Solution Formulas for Differential Sylvester and Lyapunov Equations
For researchers in control theory, system theory, and model order reduction, this work provides a more efficient numerical method for large-scale systems, though it is incremental as it builds on existing spectral and Krylov subspace techniques.
The paper addresses the computational challenges of solving large-scale differential Sylvester and Lyapunov equations, proposing a unifying spectral-based approach and a Krylov subspace algorithm with Taylor series projection. Numerical results demonstrate effective approximations for large-scale differential Lyapunov equations.
The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches if applied to large-scale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator $\mathcal S(X)=AX+XB$ and derive a formula for its norm using an induced operator norm based on the spectrum of $A$ and $B$. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.