空间分布系统的稀疏性和空间局部化测度
Sparsity and Spatial Localization Measures for Spatially Distributed Systems
摘要：
我们考虑一类空间衰减系统，其中基本的动态是空间衰减和空间分布的传感和控制。这类系统出现在各种应用中，其中存在空间距离的概念，相对于该空间距离，可以使用一类耦合权函数来量化子系统之间的耦合。我们利用系统潜在动力学的空间衰减特性，引入一类空间分布系统的稀疏性和空间局部化测度。我们开发了一种新的方法的基础上的概念的$q$ -巴拿赫代数的空间衰减运营商，使我们能够建立空间衰减系统的空间衰减特性和它们的稀疏性和空间局部化功能之间的关系。此外，它表明，逆闭性质的算子代数发挥了核心作用，利用各种结构性质的空间衰减系统。给出了$q$ - Banach代数上空间衰减系统指数稳定的条件，证明了在适当的$q$ - Banach代数上的Lyapunov方程和Riccati方程的唯一解也属于同一个$q$-巴拿赫代数.结果表明，空间衰减系统的二次型最优状态反馈控制器是稀疏和空间局部化的意义下，他们有近最优的稀疏信息结构。
英文摘要：
We consider the class of spatially decaying systems, where the underlying dynamics are spatially decaying and the sensing and controls are spatially distributed. This class of systems arise in various applications where there is a notion of spatial distance with respect to which couplings between the subsystems can be quantified using a class of coupling weight functions. We exploit spatial decay property of the underlying dynamics of the system to introduce a class of sparsity and spatial localization measures for spatially distributed systems. We develop a new methodology based on concepts of $q$-Banach algebras of spatially decaying operators that enable us to establish a relationship between spatial decay properties of spatially decaying systems and their sparsity and spatial localization features. Moreover, it is shown that the inverse-closedness property of operator algebras plays a central role in exploiting various structural properties of spatially decaying systems. We characterize conditions for exponentially stability of spatially decaying system over $q$-Banach algebras and prove that the unique solutions of the Lyapunov and Riccati equations over a proper $q$-Banach algebra also belong to the same $q$-Banach algebra. It is shown that the quadratically optimal state feedback controllers for spatially decaying systems are sparse and spatially localized in the sense that they have near-optimal sparse information structures.
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