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1

Differential equations with random Gamma distributed moments of non-instantaneous impulses and p-moment exponential stability

Agarwal R., Hristova S., O’Regan D., Kopanov P.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

151--170

EN

Nonlinear differential equations with impulses occurring at random time and acting noninstantaneously on finite intervals are studied. We consider the case when the time where the impulses occur is Gamma distributed. Lyapunov functions are applied to obtain sufficient conditions for the p-moment exponential stability of the trivial solution of the given system.

2

Stability analysis of high-order Hopfield-type neural networks based on a new impulsive differential inequality

Liu Y., Yang R., Lu J., Wu B., Cai X.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 1

201--211

EN

This paper is devoted to studying the globally exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays. In the process of impulsive effect, nonlinear and delayed factors are simultaneously considered. A new impulsive differential inequality is derived based on the Lyapunov–Razumikhin method and some novel stability criteria are then given. These conditions, ensuring the global exponential stability, are simpler and less conservative than some of the previous results. Finally, two numerical examples are given to illustrate the advantages of the obtained results.

3

Observer-based controller design of time-delay systems with an interval time-varying delay

Thuan M. V., Phat V. N., Trinh H.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 4

921-927

EN

This paper considers the problem of designing an observer-based output feedback controller to exponentially stabilize a class of linear systems with an interval time-varying delay in the state vector. The delay is assumed to vary within an interval with known lower and upper bounds. The time-varying delay is not required to be differentiable, nor should its lower bound be zero. By constructing a set of Lyapunov-Krasovskii functionals and utilizing the Newton-Leibniz formula, a delay-dependent stabilizability condition which is expressed in terms of Linear Matrix Inequalities (LMIs) is derived to ensure the closed-loop system is exponentially stable with a prescribed \alfa-convergence rate. The design of an observer based output feedback controller can be carried out in a systematic and computationally efficient manner via the use of an LMI-based algorithm. A numerical example is given to illustrate the design procedure.

4

A multi-model approach to Saint-Venant equations: A stability study by LMIs

Dos Santos Martins V., Rodrigues M., Diagne M.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 3

539-550

EN

This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities (LMIs) and an Internal Model Boundary Control (IMBC) structure. The method is applied both in simulations and real experiments through a microchannel, illustrating thus the theoretical results developed in the paper.

5

Stabilność asymptotyczna według składowych i stabilność wykładnicza dodatnich dyskretnych układów liniowych niecałkowitego rzędu

Kaczorek T., Cimochowski R.

Pomiary Automatyka Kontrola

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2010

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R. 56, nr 5

414-417

PL

Podano podstawowe definicje i twierdzenia dotyczące dodatnich układów dyskretnych niecałkowitego rzędu oraz omówiono ich stabilność asymptotyczną. Podano warunki konieczne i wystarczające stabilności asymptotycznej według składowych i stabilności wykładniczej dodatnich układów dyskretnych niecałkowitego rzędu. Przedstawiono przykłady numeryczne ilustrujące problem stabilności asymptotycznej według składowych i stabilności wykładniczej.

EN

In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear systems behaviour can be found in engineering, management science, economics, social sciences, biology and medicine, etc. Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. The concept of positive fractional discrete-time linear systems has been introduced in [6] and the reachability and controllability to zero of positive fractional system has been investigated in [10]. In this paper the problem of the componentwise asymptotic stability and exponential stability of the positive fractional systems will be solved. The paper is organized as follows. In section 2 the basic definitions and theorems concerning the positive fractional systems are recalled and their asymptotic stability is discussed. The main result of the paper is presented in section 3 and 4. Necessary and sufficient conditions for the componentwise asymptotic stability and exponential stability of the positive fractional systems are established. The considerations are illustrated by numerical examples in section 5. The algorithm in MATLAB, which allows the test of the componentwise asymptotic stability and exponential stability of the positive fractional systems is presented. How does presented procedure work is step-by step described. In section 6 the relationship between the componentwise asymptotic stability and exponential stability is presented. Concluding remarks and open problems are given in section 7.

6

Dyskretny system paraboliczny o niepewnych parametrach

Oprzędkiewicz K.

Przegląd Elektrotechniczny

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2008

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R. 84, nr 9

179-187

PL

W pracy omówiono zagadnienia dyskretyzacji równania stanu opisującego system paraboliczny o niepewnych parametrach. Rozważono system z dwuwymiarową przestrzenią niepewnych parametrów opisany abstrakcyjnym równaniem stanu w przestrzeni Hiilberta. Dla rozważanego systemu podano warunki dekompozycji widma bazujące na geometrycznej interpretacji widma systemu. Wykazano, że dyskretyzacja nie zmienia warunków dekompozycji widma dla dowolnej wartości okresu próbkowania.

EN

In the paper the discretisation problem for an uncertain - parameter parabolic system was discussed. The system under consideration is described by an abstract state – space equation in the Hilbert space. For the discussed system spectrum decomposition conditions were proposed. To the formulate these conditions the geometric interpretation of the system’s spectrum was applied. The main conclusion from the paper is, that the discretisation does not change the spectrum decomposition conditions for each value of the sample time.