  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.
