On circulant Lyapunov operators, two-variable polynomials, and DFT

In this paper we deal with a special class of Lyapunov equations which we call circulant Lyapunov equations. These are Lyapunov equations arising from system matrices that are circulant. Here we bring out new theoretical insights into such Lyapunov operators. We show that, under a suitable projection map, two-variable polynomials are related to circulant Lyapunov operators. We also give necessary and sufficient conditions for the solvability of a circulant Lyapunov equation using two dimensional Fourier transform operation (2D-DFT) performed on a specially constructed matrix. Using these links among circulant Lyapunov operators, two-variable polynomials and 2D-DFT, an algorithm to solve circulant Lyapunov equations using 2D-DFT is developed in this paper.