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Control of Higher–Dimensional PDEs pp 83–188Cite as

Spectral Approach for Time–Invariant Systems with General Spatial Domain

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Abstract

The spectral analysis of a finite– or infinite–dimensional linear operator is a well–established and profound mathematical tool for stability analysis and feedback control design. The dynamic system properties are thereby determined based on the eigenvalue distribution and the respective set of eigenvectors. For infinite–dimensional systems governed by PDEs certain restrictions apply, which are in particular related to the possible existence of continuous spectra. Fortunately, a wide class of physically important systems including, e.g., diffusion–convection–reaction, wave, Euler–Bernoulli, and Timoshenko beam equations, yields so–called Riesz spectral operators, which exhibit a purely discrete eigenvalue distribution and whose eigenvectors and adjoint eigenvectors, respectively, span a basis for the underlying function space. These properties can be advantageously exploited for the controllability and observability analysis similar to the finite–dimensional case [14]. Furthermore, Riesz spectral operators satisfy the spectrum determined growth assumption such that the stability properties of the system can be directly determined based on the eigenvalue distribution [14, 37]. This property can be in particular utilized for the stabilizability and stability analysis as well as for the design of stabilizing state–feedback controllers, see, e.g., [65, 33, 13, 48, 49, 37] and the references therein.

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  1. Automation and Control Institute / E376, Vienna University of Technology, Gusshausstrasse 27-29, 1040, Vienna, Austria

    Thomas Meurer

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Meurer, T. (2013). Spectral Approach for Time–Invariant Systems with General Spatial Domain. In: Control of Higher–Dimensional PDEs. Communications and Control Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30015-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-30015-8_6

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