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 ẋ(t) = A0x(t) +A1x(t− τ) +B0u(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(τ/2) ∈ Rn×n, i.e., the delay Lyapunov matrix at time τ/2. This linear matrix equation with n unknowns is solved by adapting a preconditioned iterative method such as GMRES. The action of the n × n 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+XN = 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 stemming from the discretization of a partial differential equation with delay. Approximate solutions to this problem can be obtained for problems of size up to n ≈ 1000, i.e., a linear system with n ≈ 10 unknowns, a dimension which is outside of the capabilities of the other existing methods for the delay Lyapunov equation.