Geometric and Stochastic System Theory (REU Supplement)

最近，PI 已经获得了非紧、一阶​​对称空间中一些可显式求解的扩散型随机最优控制问题。 建议将这些结果推广到秩大于一的非紧对称空间、紧对称空间和齐次空间。 这种流形为许多物理现象提供了模型，但已知的可解随机控制问题的例子相对较少。 建议通过研究带或不带时滞的连续时间、线性随机系统的平均成本或估计量的渐近分布和收敛速度来继续研究随机自适应控制。 建议研究一些特定模型，并扩展双线性随机系统自适应控制的工作。 通过研究上述对称空间中的一些随机自适应控制问题，提出将自适应控制与几何控制联系起来。 自回归过程的聚合经常在应用中使用，他们建议继续他们之前的一些工作。 一些估计问题已经在紧李群和对称空间中得到解决，并且建议将这项工作扩展到其他李群和对称空间。 根据这些特殊的估计结果以及非线性滤波与随机控制之间的关系，提出利用上述估计和随机控制结果来解决一些非线性滤波问题。

Recently the PI's have obtained some explicitly solvable stochastic optimal control problems of diffusion type in noncompact, rank-one symmetric spaces. It is proposed to generalize these results to noncompact symmetric spaces of rank greater than one, compact symmetric spaces, and homogeneous spaces. Such manifolds provide models for many physical phenomena, and relatively few examples of solvable stochastic control problems are known. It is proposed to continue work in stochastic adaptive control by investigating the asymptotic distribution and the rate of convergence of the average costs or the estimators for continuous-time, linear stochastic systems with and without time delays. It is proposed to investigate some specific models, and to expand work on the adaptive control of bilinear stochastic systems. It is proposed to relate adaptive control to geometric control by investigating some stochastic adaptive control problems in the aforementioned symmetric spaces. The aggregation of an autoregressive process is often used in applications, and they propose to continue some of their previous work. Some estimation problems have been solved in compact Lie groups and a symmetric space, and it is proposed to expand this work to other Lie groups and symmetric spaces. From these special estimation results and the relation between nonlinear filtering and stochastic control, it is proposed to use the aforementioned estimation and stochastic control results to solve some nonlinear filtering problems.