Solution Formulas for Differential Sylvester and Lyapunov Equations
| Autor: | Jan Heiland, Maximilian Behr, Peter Benner |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Lyapunov function
0209 industrial biotechnology MathematicsofComputing_NUMERICALANALYSIS 15A24 65F60 65L05 010103 numerical & computational mathematics 02 engineering and technology Spectral theorem 01 natural sciences symbols.namesake 020901 industrial engineering & automation Operator (computer programming) ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION FOS: Mathematics Applied mathematics Lyapunov equation Mathematics - Numerical Analysis 0101 mathematics Mathematics Algebra and Number Theory Numerical analysis Krylov subspace Numerical Analysis (math.NA) Computational Mathematics symbols Sylvester equation Operator norm |
| Zdroj: | CALCOLO : A Quarterly on Numerical Analysis and Theory of Computation |
| Popis: | 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. 30 pages, 51 figures |
| Databáze: | OpenAIRE |
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