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.
