对应扩展到多维系统。3. 有限域上线性系统的最优控制理论，将导致构造自由距离大、复杂度低、译码方案简单的卷积码的方法。卷积码用于电话线上的数据传输、无线通信领域以及来自深空的图像传输。反馈控制器在许多不同的设备中实现，例如飞机，机器人和化学过程。从数学的角度来看，线性反馈控制器和卷积编码器是由类似的一组方程控制的。提出的研究目标是发展一种数学理论，这将导致改进各种类型的反馈控制器的设计技术，以及更有效的卷积编码器和解码器。

9610389 Rosenthal This project addresses a number of issues in systems theory and coding theory. The main objectives of the proposal are: 1. A unified algebraic theory for general matrix extension problems which, in special situations, covers results of the pole placement and interpolation literature on one side and results on matrix extensions on the other side. 2. An extension of the Hermann-Martin correspondence well known for one dimensional systems to the situation of multidimensional systems. 3. An optimal control theory for linear systems over finite fields which will result in methods for constructing convolutional codes with large free distance, low complexity and simple decoding schemes. Convolutional codes are used in the data transmission over phone lines, in the area of wireless communication and in the transmission of pictures from deep space. Feedback controllers are implemented in many different devices such as aircraft, robots and chemical processes to mention a few. From a mathematical point of view linear feedback controllers and convolutional encoders are governed by similar set of equations. It is the goal of the proposed research to develop a mathematical theory which will lead to improved design techniques for feedback controllers of various type as well as more efficient convolutional encoders and decoders.