Control of Stochastic Systems

随机系统的控制是本研究的重点，它提供了一个广泛应用于许多科学和工程领域以及数学应用的研究。基于各种物理现象的经验数据，称为分数布朗运动的随机过程族适合于物理现象的随机模型。然而，分数布朗运动家族的大多数成员不具有随机模型通常假定的马尔可夫性质或半鞅性质。由于大多数连续时间受控随机系统都是用布朗运动建模的噪声来描述的，因此有必要研究具有其他分数布朗运动的随机系统的控制问题。这些控制问题需要的分析方法明显不同于发达的布朗运动系统控制方法。这些主要的差异导致了由任意分数布朗运动驱动的线性系统的最优控制几乎没有可用的结果。这项拨款的研究人员已经开始了对线性系统的这些控制问题的主要研究。这项研究不仅应该为任意分数布朗运动提供结果，而且应该为由一般的平方可积连续随机过程驱动的线性系统的控制提供结果。这项工作将进一步发展研究者。初步研究有限时间范围线性随机系统的二次成本控制。计划将这项研究扩展到其他成本函数和其他类型的系统，包括有限维和无限维。具体来说，这项工作计划确定最优控制的显式表达式和无限时间范围遍历（或长期平均）二次代价函数的最优代价。在此基础上，研究了具有二次代价泛函的有限和无限时间范围线性随机偏微分方程的控制问题。允许这些方程的噪声随机过程和控制被限制在域的边界上。被控随机系统通常具有未知参数，因此需要同时进行参数辨识和系统控制。这类问题被称为自适应控制，研究人员计划将他们对标量线性系统的自适应控制的初步工作扩展到具有任意分数布朗运动的多维线性系统。许多物理系统都是受控的，因此就产生了确定最佳或最优控制的问题。这一研究领域被称为最优控制。通常，物理现象的数学模型必须考虑系统的扰动或未建模的动力学，因此在模型中引入噪声过程。这些模型被称为随机系统。给定一个被控制的随机系统，以及使用控制的成本和系统的行为，通常很难获得一个明确的最优控制和相关的最优成本。此外，连续时间的受控随机系统被限制为使用一个特定的随机过程布朗运动（白噪声）作为噪声。然而，来自各种物理现象的经验证据表明，需要其他噪声过程，特别是分数布朗运动家族。这项拨款的研究人员最近开始研究线性系统的控制，其成本函数在系统状态和控制中是二次的。对于具有任意分数布朗运动的线性系统，他们得到了一些显式最优控制的结果。这些研究人员计划将这项工作扩展到更一般的线性系统，以及具有任意分数布朗运动或更一般随机过程的模型的无限时间范围控制问题。这些模型的最优控制可用于许多领域的控制应用。分数布朗运动的有用性已经在水文学、电信、湍流、癫痫和认知以及其他领域得到了证明。最优控制结果对于有效控制这些物理现象具有重要意义。这些结果将对许多领域产生重要影响。

The control of stochastic systems is the focus of this research and it provides a study that has wide applications to many areas of science and engineering as well as applications within mathematics. Based on empirical data from a wide variety of physical phenomena, the family of stochastic processes called fractional Brownian motions is appropriate for stochastic models of physical phenomena. However most of the members of this family of fractional Brownian motions do not possess the Markovian property or the semimartingale property that are usually assumed for stochastic models. Since most controlled stochastic systems in continuous time have been described with a noise modeled by a Brownian motion there is demand to study control problems for stochastic systems with other fractional Brownian motions. These control problems require significantly different methods for analysis than the well developed methods for the control of systems with Brownian motions. These major differences have resulted in having few results available for the optimal control of linear systems driven by an arbitrary fractional Brownian motion. The investigators for this grant have initiated a major study of these control problems for linear systems. This study should not only provide results for an arbitrary fractional Brownian motion but also provide results for the control of linear systems driven by processes from a general family of square integrable continuous stochastic processes. This work will develop further the investigators? initial work on finite time horizon, quadratic cost control of linear stochastic systems. It is planned to expand this study to other cost functionals and to other types of systems, both finite and infinite dimensional. Specifically this work is planned to determine explicit expressions for the optimal control and the optimal cost for an infinite time horizon ergodic (or long run average) quadratic cost functional. Furthermore it is planned to study the control of linear stochastic partial differential equations for both finite and infinite time horizon control problems with a quadratic cost functional. The noise stochastic process and the control for these equations are allowed to be restricted to the boundary of the domain. Typically the controlled stochastic systems have unknown parameters so it is necessary to identify the parameters and control the system simultaneously. This class of problems is called adaptive control and the investigators plan to extend their initial work on adaptive control for a scalar linear system to multidimensional linear systems with an arbitrary fractional Brownian motion. Many physical systems are controlled so the question of the determination of a best or optimal control arises. This area of research is called optimal control. Typically a mathematical model of a physical phenomena must account for perturbations of the system or unmodelled dynamics so a noise process is introduced in the model. These models are called stochastic systems. Given a controlled stochastic system and a cost for the use of control and the behavior of the system it is usually very difficult to obtain an explicit optimal control and the associated optimal cost. Furthermore the controlled stochastic systems in continuous time have been restricted to using one specific stochastic process, Brownian motion (white noise) as the noise. However empirical evidence from a wide variety of physical phenomena demonstrates a need for other noise processes, particularly for the family of fractional Brownian motions. The investigators for this grant have recently initiated a study of the control of linear systems with cost functionals that are quadratic in the system state and the control. They have obtained some results for explicit optimal controls for linear systems with an arbitrary fractional Brownian motion. These investigators plan to extend this work to more general linear systems and to infinite time horizon control problems with models that have an arbitrary fractional Brownian motion or a more general stochastic process. The optimal controls for these models can be used for control applications in many fields. The usefulness of fractional Brownian motions has been demonstrated in hydrology, telecommunications, turbulence, epilepsy and cognition as well as other areas. The optimal control results can be important in the use of effective controls for these physical phenomena. These results should have an important impact on many fields.