Determination of transient effect in time-dependent linear dynamical system using condition spectrum. (English) Zbl 07878125
Summary: Condition spectrum is an essential generalization of the spectrum. This article considers the condition spectrum of bounded linear operators on Banach space and develops certain topological properties. It is observed that the condition spectrum is useful than the spectrum and pseudospectrum for identifying the norm behavior of non-normal matrices. For a bounded linear operator \(A\) on a Banach space, we find upper and lower bounds for \(\Vert e^{tA}\Vert, \, t\geq 0\) and \(\Vert A^n\Vert,\, n=1,2,\ldots\) using the condition spectrum of \(A\). These bounds are used to identify the transient effect of the quantities appearing in the time dependent linear dynamical system.
MSC:
| 47A10 | Spectrum, resolvent |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 37C75 | Stability theory for smooth dynamical systems |
| 47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
Keywords:
spectrum; resolvent norm; pseudospectrum; condition spectrum; time dependent linear dynamical system; transient effectReferences:
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