On linear-quadratic dissipative control processes with time-varying coefficients
Roberta Fabbri Russell Johnson Sylvia Novo Carmen Núñez
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 193-210 doi: 10.3934/dcds.2013.33.193
Yakubovich, Fradkov, Hill and Proskurnikov have used the Yaku-bovich Frequency Theorem to prove that a strictly dissipative linear-quadratic control process with periodic coefficients admits a storage function, and various related results. We extend their analysis to the case when the coefficients are bounded uniformly continuous functions.
keywords: dissipativity Linear-quadratic control system supply rate storage function.
On the spectrum of the one-dimensional Schrödinger operator
Nguyen Dinh Cong Roberta Fabbri
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 541-554 doi: 10.3934/dcdsb.2008.9.541
The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
keywords: Quasi-periodic Schrödinger operator Cantor spectrum exponential dichotomy.
Roberta Fabbri Carmen Núñez
Communications on Pure & Applied Analysis 2011, 10(3): i-iii doi: 10.3934/cpaa.2011.10.3i
This special issue collects eleven papers in the general area of nonautonomous dynamical systems. They contain a rich selection of new results on pure and applied aspects of the eld.
keywords: no
Roberta Fabbri Russell Johnson Carmen Núñez Rafael Obaya
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): i-ii doi: 10.3934/dcdsb.2008.9.3i
In recent years, the area of nonautonomous dynamical systems has matured into a field with recognizable contours together with well-defined themes and methods. Its development has been strongly stimulated by various problems of applied mathematics, and it has in its turn influenced such areas of applied and pure mathematics as spectral theory, stability theory, bifurcation theory, the theory of bounded/recurrent motions, etc. Much work in this field concerns the asymptotic properties of the solutions of a nonautonomous differential or discrete system. However, that is by no means always the case, and the reader will find papers in this volume which are concerned only at a distance or not at all with asymptotic matters.
   There is a close relation between the field of nonautonomous dynamical systems and that of stochastic dynamical systems. They can be distinguished to a certain extent by the observation that a nonautonomous dynamical system often arises from the study of a differential or discrete system whose coefficients depend on time, but in a non-stochastic way. The limiting case is that of periodic coefficients, but one is also interested in equations whose coefficients exhibit weaker recurrence properties; for example almost periodicity, Birkhoff recurrence, Poisson recurrence, etc. A distinction also occurs on the methodological level in that topological methods tend to find more application in the former field as compared to the latter (while analytical and ergodic tools are heavily used in both). In any case, some people use the term “random dynamics” to refer to both fields in a more or less interchangeable way.

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A perturbation theorem for linear Hamiltonian systems with bounded orbits
Roberta Fabbri Carmen Núñez Ana M. Sanz
Discrete & Continuous Dynamical Systems - A 2005, 13(3): 623-635 doi: 10.3934/dcds.2005.13.623
This paper concerns the Sacker-Sell spectral decomposition of a one-parametric perturbation of a non-autonomous linear Hamiltonian system with bounded solutions. Conditions ensuring the continuous variation with respect to the parameter of the spectral intervals and subbundles are established. These conditions depend on the perturbation direction and are closely related to the topological structure of the flows induced by the initial system on the real and complex Lagrange bundles.
keywords: Linear Hamiltonian systems Sacker-Sell spectral decomposition perturbation theorems.
On the Yakubovich frequency theorem for linear non-autonomous control processes
Roberta Fabbri Russell Johnson Carmen Núñez
Discrete & Continuous Dynamical Systems - A 2003, 9(3): 677-704 doi: 10.3934/dcds.2003.9.677
Using methods of the theory of nonautonomous linear differential systems, namely exponential dichotomies and rotation numbers, we generalize some aspects of Yakubovich's Frequency Theorem from periodic control systems to systems with bounded uniformly continuous coefficients.
keywords: exponential dichotomy rotation number. Frequency Theorem

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