Approximate Controllability of Infinite Dimensional Systems of the n-th Order

Authors:
Jerzy Respondek
Affiliations:
Institute of Computer Science, Silesian University of Technology, ul. Akademicka 16, 44-100 Gliwice, Poland
Venue:
International Journal of Applied Mathematics and Computer Science - Special Section: Selected Topics in Biological Cybernetics, Special Editors: Andrzej Kasiński and Filip Ponulak
Year:
2008

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Abstract

The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation. This innovative approach was possible owing to analyzing the n-th order linear system in the Frobenius form which generates a Jordan transition matrix of the Vandermonde form. We extensively used the fact that the knowledge of the inverse of a Jordan transition matrix enables us to directly verify the controllability by Chen's theorem. We used the explicit analytical form of the inverse Vandermonde matrix. This enabled us to obtain more general conditions for different types of controllability for infinite dimensional systems than the conditions existing in the literature so far. The methods introduced can be easily adapted to the analysis of other dynamic properties of the systems considered.