Distributed System Observability and Controllability with Finite-Dimensional Sensors and Actuators: Application to Thermal and Fluid Dynamics

9312745 Mohler本研究的目的是为一些线性时变和非线性分布参数系统开发新的可观测性和可控性技术，这些系统具有内部离散时间有限维观测和控制。 重点是有限维传感器和执行器的位置。 这些问题，作为传统的兴趣偏微分方程的反问题，在文献中得到了相当大的关注，主要是在线性时不变的设置。 在谐波和非谐波分析的基础上，它表明，在这种情况下，系统的可观测性和可控性的问题取决于至关重要的相关特征值（特别是，在他们的增长和多重性），因此，固定有限维传感器和执行器往往失败时，空间维度高于1。 这里提出的研究的目的是克服这个困难的基础上引入空间扫描观测和控制。 在上面的上下文中，所提出的方法主要处理本征函数，而在热方程的情况下，例如，只使用本征值的符号。 计划研究上述系统的连续时间和离散时间观测与控制之间的相互关系，将离散时间设置解释为连续时间设置的骨架。 这里的研究的目的是开发一种方法（在我们以前的研究中介绍），使我们能够取代连续时间的传感器和执行器，空间平均的，与集值映射也将进行研究。 后者是不连续广义解情况下逐点观察/控制概念的自然推广，使我们能够确保有限维观察和控制的适定性。 ***

9312745 Mohler The object of this research is to develop new observability and controllability techniques for a number of both linear time-varying and nonlinear distributed-parameter systems with internal discrete- time finite-dimensional observation and control. Emphasis is given to location of finite-dimensional sensors and actuators. These problems, being the inverse problems for partial differential equations of traditional interest, have received considerable attention in the literature, mostly in the linear time-invariant setting. On the basis of harmonic and nonharmonic analysis it was shown that in this case observability and controllability of the system in question depend crucially upon the associated eigenvalues (in particular, upon their growth and multiplicity) and, consequently, stationary finite-dimensional sensors and actuators often fail when the space dimension is higher than 1. The research proposed here is aimed to overcome this difficulty on the basis of the introduction of spatially scanning observations and control. In the above context, the proposed methodology mostly deals with the eigenfunctions, while employing in the case of the heat equation, for example, only the sign of the eigenvalues. It is planned to study the interrelation between continuous-time and discrete-time observations and controls for the above systems, interpreting the discrete-time setting as a skeleton for the continuous-time one. The objective of research here is to develop a methodology (introduced in our previous studies) that allows us to replace continuous-time sensors and actuators, the spatially- averaged ones, associated with the set-valued mappings will also be studied. The latter is a natural generalization of the notion of pointwise observations/control for the case of discontinuous generalized solutions, allowing us to ensure the well-posedness of the finite-dimensional observation and control. ***