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Controllability of partial differential equations on graphs

100%
Avdonin S., Mikhaylov V.
Applicationes Mathematicae
|
2008
|
tom 35
|
nr 4
379-393
EN
We study boundary control problems for the wave, heat, and Schrödinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. Exact controllability in L₂-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. Null controllability for the heat equation and exact controllability for the Schrödinger equation in any time interval are obtained.
2
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Ingham-type inequalities and Riesz bases of divided differences

100%
Avdonin S., Moran W.
International Journal of Applied Mathematics and Computer Science
|
2001
|
tom 11
|
nr 4
803-820
EN
We study linear combinations of exponentials e^{iλ_nt} , λ_n ∈ Λ in the case where the distance between some points λ_n tends to zero. We suppose that the sequence Λ 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 Λ is less than T/(2π), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to {e^{iλ_nt}} a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2π), the family of divided differences can be made into a Riesz basis by removing from {e^{iλ_nt}} a suitable collection of functions e^{iλ_nt}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.
3
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Construction of sampling and interpolating sequences for multi-band signals. the two-band case

81%
Avdonin S., Bulanova A., Moran W.
International Journal of Applied Mathematics and Computer Science
|
2007
|
tom 17
|
nr 2
143-156
EN
Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and the controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence, and prove the invertibility of this operator.
4
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Exact boundary controllability of coupled hyperbolic equations

81%
Avdonin S., Choque Rivero A., de Teresa L.
International Journal of Applied Mathematics and Computer Science
|
2013
|
tom 23
|
nr 4
701-710
EN
We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.
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