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Preprint 14-2011

Spectra and leading directions for differential-algebraic equations

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Author(s) : Vu Hoang Linh , Volker Mehrmann

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 14-2011

MSC 2000

65L07 Numerical investigation of stability of solutions
65L80 Methods for differential-algebraic equations

Abstract :
The state of the art in the spectral theory of linear time-varying differential-algebraic equations (DAEs) is surveyed. To characterize the asymptotic behavior and the growth rate of solutions, basic spectral notions such as Lyapunov- and Bohl exponents, and Sacker-Sell spectra are discussed. For DAEs in strangeness-free form, the results extend those for ordinary differential equations, but only under additional conditions. This has consequences concerning the boundedness of solutions of inhomogeneous equations. Also, linear subspaces of leading directions are characterized, which are associated with spectral intervals and which generalize eigenvectors and invariant subspaces as they are used in the linear time-invariant setting.

Keywords : differential-algebraic equation, strangeness index, Lyapunov exponent, Bohl exponent, Sacker-Sell spectrum, exponential dichotomy, leading direction

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