Stabilizability of linear systems defined overC *-algebras

Sparsity measures for spatially decaying systems

“…Therefore, we develop a constructive proof based on finite covering of compact sets to show that the unique solution of an algebraic Riccati equation over a q-Banach algebra belongs to that q-Banach algebra. The following result is the most general known result about structural properties of solutions of algebraic Riccati equations and all previous known results, (e.g., see [1]- [4], [10], [12], [13], [30]- [32] and references in there), turn out to be special cases of our result.…”

“…The range of exponents 0 < q < 1 is extremely important for us as we can asymptotically approximate sparsity and spatial localization features of spatially decaying systems as q tends to zero (see Section VI). We characterize the class of proper q-Banach algebras and show that the unique solutions of the algebraic Lyapunov and Riccati equations which are defined over a proper q-Banach algebra also belong to that q-Banach algebra, significantly generalizing all existing works in the literature (e.g., see [1]- [4], [10], [12], [13], [30]- [32] and reference in there) to consider spatially distributed systems that are defined over q-Banach algebras for 0 < q ≤ ∞.…”

Sparsity and Spatial Localization Measures for Spatially Distributed Systems

“…Our goal is to determine degrees of sparsity and spatial localization for the solution of the algebraic Lyapunov equation. We recall a well-known result about solving the algebraic Lyapunov equation in B(ℓ 2 (G)); for more details see [39, P. 76] and [12,Theorem 1]. Suppose that Q ∈ B(ℓ 2 (G)) is strictly positive on ℓ 2 (G), and A ∈ B(ℓ 2 (G)) is exponentially stable on ℓ 2 (G).…”

“…The existing traditional methods to study this class of problems are usually based on notions of Banach algebras. One of earliest works in this area is [12] where the algebraic properties of Riccati equations is studied over C * -subalgebras of the space of bounded linear operators on some Hilbert space. More recent effort is reported [16] where the algebraic properties of Riccati equations is investigated over noncommutative involutive Banach algebras, which can be considered as a generalization of earlier results of [12].…”

“…A basic fundamental property of a Banach algebra is that it is a Banach space and locally convex. These nice properties allow us to apply existing methods in the literature to study the class of spatially distributed systems over Banach algebras; for example see [12,13,1,2,3,5,14,16,15].…”