104 research outputs found
Finite Frequency H
This paper investigates the problem of finite frequency (FF) Hâ filtering for time-delayed singularly perturbed systems. Our attention is focused on designing filters guaranteeing asymptotic stability and FF Hâ disturbance attenuation level of the filtering error system. By the generalized Kalman-Yakubovich-Popov (KYP) lemma, the existence conditions of FF Hâ filters are obtained in terms of solving an optimization problem, which is delay-independent. The main contribution of this paper is that systematic methods are proposed for designing Hâ filters for delayed singularly perturbed systems
An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection
We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV
transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These
qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This
method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic
diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the
other conventional approaches that are routinely used for such problems.IS
Stability of singular jump-linear systems with a large state space : a two-time-scale approach
This paper considers singular systems that involve both continuous dynamics and discrete events with the coefficients being modulated by a continuous-time Markov chain. The underlying systems have two distinct characteristics. First, the systems are singular, that is, characterized by a singular coefficient matrix. Second, the Markov chain of the modulating force has a large state space. We focus on stability of such hybrid singular systems. To carry out the analysis, we use a two-time-scale formulation, which is based on the rationale that, in a large-scale system, not all components or subsystems change at the same speed. To highlight the different rates of variation, we introduce a small parameter Δ>0. Under suitable conditions, the system has a limit. We then use a perturbed Lyapunov function argument to show that if the limit system is stable then so is the original system in a suitable sense for Δ small enough. This result presents a perspective on reduction of complexity from a stability point of view
Fitted numerical methods for delay differential equations arising in biology
Philosophiae Doctor - PhDFitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape.
This thesis deals with the design and analysis of tted numerical methods
for some delay di erential models that arise in biology. Very often such
di erential equations are very complex in nature and hence the well-known
standard numerical methods seldom produce reliable numerical solutions
to these problems. Ine ciencies of these methods are mostly accumulated
due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are
initially designed to solve a class of general problems without considering
the structure of any individual problems. In this thesis, issues like these
are resolved for a set of delay di erential equations. Though the developed
approaches are very simplistic in nature, they could solve very complex
problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better.
Finally, wherever appropriate, we have indicated possible extensions of
our approaches to cater for other classes of problems. May 2009
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