Spectral Approach for Time–Invariant Systems with General Spatial Domain

  • Thomas Meurer
Part of the Communications and Control Engineering book series (CCE)


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.


Entire Function Invariant System Riesz Basis Feedforward Control Approximate Controllability 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Automation and Control Institute / E376Vienna University of TechnologyViennaAustria

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