Parts Of Eigenvalues Research Articles

Polynomial decay rate of C 0 semigroup in Hilbert spaces

In the paper, we characterize the polynomial stability of semigroup with generator A on Hilbert space . Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for . By the asymptotic relation of the real part and imaginary part of eigenvalues of A, we give the optimal decay rate of polynomial stability of . Moreover, we give the zeros distribution of certain equations. As an application, we illustrate our general results by an acoustical system.

The inverse problem for differential pencils on a star-shaped graph with mixed spectral data

The partial inverse problem for differential pencils on a star-shaped graph is studied from mixed spectral data. More precisely, we show that if the potentials on all edges on the star-shaped graph but one are known a priori then the unknown potential on the remaining edge can be uniquely determined by partial information on the potential and a part of eigenvalues.

Fuzzy multi-objective approach-based small signal stability analysis and optimal control of a PMSG-based wind turbine

The objective of this manuscript is to design a controller to enhance the degree of stability through small signal analysis in case of a grid connected permanent magnet synchronous generator (PMSG)-based wind turbine and to ensure an optimal set of control parameters to achieve an enhanced performance. The optimal control parameters are computed by optimising the placement of system eigenvalues and net errors by formulating a fuzzy-based multi-objective approach. The idea behind the formulation of the objective function through a multi-objective approach involves the association of error with relative stability of the system through computation of the real parts of eigenvalues. To find the optimal control gains, a two-fold mutation-based differential evolution optimisation is used. Results from a MATLAB-based model are presented for validation of the proposed technique to demonstrate the system stability when subjected to wind speed variation.

Spectral properties of discrete models of multi-dimensional elliptic problems with mixed derivatives

The influence of the spectrum of original and preconditioned matrices on a convergence rate of iterative methods for solving systems of finite-difference equations applicable to two-dimensional elliptic equations with mixed derivatives is investigated. It is shown that the efficiency of the bi-conjugate gradient iterative methods for systems with asymmetric matrices significantly depends not only on the matrix spectrum boundaries, but also on the heterogeneity of the distribution of the spectrum components, as well as on the magnitude of the imaginary part of complex eigenvalues. For test matrices with a fixed condition number, three variants of the spectral distribution were studied and the dependences of the number of iterations on the dimension of matrices were estimated. It is shown that the non-uniformity in the eigenvalue distribution within the fixed spectrum boundaries leads to a significant increase in the number of iterations with increasing dimension of the matrices. The increasing imaginary part of the eigenvalues has a similar effect on the convergence rate. Using as an example the model potential distribution problem in a square domain, including anisotropic ring inhomogeneity, a comparative analysis of the matrix structure and the convergence rate of the bi-conjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners is performed. It is shown that the advantages of the Fourier – Jacobi preconditioner are associated with a more uniform distribution of the spectrum of the preconditioned matrix along the real axis and a better suppression of the imaginary part of the spectrum compared to the preconditioner based on the incomplete LU factorization.

Extremal iota energy of a subclass of bicyclic digraphs and sidigraphs

The iota energy of an [Formula: see text]-vertex digraph [Formula: see text] is defined by [Formula: see text], where [Formula: see text] are eigenvalues of [Formula: see text] and [Formula: see text] is the imaginary part of eigenvalue [Formula: see text] Recently, this concept of iota energy of digraphs is extended to sidigraphs. The iota energy of a sidigraph is defined in an analogous way. In this paper, we consider a class of digraphs with two linear subdigraphs of equal length and find digraphs in this class with extremal iota energy. Furthermore, we take a similar class of sidigraphs and find minimal and maximal iota energy among the sidigraphs in this class.

Inverse problems for Sturm–Liouville operators on a star-shaped graph with mixed spectral data

The partial inverse spectral problem for Sturm–Liouville operators on a star-shaped graph was studied. The authors showed that if the potentials but one were known a priori, then the unknown potential on the whole interval can be uniquely determined by part of information of the potential and part of eigenvalues. The methods employed rest on the Weyl's m-function and theory concerning densities of zeros of entire functions.

Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays

In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.

Many-Objective Optimal and Robust Design of Proportional-Integral-Derivative Controls With a State Observer

This paper presents a many-objective optimization (MaOO) approach to design control and observer gains simultaneously. The many-objective optimization (MaOO) approach finds trade-offs between different nonagreeable design goals. We report the MaOO results of a proportional-integral-derivative (PID) control with a state estimator applied to a second-order oscillator. The overall system is optimized to minimize the peak time, overshoot, tracking error, and control energy, and maximize the rejection of external disturbance and measurement noise, while the relative stability, which is defined by maximum real part of eigenvalues of the closed-loop system, is constrained to be less than or equal to −1. The numerical simulations show promising findings of the proposed method.

An iterative method for computing a symplectic SVD-like decomposition

In this paper, we present an effective iterative method for computing symplectic SVD-like decomposition for a 2n-by-m rectangular real matrix A. The main purpose here is a block-power iterative method with the ortho-symplectic SR decomposition in the normalization step. We compute an k-block symplectic SVD-like decomposition, namely \(A_k=S_{k} \Sigma _{k} V_{k}^{T}\) where \(S_{k}\in \mathbb {R}^{2n\times 2k}\) is symplectic and \(V_{k}\in \mathbb {R}^{m\times 2k}\) is orthogonal. For large matrices, we usually have to compute only part of eigenvalues. The main interest of the proposed method is to efficiently compute the desired number of ordered in magnitude eigenvalues of structured matrices.

Distribution of transmission eigenvalues and inverse spectral analysis with partial information on the refractive index

In this work, we consider the interior transmission eigenvalue problem for a spherically stratified medium, which can be formulated as y′′(r) + k2η(r)y(r) = 0 endowed with boundary conditions , where the refractive index η(r) is positive and real. We obtain the distribution of transmission eigenvalues under assumptions that and one of these conditions that (i) η(1) ≠ 1, (ii) η(1) = 1,η′(1) ≠ 0, and (iii) η(1) = 1,η′(1) = 0,η″(1) ≠ 0, respectively. Moreover, in the case a = 1, we prove that if partial information on η(r) is known on subdomain, then only a part of eigenvalues can uniquely determine η(r) on the whole interval, and the relationship between the proportion of the missing eigenvalues and the subinterval of the known information on η(r) is revealed. Copyright © 2016 John Wiley & Sons, Ltd.

On the Eigenvalue Based Detection for Multiantenna Cognitive Radio System

Eigenvalue based spectrum sensing can make detection by catching correlation features in space and time domains, which can not only reduce the effect of noise uncertainty, but also achieve high detection probability. Hence, the eigenvalue based detection is always a hot topic in spectrum sensing area. However, most existing algorithms only consider part of eigenvalues rather than all the eigenvalues, which does not make full use of correlation of eigenvalues. Motivated by this, this paper focuses on multiantenna system and makes all the eigenvalues weighted for detection. Through the analysis of system model, we transfer the eigenvalue weighting issue to an optimal problem and derive the theoretical expression of detection threshold and probability of false alarm and obtain the close form expression of optimal solution. Finally, we propose new weighting schemes to give promotions of the detection performance. Simulations verify the efficiency of the proposed algorithms.

Dynamical invariants of open quantum systems

The dynamical invariant, whose expectation value is constant, is generalized to open quantum system. The evolution equation of dynamical invariant (the dynamical invariant condition) is presented for Markovian dynamics. Different with the dynamical invariant for the closed quantum system, the evolution of the dynamical invariant for the open quantum system is no longer unitary, and the eigenvalues of it are time-dependent. Since any hermitian operator fulfilling dynamical invariant condition is a dynamical invariant, we propose a sort of special dynamical invariant (decoherence free dynamical invariant) in which a part of eigenvalues are still constant. The dynamical invariant in the subspace spanned by the corresponding eigenstates evolves unitarily. Via the dynamical invariant condition, the results demonstrate that this dynamical invariant exists under the circumstances of emergence of decoherence free subspaces.

