Iterative solution of operator Lyapunov equations arising in heat transfer

We consider an iterative method for the numerical solution of Lyapunov equations of infinite-dimensional control systems governed by an heat equation. Inspired by the `alternating direction implicit (ADI)' iteration, which has been successfully applied to the solution of matrix Lyapunov equations, we present a method to determine approximations of the Gramian operator corresponding to the heat equation system. This method provides approximations of finite rank and is shown to be convergent in the nuclear norm.