A Study on the Spectrum of the Sampled-Data Transfer Operator with Application to Robust Exponential Stability Problems
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Abstract:
This paper begins by studying some spectral properties of the transfer operators of sampled-data systems described by applying the lifting technique. Through a "nonasymptotic" characterization of the transfer operator, its spectrum is determined in terms of finite-dimensional eigenvalue problems. Then, it is shown that a close connection with such eigenvalue problems and the exponential stability condition can be exploited to study the robust internal (exponential) stability problem of sampled-data systems. Since the transfer operator is relevant to input-output characteristics, the relationship between input-output stability and internal stability is also discussed in the context of sampled-data systems.Keywords:
Transfer operator
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Transfer operator-based approaches have been successfully applied to the extraction of coherent features in flows. Transfer operators describe the evolution of densities under the action of the flow. They can be efficiently approximated within a set-oriented numerical framework and spectral properties of the resulting stochastic matrices are used to extract finite-time coherent sets. Also finite-time entropy, a density-based stretching quantity similar to finite-time Lyapunov exponents, is conveniently approximated by means of the discretized transfer operator. Transfer operator-based computational methods are purely probabilistic and derivative-free. Therefore, they can also be applied in settings where derivatives of the flow map are hardly accessible. In this paper, we summarize the theory and numerics behind the transfer operator approach and then introduce a straightforward extension, which allows us to study coherent structures in complex flows on triangulated surfaces. We illustrate our general computational framework with the well-known periodically driven double-gyre flow. To demonstrate the applicability of the approach for complex flows, we consider an approximation of the surface ocean flow, obtained by a numerical solution of the incompressible surface Navier-Stokes equation in a complicated geometry on the sphere.
Transfer operator
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We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.
Transfer operator
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Smoothness
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The stability of equilibrium states for Cohen-Grossberg generalized neural networks with time-varying delays is discussed. By employing the extended Halanay's delay differential inequality, constructing suitable Lyapunov functions and the inequality technique discussed the exponential stability of equilibrium states of generalized neural networks with time-varying delays. Some sufficient criteria for the asymptotic exponential asymptotic stability of the system are obtained.
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Investment
Transfer operator
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In this short note, we propose to extend differentiability (with respect to a multidimensional parameter) of a normalized eigenfunction associated to the simple, dominating eigenvalue of the weighted transfer operator for a uniformly expanding map, to the whole discrete spectrum. We do so by studying directly the regularity (with respect to the parameter) of the resolvent operator, applying a general regularity result for fixed points of maps having loss of regularity.
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Transfer operator
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This brief studies the global asymptotic stability and the global exponential stability of neural networks with unbounded time-varying delays and with bounded and Lipschitz continuous activation functions. Several sufficient conditions for the global exponential stability and global asymptotic stability of such neural networks are derived. The new results given in the brief extend the existing relevant stability results in the literature to cover more general neural networks.
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Transfer operator
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We analyze the spectrum of a discrete Schrodinger operator with a potential given by a periodic variant of the Anderson Model. In order to do so, we study the uniform hyperbolicity of a Schrodinger cocycle generated by the SL(2,R) transfer matrices. In the specific case of the potential generated by an alternating sequence of random values we show that the almost sure spectrum consists of at most 4 intervals.
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Sequence (biology)
Transfer operator
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