Iterative methods for the delay Lyapunov equation with T-Sylvester preconditioning

The delay Lyapunov equation is an important matrix boundary-value problem which arises as an analogue of the Lyapunov equation in the study of time-delay systems $\dot{x}(t) = A_0x(t)+A_1x(t-\tau)+B_0u(t)$. We propose a new algorithm for the solution of the delay Lyapunov equation. Our method is based on the fact that the delay Lyapunov equation can be expressed as a linear system of equations, whose unknown is the value $U(\tau/2)\in\mathbb{R}^{n\times n}$, i.e., the delay Lyapunov matrix at time $\tau/2$. This linear matrix equation with $n^2$ unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the $n^2\times n^2$ matrix associated to this linear system can be computed by solving a coupled matrix initial-value problem. A preconditioner for the iterative method is proposed based on solving a T-Sylvester equation $MX+X^TN=C$, for which there are methods available in the literature. We prove that the preconditioner is effective under certain assumptions. The efficiency of the approach is illustrated by applying it to a time-delay system stemmingfrom the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to $n\approx 1000$, i.e., a linear system with $n^2\approx 10^6$ unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation.