scholarly journals Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2617
Author(s):  
Natalia P. Bondarenko ◽  
Andrey V. Gaidel
Keyword(s):  
Inverse Problem ◽  
Inverse Spectral Problem ◽  
Sufficient Conditions ◽  
Local Solvability ◽  
Global Solvability ◽  
Multiple Eigenvalues ◽  
Eigenvalue Multiplicities ◽  
Complex Valued ◽  
Theoretical Results ◽  
Infinite Sequences

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

scholarly journals Inverse Spectral Problem for Integrodifferential Sturm-Liouville Operators with Discontinuity Conditions

2018 ◽  
Vol 64 (3) ◽  
pp. 427-458 ◽  
Author(s):  
S A Buterin
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Global Solvability ◽  
Convolution Integral ◽  
Main Equation ◽  
Convolution Integral Operator ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Discontinuity Conditions ◽  
Liouville Operators

We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a finite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodifferential expression under consideration. We prove the global solvability of the main equation, this implies the uniqueness of solution of the inverse problem and leads to necessary and sufficient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem.

scholarly journals Passivity analysis of multiple time-varying time delayed complex variable neural networks in finite-time

2021 ◽  
Vol 8 (4) ◽  
pp. 842-854
Author(s):  
N. Jayanthi ◽  
◽  
R. Santhakumari ◽  
Keyword(s):  
Neural Networks ◽  
Finite Time ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Linear Matrix ◽  
Multiple Time ◽  
Time Varying ◽  
Matrix Inequalities ◽  
Complex Valued ◽  
Theoretical Results

In this article, we investigate the problem of finite-time passivity for the complex-valued neural networks (CVNNs) with multiple time-varying delays. To begin, many definitions relevant to the finite-time passivity of CVNNs are provided; then the suitable control inputs are designed to guarantee the class of CVNNs are finite-time passive. In the meantime, some sufficient conditions of linear matrix inequalities (LMIs) are derived by using inequalities techniques and Lyapunov stability theory. Finally, a numerical example is presented to illustrate the usefulness of the theoretical results.

scholarly journals Inverse spectral problem for Dirac operators by spectral data

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1065-1077
Author(s):  
Ozge Akcay ◽  
Khanlar Mamedov
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Spectral Problem ◽  
Spectral Data ◽  
Spectral Parameter ◽  
Inverse Spectral Problem ◽  
Sufficient Conditions ◽  
Dirac Operators ◽  
Piecewise Continuous ◽  
Necessary And Sufficient

This work deals with the solution of the inverse problem by spectral data for Dirac operators with piecewise continuous coefficient and spectral parameter contained in boundary condition. The main theorem on necessary and sufficient conditions for the solvability of inverse problem is proved. The algorithm of the reconstruction of potential according to spectral data is given.

Periodic oscillation for a complex-valued neural network model with discrete and distributed delays

2020 ◽  
Vol 8 (5) ◽  
Author(s):  
Chunhua Feng
Keyword(s):  
Neural Network ◽  
Network Model ◽  
Neural Network Model ◽  
Sufficient Conditions ◽  
Periodic Oscillation ◽  
Activation Function ◽  
Distributed Delays ◽  
Oscillatory Solutions ◽  
Complex Valued ◽  
Theoretical Results

In this paper, a complex-valued neural network model with discrete and distributed delays is investigated under the assumption that the activation function can be separated into its real and imaginary parts. Based on the mathematical analysis method, some sufficient conditions to guarantee the existence of periodic oscillatory solutions are established. Computer simulation is given to illustrate the validity of the theoretical results.

On recovering a Sturm–Liouville-type operator with the frozen argument rationally proportioned to the interval length

2019 ◽  
Vol 27 (3) ◽  
pp. 429-438 ◽  
Author(s):  
Sergey A. Buterin ◽  
Sergey V. Vasiliev
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Interval Length ◽  
Liouville Type ◽  
Type Property ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Complex Valued ◽  
The Uniqueness Theorem

