Measuring sparsity in spatially interconnected systems

The goal of this paper is to develop a mathematical framework to measure sparsity of state feedback controllers for spatially interconnected systems. We introduce a new algebra of infinite-dimensional matrices equipped with a matrix quasi-norm which is defined using ℓ^q quasi-norm for 0 <; q ≤ 1. When q = 0, the value of the matrix quasi-norm is equal to the maximum number of nonzero entries in rows or columns of a matrix. When 0 <; q ≤ 1, the proposed matrix algebra forms a mathematical object so called q-Banach algebra, which is not a Banach algebra. We show that this matrix algebra is inverse-closed. Moreover, we prove that the unique solutions of Lyapunov and Riccati equations belong to this matrix algebra. We show that there exists a nonzero q for which the value of the matrix quasi-norm reflects a reasonable estimate for sparsity of a spatially decaying matrix.