Sparsity measures for spatially decaying systems

Abstract

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.