Problems in Stochastic Control and Their Numerical Approximations

将发展控制非线性随机延迟方程的数值方法。基本的逼近方案是所谓的马尔可夫链逼近法，这是目前非延迟问题的首选方法。这将扩大到包括新系统的兴趣。将发展关于底层系统的定性和近似性质的基本定理，并分析其长期行为。将开发数字高效算法。减少内存需求是一项主要任务，也许是最关键的任务，因为此类系统通常需要超出当前计算机能力的内存。有很多很有前途的数据结构、系统表示和对偶公式。此外，我们将开发有效的方法来管理在连接信道随机时变的环境中运行的移动网络，因为由于移动设备的运动散射。基本技术将基于摄动李雅普诺夫函数方法，这是一种在系统非马尔可夫时进行稳定性分析的有效方法。非线性随机延迟方程在应用中无处不在。特别令人感兴趣的是高速通信的应用。这样的系统总是受到延迟的影响，并且是随机的和非线性的。这是一个非常有趣的主题，但是当前的控制方法试图通过各种技巧来绕过延迟的影响，从而导致保守（通常是不稳定的）系统。但很明显，利用最优随机控制理论的全部力量，考虑真正的延迟，可以大大改善操作。这只有在数值近似被很好地理解和有效算法可用的情况下才能做到。

Numerical methods for controlled nonlinear stochastic delay equations will be developed. The basic approximating scheme is the so-called Markov chain approximation method which is the current method of choice for the non-delay problem. This will be extended to cover the new systems of interest. Basic theorems on qualitative and approximation properties of the underlying systems will be developed, and the long-term behavior analyzed. Numerically efficient algorithms will be developed. Reducing memory requirements is a major task, perhaps the most crucial, since such systems typically require memory that is beyond the capabilities of current computers. There are various data structures, system representations, and dual formulations that are very promising. In addition, we will develop efficient methods of managing networks of mobiles that operate in an environment where the connecting channels are randomly time varying because of the scattering due to the motions of the mobiles. The basic technique will be based on the perturbed Liapunov function method, a powerful approach for stability analysis when the system is not Markovian.Nonlinear stochastic delay equations are ubiquitous in applications. Of particular interest are applications to high-speed communications. Such systems are always subject to delays, and are stochastic and nonlinear. This has been the subject of much interest, but current methods of control try to get around the effects of the delays with various tricks that lead to conservative (and often unstable) systems. But it is apparent that the use of the full power of optimal stochastic control theory, taking the true delays into account, can greatly improve the operation. This can only be done if numerical approximations are well-understood and efficient algorithms available.