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1

Algorithm for solution of systems of singularly perturbed differential equations with adifferential turning point

Sobchuk Valentyn, Zelenska Iryna, Laptiev Oleksandr

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 3

art. no. e145682

EN

The dynamic development of science requires constant improvement of approaches to modeling physical processes and phenomena. Practically all scientific problems can be described by systems of differential equations. Many scientific problems are described by systems of differential equations of a special class, which belong to the group of so-called singularly perturbed differential equations. Mathematical models of processes described by such differential equations contain a small parameter near the highest derivatives, and it was the presence of this small factor that led to the creation of a large mathematical theory. The work proposes a developed algorithm for constructing uniform asymptotics of solutions to systems of singularly perturbed differential equations.

2

Pontryagin Maximum Principle for coupled slow and fast systems

Artstein Z.

Control and Cybernetics

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2009

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Vol. 38, no 4A

1003-1019

EN

When slow and fast controlled dynamics are coupled, the variational limit, as the ratio of time scales grows, is best depicted as a trajectory in a probability measures space. The effective control is then an invariant measure on the fast state-control space. The paper presents the form of the Pontryagin Maximum Principle for this variational limit and examines its relation to the Maximum Principle of the perturbed system.

3

Slip flow in the gas-lubricated Rayleigh step-slider bearing

Farmer D. G., Shepherd J. J.

International Journal of Applied Mechanics and Engineering

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2006

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Vol. 11, no 3

593-608

EN

Singular perturbation methods are applied to analyse the isothermal operation of the Rayleigh step slider bearing of narrow geometry, when the bearing number is moderate and the gas lubricant is rarefied, so that 'slip flow' occurs. Approximations to the pressure field and load-carrying capacity of such a bearing are obtained; and the influence of step geometry and degree of slip on those quantities is discussed.

4

Fale biegnące w ośrodkach z dyfuzją

Kaźmierczak B.

Matematyka Stosowana : matematyka dla społeczeństwa

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2005

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Nr 6

29-47

PL

W pracy omawiamy rozwiązania w postaci fal biegnących dla równań modelujących ośrodki nieliniowe z efektywną dyfuzją. Rozwiązania takie mogą opisywać np. propagację frontów heteroklinicznych oraz impulsów (wzbudzeń ośrodka). W pracy przedstawiamy kilka przykładów procesów, w których pojęcie fali biegnącej dobrze oddaje istotę analizowanych zjawisk. Są to: plazma podtrzymywana promieniowaniem laserowym, przejścia fazowe w płynach van der Waalsa, morfogeneza skóry oraz transkrypcja materiału genetycznego z DNA na RNA.

EN

We discuss the travelling wave solutions to equations modelling nonlinear media with diffusion. Such solutions may describe, e.g., propagation of heteroclinic fronts or impulses (medium excitations). We present several examples of processes where the notion of the travelling wave is especially useful, including plasma sustained by a laser beam, phase changes in van der Waals fluids, skin morphogenesis and DNA-RNA transcription process.

5

Bang-bang controls in the singular perturbations limit

Artstein Z.

Control and Cybernetics

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2005

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Vol. 34, no 3

645-663

EN

A general form of the dynamics obtained as a limit of trajectories of singularly perturbed linear control systems is presented. The limit trajectories are described in terms of probability measure-valued maps. This allows to determine the extent to which the bang-bang principle for linear control systems is carried over to the singular limit.

6

High-gain feedback and sliding modes in infinite dimensional systems

Levaggi L.

Control and Cybernetics

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2004

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Vol. 33, no 1

33-50

EN

This paper focuses on the connection between sliding motions and low frequency modes of high-gain feedback systems in an infinite dimensional framework. We study a particular class of abstract control systems in a Hilbert space setting and analyse their high-gain behaviour through singular perturbations. We show that the "slow" motion derived from the reduced model approximates the evolution of the closed loop after a fast transient. Moreover we prove a relation between this slow component of the high-gain feedback system and sliding motions, in the spirit, of the analogous result in the finite dimensional setting by Young, Kokotovic and Utkin (Young et al., 1977).

7

Anoccupational measure solution to a singularly perturbed optimal control problem

Artstein Z.

Control and Cybernetics

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2002

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Vol. 31, no 3

623-642

EN

Employing limit occupational measures, we provide an explicit solution to a singularly perturbed optimal control problem for which the order reduction method does not apply.

8

The existence of moments of solutions to transport equations with inelastic scattering and thie applications in the asymptotic analysis

Banasiak J.

Journal of Applied Analysis

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2000

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Vol. 6, nr 2

187-211

EN

In this paper we prove the existence of all moments of the solutions to the time-dependent spatially homogeneous transport equation describing elastic and inelastic scattering of particles. The proof uses the theory of resolvent positive operators and Desch's perturbation theorem. As an application we carry out the asymptotic analysis of the full transport equation with dominant elastic scattering and, using the results of the first part of the paper, we show that its solution can be approximated in the L1-norm by the solution of the limit equation obtained by formal asymptotic expansion.