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1
Content available remote

An unconditionally positive and global stability preserving NSFD scheme for an epidemic model with vaccination

100%
Ding D., Ma Q., Ding X.
International Journal of Applied Mathematics and Computer Science
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2014
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tom 24
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nr 3
635-646
EN
In this paper, a NonStandard Finite Difference (NSFD) scheme is constructed, which can be used to determine numerical solutions for an epidemic model with vaccination. Here the NSFD method is employed to derive a set of difference equations for the epidemic model with vaccination. We show that difference equations have the same dynamics as the original differential system, such as the positivity of the solutions and the stability of the equilibria, without being restricted by the time step. Our proof of global stability utilizes the method of Lyapunov functions. Numerical simulation illustrates the effectiveness of our results.
2
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A constructive method for solving stabilization problems

100%
Azhmyakov V.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2000
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tom 20
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nr 1
51-62
EN
The problem of asymptotic stabilization for a class of differential inclusions is considered. The problem of choosing the Lyapunov functions from the parametric class of polynomials for differential inclusions is reduced to that of searching saddle points of a suitable function. A numerical algorithm is used for this purpose. All the results thus obtained can be extended to cover the discrete systems described by difference inclusions.
3
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The choice of the forms of Lyapunov functions for a positive 2D Roesser model

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2007
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tom 17
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nr 4
471-475
EN
The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix P such that the matrix A^{T}PA-P is negative definite. The theoretical deliberations will be illustrated by numerical examples.
4
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Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

100%
Gluzman M. O., Gorban N. V., Kasyanov P. O.
Nonautonomous Dynamical Systems
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2015
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tom 2
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nr 1
EN
In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction functions; (ii) convergence results for all weak solutions in the strongest topologies; (iii) new structure and regularity properties for global and trajectory attractors. The obtained results allow investigating the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; (c) a climate energy balance model; (d) a parabolic feedback control problem.
5
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Gradient systems of closed operators

100%
Pata V.
Open Mathematics
|
2009
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tom 7
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nr 3
487-492
EN
A classical result on the existence of global attractors for gradient systems is extended to the case of a semigroup S(t) lacking strong continuity, but satisfying the weaker property of being a closed map for every fixed t ≥ 0.
6
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Stability of impulsive hopfield neural networks with Markovian switching and time-varying delays

88%
Raja R., Sakthivel R., Anthoni S., Kim H.
International Journal of Applied Mathematics and Computer Science
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2011
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tom 21
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nr 1
127-135
EN
The paper is concerned with stability analysis for a class of impulsive Hopfield neural networks with Markovian jumping parameters and time-varying delays. The jumping parameters considered here are generated from a continuous-time discrete-state homogenous Markov process. By employing a Lyapunov functional approach, new delay-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities (LMIs). The proposed criteria can be easily checked by using some standard numerical packages such as the Matlab LMI Toolbox. A numerical example is provided to show that the proposed results significantly improve the allowable upper bounds of delays over some results existing in the literature.
7
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A backstepping approach to ship course control

88%
Witkowska A., Tomera M., Smierzchalski R.
International Journal of Applied Mathematics and Computer Science
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2007
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tom 17
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nr 1
73-85
EN
As an object of course control, the ship is characterised by a nonlinear function describing static manoeuvring characteristics that reflect the steady-state relation between the rudder deflection and the rate of turn of the hull. One of the methods which can be used for designing a nonlinear ship course controller is the backstepping method. It is used here for designing two configurations of nonlinear controllers, which are then applied to ship course control. The parameters of the obtained nonlinear control structures are tuned to optimise the operation of the control system. The optimisation is performed using genetic algorithms. The quality of operation of the designed control algorithms is checked in simulation tests performed on the mathematical model of a tanker. In order to obtain reference results to be used for comparison with those recorded for nonlinear controllers designed using the backstepping method, a control system with the PD controller is examined as well.
8
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Pullback incremental attraction

88%
Kloeden P. E., Lorenz T.
Nonautonomous Dynamical Systems
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2014
|
tom 1
EN
A pullback incremental attraction, a nonautonomous version of incremental stability, is introduced for nonautonomous systems that may have unbounded limiting solutions. Its characterisation by a Lyapunov function is indicated.
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