Eigenvalues of Jacobian Matrices Report on Steps of Metabolic Reprogramming in a Complex Plant-Environment Interaction

Mathematical modeling of biochemical systems aims at improving the knowledge about complex regulatory networks. The experimental high-throughput measurement of levels of biochemical components, like metabolites and proteins, has become an integral part for characterization of biological systems. Yet, strategies of mathematical modeling to functionally integrate resulting data sets is still challenging. In plant biology, regulatory strategies that determine the metabolic output of metabolism as a response to changes in environmental conditions are hardly traceable by intuition. Mathematical modeling has been shown to be a promising approach to address such problems of plant-environment interaction promoting the comprehensive understanding of plant biochemistry and physiology. In this context, we recently published an inversely calculated solution for first-order partial derivatives, i.e. the Jacobian matrix, from experimental high-throughput data of a plant biochemical model system. Here, we present a biomathematical strategy, comprising 1) the inverse calculation of a biochemical Jacobian; 2) the characterization of the associated eigenvalues and 3) the interpretation of the results with respect to biochemical regulation. Deriving the real parts of eigenvalues provides information about the stability of solutions of inverse calculations. We found that shifts of the eigenvalue real part distributions occur together with metabolic shifts induced by short-term and long-term exposure to low temperature. This indicates the suitability of mathematical Jacobian characterization for recognizing perturbations in the metabolic homeostasis of plant metabolism. Together with our previously published results on inverse Jacobian calculation this represents a comprehensive strategy of mathematical modeling for the analysis of complex biochemical systems and plant-environment interactions from the molecular to the ecosystems level.

Stability of the Jeffcott Rotor with Local Rubbing

The stability and bifurcation behavior of Jeffcott rotor with local rubbing are investigated in terms of Hartman-Grobman theorem in this paper. The case with double zero real part of eigenvalues is analyzed by means of the theory of center manifold and n-dimension Hopf bifurcation. Along with discussion for the effects of parameters on system stability and bifurcation behavior, numerical simulation of rotor locus is conducted and the stability condition is derived.

Identification of Damping and Dynamic Young’s Modulus of a Structural Adhesive Using Radial Basis Function Neural Networks and Modal Data

In this paper, the radial basis function neural networks (RBFNN) was applied to the problem of identifying dynamic Young’s modulus and damping characteristic of a structural adhesive, using modal data. To identify Young’s modulus from undamped model, an appropriate RBFNN using modal data (mode shape and natural frequency) in each mode is developed. Based on a previous work, in order to identify loss factor, two approaches adopted in the identification process. In the first one, a two stage RBFNN is developed. In stage I, Young’s modulus is identified from undamped model and in stage II using the results of stage I an appropriate RBFNN is developed in each mode for identification of loss factor by implementing real parts of eigenvalues of damped model. In the second approach, a one stage RBFNN is developed using real and imaginary parts of eigenvalues of damped model to identify Young’s moduli and loss factors simultaneously. The repeatability and consistency of the method is proved by repeating the identification process for several times. The validity of results is proved by comparing the results with those identified in a previous work.

Characterization of stationary mixing patterns in a three-dimensional open Stokes flow: spectral properties, localization and mixing regimes

