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2 International Journal of Applied Mathematics and Computer Science

2 2001

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Well-Posed Linear Systems - a Survey With Emphasis on Conservative Systems

Weiss G., Staffans O. J., Tucsnak M.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 1

7-33

We survey the literature on well-posed linear systems, which has been an area of rapid development in recent years. We examine the particular subclass of conservative systems and its connections to scattering theory. We study some transformations of well-posed systems, namely duality and time-flow inversion, and their effect on the transfer function and the generating operators. We describe a simple way to generate conservative systems via a second-order differential equation in a Hilbert space. We give results about the stability, controllability and observability of such conservative systems and illustrate these with a system modeling a controlled beam.

Ingham-Type Inequalities and Riesz Bases of Divided Differences

Avdonin S., Moran W.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 4

803-820

We study linear combinations of exponentials e^{i lambda_n t}, lambda_n in Lambda in the case where the distance between some points lambda_n tends to zero. We suppose that the sequence Lambda is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L^2 (0,T). Here we prove that if the upper uniform density of Lambda is less than T/(2 pi), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to {e^{i lambda_n t} } a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2 pi), the family of divided differences can be made into a Riesz basis by removing from {e^{i lambda_n t} } a suitable collection of functions e^{i lambda_n t}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.