Abstract We consider the operator {\ell y\mathrel{\mathop{:}}=-y^{\prime\prime}(x)+q(x)y(a)} , {0<x<\pi} , {y(0)=y(\pi)=0} , where {q(x)\in L_{2}(0,\pi)} is a complex-valued function and {a/\pi\in[0,1]} is a rational number. The inverse problem of recovering the potential {q(x)} from the spectrum of {\ell} is studied. We describe the sets of iso-spectral potentials and prove the uniqueness theorem in the class of potentials possessing some symmetry-type property. Moreover, we obtain a constructive procedure for solving this inverse problem along with necessary and sufficient conditions of its solvability, which in turn give the characterization of the spectrum. In parallel, we establish that the informativity of the spectrum is severely unstable with respect to the parameter a.

scholarly journals Stability Analysis for Memristor-Based Complex-Valued Neural Networks with Time Delays

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 120
Author(s):  
Ping Hou ◽  
Jun Hu ◽  
Jie Gao ◽  
Peican Zhu
Keyword(s):  
Numerical Simulation ◽  
Neural Networks ◽  
Stability Analysis ◽  
Exponential Stability ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Activation Functions ◽  
Complex Valued ◽  
Theoretical Results

In this paper, the problem of stability analysis for memristor-based complex-valued neural networks (MCVNNs) with time-varying delays is investigated extensively. This paper focuses on the exponential stability of the MCVNNs with time-varying delays. By means of the Brouwer’s fixed-point theorem and M-matrix, the existence, uniqueness, and exponential stability of the equilibrium point for MCVNNs are studied, and several sufficient conditions are obtained. In particular, these results can be applied to general MCVNNs whether the activation functions could be explicitly described by dividing into two parts of the real parts and imaginary parts or not. Two numerical simulation examples are provided to illustrate the effectiveness of the theoretical results.

scholarly journals Inverse Sturm-Liouville problem with analytical functions in the boundary condition

2020 ◽  
Vol 18 (1) ◽  
pp. 512-528 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problem ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Liouville Problem ◽  
Local Solvability ◽  
Cauchy Data ◽  
Problem Solution ◽  
Analytical Functions ◽  
Half Inverse Problem ◽  
Sturm Liouville

Abstract The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

scholarly journals Spectral analysis for the matrix Sturm-Liouville operator on a finite interval

2011 ◽  
Vol 42 (3) ◽  
pp. 305-327 ◽  
Author(s):  
Natalia Bondarenko
Keyword(s):  
Inverse Problem ◽  
Spectral Analysis ◽  
Liouville Equation ◽  
Inverse Spectral Problem ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Sufficient Conditions ◽  
The Matrix ◽  
Sturm Liouville ◽  
Necessary And Sufficient

The inverse spectral problem is investigated for the matrix Sturm-Liouville equation on a finite interval. Properties of spectral characteristics are provided, a constructive procedure for the solution of the inverse problem along with necessary and sufficient conditions for its solvability is obtained.

scholarly journals Solving An Inverse Problem For The Sturm-Liouville Operator With A Singular Potential By Yurko's Method

2020 ◽  
Vol 52 (1) ◽  
Author(s):  
Natalia P. Bondarenko
Keyword(s):  
Inverse Problem ◽  
Liouville Operator ◽  
Singular Potential ◽  
Inverse Spectral Problem ◽  
Sufficient Conditions ◽  
The Self ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
The Uniqueness Theorem

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases.

Stability and Hopf Bifurcation of Fractional-Order Complex-Valued Single Neuron Model with Time Delay

2017 ◽  
Vol 27 (13) ◽  
pp. 1750209 ◽  
Author(s):  
Zhen Wang ◽  
Xiaohong Wang ◽  
Yuxia Li ◽  
Xia Huang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Fractional Order ◽  
Single Neuron ◽  
Neuron Model ◽  
Sufficient Conditions ◽  
Order Complex ◽  
The Stability ◽  
Complex Valued ◽  
Theoretical Results

In this paper, the problems of stability and Hopf bifurcation in a class of fractional-order complex-valued single neuron model with time delay are addressed. With the help of the stability theory of fractional-order differential equations and Laplace transforms, several new sufficient conditions, which ensure the stability of the system are derived. Taking the time delay as the bifurcation parameter, Hopf bifurcation is investigated and the critical value of the time delay for the occurrence of Hopf bifurcation is determined. Finally, two representative numerical examples are given to show the effectiveness of the theoretical results.

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