This article analyses stationary scalar mixing downstream an open flow Couette device operating in the creeping flow regime. The device consists of two coaxial cylinders of finite length Lz, and radii κ R and R (κ < 1), which can rotate independently. At relatively large values of the aspect ratio α = Lz/R ≫ 1, and of the Péclet number Pe, the stationary response of the system can be accurately described by enforcing the simplifying assumption of negligible axial diffusion. With this approximation, homogenization along the device axis can be described by a family of generalized one-dimensional eigenvalue problems with the radial coordinate as independent variable. A variety of mixing regimes can be observed by varying the geometric and operating parameters. These regimes are characterized by different localization properties of the eigenfunctions and by different scaling laws of the real part of the eigenvalues with the Péclet number. The analysis of this model flow reveals the occurrence of sharp transitions between mixing regimes, e.g. controlled by the geometric parameter κ. The eigenvalue scalings can be theoretically predicted by enforcing eigenfunction localization and simple functional equalities relating the behaviour of the eigenvalues to the functional form of the associated eigenfunctions. Several flow protocols corresponding to different geometric and operating conditions are considered. Among these protocols, the case where the inner and the outer cylinders counter-rotate exhibits a peculiar intermediate scaling regime where the real part of the dominant eigenvalue is independent of Pe over more than two decades of Pe. This case is thoroughly analysed by means of scaling analysis. The practical relevance of the results deriving from spectral analysis for fluid mixing problems in finite-length Couette devices is addressed in detail.

Multi-stage design procedure for modal controllers of multi-input defective systems

The modal controller of single-input system cannot stabilize the defective system with positive real part of repeated eigenvalues, because some of the generalized modes are uncontrollable. In order to stabilize the uncontrollable modes with positive real part of eigenvalues, the multi-input system should be introduced. This paper presents a recursive procedure for designing the feedback controller of the multi-input system with defective repeated eigenvalues. For a nearly defective system, we first transform it into a defective one, and apply the same method to manage. The proposed methods are based on the modal coordinate equations, to avoid the tedious mathematic manipulation. As an application of the presented procedure, two numerical examples are given at end of the paper.

항공기용 하니콤 트림판넬의 다채널 능동제어

This paper summarizes theoretical work on the multichannel decentralized feedback control of sound radiation from aircraft trim panels using piezoceramic actuators. The aircraft trim panels are generally honeycomb structures designed to meet the design requirement of low weight and high stiffness. They are resiliently-mounted to the fuselage for the passive reduction of noise transmission. It is motivated by the localization of reduction in vibration of single channel active trim panels. 12-channel decentralized feedback control systems are investigated in terms of the reduction of noise and vibration for three configurations of sensor actuator pairs. Local coupling of the closely-spaced sensor and actuator pairs was modeled using single degree of freedom systems. The multichannel control system is characterized using the state-space model. For the stability point of view, the relative stability or robustness is evaluated by comparing the real part of eigenvalues of the system matrix for the three configurations. The control performance is also evaluated and compared for the three configurations. It is found that the multichannel system can lead to the globalization of the reduction in vibration and radiated noise. It does not appear to yield a significant improvement in the vibration because of decreased gain margin. However, the reduction in the radiated noise is remarkably improved due to the variation of the vibration pattern with the actuation configurations.

Three-dimensional asymptotic stress field in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity

An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the circumference of a bimaterial penny-shaped interfacial discontinuity, e.g., crack, anticrack (infinitely rigid lamella), etc., located at the center, edge or corner, and subjected to the far-field torsion (mode III), extension/bending (mode I), and sliding shear/twisting (mode II) loadings. Five different discontinuity-surface boundary conditions are considered: (1) bimaterial penny-shaped interface anticrack or perfectly bonded thin rigid inclusion, (2) bimaterial penny-shaped interfacial jammed contact, (3) bimaterial penny-shaped interface crack, (4) bimaterial penny-shaped interface crack with partial axisymmetric frictionless slip, and (5) bimaterial penny-shaped interface thin rigid inclusion alongside penny-shaped crack. Solutions to these cases except (3) are hitherto unavailable in the literature. Closed-form expressions for stress intensity factors subjected to various far-field loadings are also presented. Numerical results presented include the effect of the ratio of the shear moduli of the layer materials, and also Poisson’s ratios on the computed lowest real parts of eigenvalues for the case (5). Interesting and physically meaningful conclusions are also presented, especially with regard to cases (1) and (2).

On the exponential decay of magnetic stark resonances

We study the time decay of magnetic Stark resonant states. As a main result we prove that for sufficiently large time these states decay exponentially with the rate given by the imaginary parts of eigenvalues of certain non-selfadjoint operator. The proof is based on the method of complex translations.