Well-Posedness of the Gnedenko System with Multiple Vacations

2012 ◽  
Vol 522 ◽  
pp. 902-909
Author(s):  
Bilikiz Yunus ◽  
Abdukerim Haji
Keyword(s):  
Semigroup Theory ◽  
Linear Operators ◽  
Well Posedness ◽  
Multiple Vacations ◽  
Dynamic Solution ◽  
Multiple Vacation ◽  
Gnedenko System

We investigate the solution of the Gnedenko system with multiple vacation of a repairman. By using-semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

Stability of the Redundant System with Two Correlated Units

2012 ◽  
Vol 198-199 ◽  
pp. 973-977 ◽  
Author(s):  
Xing Qiao ◽  
Ming Fang
Keyword(s):  
Banach Space ◽  
Functional Analysis ◽  
Steady State ◽  
Semigroup Theory ◽  
Steady State Solution ◽  
Linear Operators ◽  
Well Posedness ◽  
Redundant System ◽  
Bounded Linear ◽  
System Operator

In this paper the qualitative behaviors of the two correlated units redundant system with two types of failure were discussed. By using the method of functional analysis, especially, the semigroup theory of bounded linear operators on Banach space, the well-posedness and the existence of positive solution of the system was obtained. By analyzing the spectra distribution of the system operator, the dynamic solution of the system asymptotically converges to the nonnegative steady-state solution which is the eigenfunction corresponding to eigenvalue 0 of the system operator was proved.

scholarly journals Exponential Stability and Availability Analysis of a Complex Standby System

2014 ◽  
Vol 2 (3) ◽  
pp. 279-288 ◽  
Author(s):  
Chunli Wang ◽  
Mingyan Teng ◽  
Fu Zheng
Keyword(s):  
Exponential Stability ◽  
Semigroup Theory ◽  
Steady State Solution ◽  
Linear Operators ◽  
Well Posedness ◽  
Redundant System ◽  
Bounded Linear ◽  
Reliability Indices ◽  
Availability Analysis ◽  
System Operator

Abstract In this paper, we first investigate the solution of the two correlated units redundant system with two types of failure. By using the method of functional analysis, especially, the c0 semigroup theory of bounded linear operators on Banach space, we prove the well-posedness and the existence of positive solution of the system. By analyzing the spectra distribution of the system operator, we prove that the dynamic solution of the system asymptotically converges to the nonnegative steady-state solution which is the eigenfunction corresponding to eigenvalue 0 of the system operator. Furthermore, we discuss the exponential stability of the system. Finally, we analyze the reliability the system with the help of our main results and present some reliability indices of the system at the end of the paper.

scholarly journals Hyperbolic Maxwell variational inequalities of the second kind

2020 ◽  
Vol 26 ◽  
pp. 34 ◽  
Author(s):  
Irwin Yousept
Keyword(s):  
Variational Inequalities ◽  
Existence Result ◽  
Semigroup Theory ◽  
Regularity Property ◽  
Well Posedness ◽  
Test Functions ◽  
Local Boundedness ◽  
Subdifferential Calculus ◽  
Faraday’S Law ◽  
Yosida Regularization

We analyze a class of hyperbolic Maxwell variational inequalities of the second kind. By means of a local boundedness assumption on the subdifferential of the underlying nonlinearity, we prove a well-posedness result, where the main tools for the proof are the semigroup theory for Maxwell’s equations, the Yosida regularization and the subdifferential calculus. The second part of the paper focuses on a more general case omitting the local boundedness assumption. In this case, taking into account more regular initial data and test functions, we are able to prove a weaker existence result through the use of the minimal section operator associated with the Nemytskii operator of the governing subdifferential. Eventually, we transfer the developed well-posedness results to the case involving Faraday’s law, which in particular allows us to improve the regularity property of the electric field in the weak existence result.

scholarly journals An age-dependent population equation with diffusion and delayed birth process

2005 ◽  
Vol 2005 (20) ◽  
pp. 3273-3289 ◽  
Author(s):  
G. Fragnelli
Keyword(s):  
Asymptotic Behavior ◽  
Semigroup Theory ◽  
New Age ◽  
Well Posedness ◽  
Birth Process ◽  
Age Dependent ◽  
Population Equation

We propose a new age-dependent population equation which takes into account not only a delay in the birth process, but also other events that may take place during the time between conception and birth. Using semigroup theory, we discuss the well posedness and the asymptotic behavior of the solution.

Feedback theory to the well-posedness of evolution equations

2020 ◽  
Vol 378 (2185) ◽  
pp. 20190616
Author(s):  
S. Boulite ◽  
S. Hadd ◽  
L. Maniar
Keyword(s):  
Evolution Equations ◽  
Semigroup Theory ◽  
Well Posedness ◽  
Boundary Problems ◽  
Neutral Equations ◽  
Infinite Dimensional ◽  
Research Activities ◽  
Semigroup Approach ◽  
History Of ◽  
And Control

In this paper, we cross the boundary between semigroup theory and general infinite-dimensional systems to bridge the isolated research activities in the two areas. Indeed, we first give a chronological history of the development of the semigroup approach for control theory. Second, we use the feedback theory to prove the well-posedness of a class of dynamic boundary problems. Third, the obtained results are applied to the well-posedness of neutral equations with non-autonomous past. We will also see that the strong connection between semigroup and control theories lies in feedback theory, where different kinds of perturbations appear. This article is part of the theme issue ‘Semigroup applications everywhere’.

scholarly journals Semigroup Methods for the M/G/1 Queueing Model with Working Vacation and Vacation Interruption

2016 ◽  
Vol 8 (5) ◽  
pp. 56 ◽  
Author(s):  
Ehmet Kasim
Keyword(s):  
Semigroup Theory ◽  
Steady State Solution ◽  
Linear Operators ◽  
Time Dependent ◽  
Queueing Model ◽  
Working Vacation ◽  
Positive Time ◽  
Vacation Interruption ◽  
Vacation Period ◽  
Time Dependent Solution

By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with working vacation and vacation interruption has a unique positive time dependent solution which satisfies probability conditions. When the both service completion rate in a working vacation period and in a regular busy period are constant, by investigating the spectral properties of an operator corresponding to the model we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.

scholarly journals Uniform stability for a type III thermoelastic laminated beam with structural damping

2022 ◽  
Author(s):  
Fayssal Djellali
Keyword(s):  
Heat Flux ◽  
Exponential Stability ◽  
Semigroup Theory ◽  
Stability Result ◽  
Uniform Stability ◽  
Well Posedness ◽  
Structural Damping ◽  
Laminated Beam ◽  
Wave Speeds ◽  
Lack Of Exponential Stability

In this work, we consider a thermoelastic laminated beam with structural damping, where the heat flux is given by Green and Naghdi theories. We establish the well-posedness of the system using semigroup theory. Moreover, under the condition of equal wave speeds, we prove an exponential stability result for the considered system. In the case of lack of exponential stability we show that the solution decays polynomially.

Well-posedness of fractional degenerate differential equations in Banach spaces

2019 ◽  
Vol 22 (2) ◽  
pp. 379-395
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Differential Equation ◽  
Fourier Multiplier ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Well Posedness ◽  
Multiplier Theorems ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Closed Linear Operators

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

Exponential stabilization of an ODE–linear KdV cascaded system with boundary input delay

2020 ◽  
Vol 37 (4) ◽  
pp. 1506-1523
Author(s):  
Habib Ayadi
Keyword(s):  
Differential Equation ◽  
Kdv Equation ◽  
Semigroup Theory ◽  
Functional Space ◽  
Input Delay ◽  
Exponential Stabilization ◽  
Target System ◽  
Control Input ◽  
Well Posedness ◽  
Bounded Interval

Abstract This paper considers the well posedness and the exponential stabilization problems of a cascaded ordinary differential equation (ODE)–partial differential equation (PDE) system. The considered system is governed by a linear ODE and the one-dimensional linear Korteweg–de Vries (KdV) equation posed on a bounded interval. For the whole system, a control input delay acts on the left boundary of the KdV domain by Dirichlet condition. Whereas, the KdV acts back on the ODE by Dirichlet interconnection on the right boundary. Firstly, we reformulate the system in question as an undelayed ODE-coupled KdV-transport system. Secondly, we use the so-called infinite dimensional backstepping method to derive an explicit feedback control law that transforms system under consideration to a well-posed and exponentially stable target system. Finally, by invertibility of such design, we use semigroup theory and Lyapunov analysis to prove the well posedness and the exponential stabilization in a suitable functional space of the original plant, respectively.

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