Geometric and Stochastic System Theory (REU Supplement)

最近PI的已经获得了一些显式可解随机 非紧秩一扩散型最优控制问题 对称空间 建议将这些结果推广到 秩大于1的非紧对称空间，紧的 对称空间和齐性空间。 这种歧管提供 许多物理现象的模型，和相对较少的例子， 已知可解的随机控制问题。 提出要 继续在随机自适应控制的研究工作， 平均值的渐近分布和收敛速度 连续时间线性随机系统的估计量 有和没有时间延迟。 建议调查一些 具体模型，并扩大工作的自适应控制 双线性随机系统 建议将自适应 控制几何控制的研究一些随机自适应 在上述对称空间中的控制问题。 的 自回归过程的聚合经常被用于 申请，他们建议继续他们以前的一些 工作 在紧李群中解决了一些估计问题 和对称空间，并建议将这项工作扩展到其他 李群与对称空间。 从这些特殊的估计 结果以及非线性滤波与随机滤波的关系 控制，建议使用上述估计和 随机控制的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.