Research

20 pages, 385 KiB

Open AccessArticle

Homogenization of Smoluchowski Equations in Thin Heterogeneous Porous Domains

by Reine Gladys Noucheun and Jean Louis Woukeng

Mathematics 2023, 11(17), 3796; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173796 - 04 Sep 2023

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In a thin heterogeneous porous layer, we carry out a multiscale analysis of Smoluchowski’s discrete diffusion–coagulation equations describing the evolution density of diffusing particles that are subject to coagulation in pairs. Assuming that the thin heterogeneous layer is made up of microstructures that [...] Read more.

In a thin heterogeneous porous layer, we carry out a multiscale analysis of Smoluchowski’s discrete diffusion–coagulation equations describing the evolution density of diffusing particles that are subject to coagulation in pairs. Assuming that the thin heterogeneous layer is made up of microstructures that are uniformly distributed inside, we obtain in the limit an upscaled model in the lower space dimension. We also prove a corrector-type result very useful in numerical computations. In view of the thin structure of the domain, we appeal to a concept of two-scale convergence adapted to thin heterogeneous media to achieve our goal. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

11 pages, 1698 KiB

Open AccessArticle

Robust Stabilization and Observer-Based Stabilization for a Class of Singularly Perturbed Bilinear Systems

by Ding-Horng Chen, Chun-Tang Chao and Juing-Shian Chiou

Mathematics 2021, 9(19), 2380; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192380 - 25 Sep 2021

Viewed by 1212

Abstract

An infinite-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. First, we present a Lyapunov equation approach for the stabilization of singularly perturbed bilinear systems for all ε∈(0, ∞). The method is based on the Lyapunov stability theorem. [...] Read more.

An infinite-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. First, we present a Lyapunov equation approach for the stabilization of singularly perturbed bilinear systems for all ε∈(0, ∞). The method is based on the Lyapunov stability theorem. The state feedback constant gain can be determined from the admissible region of the convex polygon. Secondly, we extend this technique to study the observer and observer-based controller of singularly perturbed bilinear systems for all ε∈(0, ∞). Concerning this problem, there are two different methods to design the observer and observer-based controller: one is that the estimator gain can be calculated with known bounded input, the other is that the input gain can be calculated with known observer gain. The main advantage of this approach is that we can preserve the characteristic of the composite controller, i.e., the whole dimensional process can be separated into two subsystems. Moreover, the presented stabilization design ensures the stability for all ε∈(0, ∞). A numeral example is given to compare the new ε-bound with that of previous literature. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

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16 pages, 594 KiB

Open AccessArticle

Characterization of Traveling Waves Solutions to an Heterogeneous Diffusion Coupled System with Weak Advection

by José Luis Díaz Palencia

Mathematics 2021, 9(18), 2300; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182300 - 17 Sep 2021

Cited by 3 | Viewed by 1292

Abstract

The aim of this work is to characterize Traveling Waves (TW) solutions for a coupled system with KPP-Fisher nonlinearity and weak advection. The heterogeneous diffusion introduces certain instabilities in the TW heteroclinic connections that are explored. In addition, a weak advection reflects the [...] Read more.

The aim of this work is to characterize Traveling Waves (TW) solutions for a coupled system with KPP-Fisher nonlinearity and weak advection. The heterogeneous diffusion introduces certain instabilities in the TW heteroclinic connections that are explored. In addition, a weak advection reflects the existence of a critical combined TW speed for which solutions are purely monotone. This study follows purely analytical techniques together with numerical exercises used to validate or extent the contents of the analytical principles. The main concepts treated are related to positivity conditions, TW propagation speed and homotopy representations to characterize the TW asymptotic behaviour. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

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11 pages, 939 KiB

Open AccessArticle

Stabilization and the Design of Switching Laws of a Class of Switched Singularly Perturbed Systems via the Composite Control

by Chun-Tang Chao, Ding-Horng Chen and Juing-Shian Chiou

Mathematics 2021, 9(14), 1664; https://0-doi-org.brum.beds.ac.uk/10.3390/math9141664 - 15 Jul 2021

Cited by 3 | Viewed by 1134

Abstract

This paper proves that the controller design for switched singularly perturbed systems can be synthesized from the controllers of individual slow–fast subsystems. Under the switching rules of individual slow–fast subsystems, switched singularly perturbed systems can be stabilized under a small value of

ε

[...] Read more.

This paper proves that the controller design for switched singularly perturbed systems can be synthesized from the controllers of individual slow–fast subsystems. Under the switching rules of individual slow–fast subsystems, switched singularly perturbed systems can be stabilized under a small value of

ε

. The switching rule is designed on the basis of state transformation of the individual subsystems. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

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14 pages, 329 KiB

Open AccessArticle

Homogenization of a 2D Tidal Dynamics Equation

by Giuseppe Cardone, Aurelien Fouetio and Jean Louis Woukeng

Mathematics 2020, 8(12), 2209; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122209 - 12 Dec 2020

Cited by 1 | Viewed by 1279

Abstract

This work deals with the homogenization of two dimensions’ tidal equations. We study the asymptotic behavior of the sequence of the solutions using the sigma-convergence method. We establish the convergence of the sequence of solutions towards the solution of an equivalent problem of [...] Read more.

This work deals with the homogenization of two dimensions’ tidal equations. We study the asymptotic behavior of the sequence of the solutions using the sigma-convergence method. We establish the convergence of the sequence of solutions towards the solution of an equivalent problem of the same type. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

23 pages, 360 KiB

Open AccessArticle

A Ginzburg–Landau Type Energy with Weight and with Convex Potential Near Zero

by Rejeb Hadiji and Carmen Perugia

Mathematics 2020, 8(6), 997; https://0-doi-org.brum.beds.ac.uk/10.3390/math8060997 - 18 Jun 2020

Cited by 1 | Viewed by 1406

Abstract

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy [...] Read more.

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis–Merle–Rivière. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

17 pages, 302 KiB

Open AccessArticle

Complete Asymptotics for Solution of Singularly Perturbed Dynamical Systems with Single Well Potential

by Denis I. Borisov and Oskar A. Sultanov

Mathematics 2020, 8(6), 949; https://0-doi-org.brum.beds.ac.uk/10.3390/math8060949 - 10 Jun 2020

Cited by 1 | Viewed by 1523

Abstract

We consider a singularly perturbed boundary value problem

(- ε^{2} ∆ + \nabla V \cdot \nabla) u_{ε} = 0

[...] Read more.

We consider a singularly perturbed boundary value problem

(- ε^{2} ∆ + \nabla V \cdot \nabla) u_{ε} = 0

in Ω, u_{ε} = f on \partial Ω, f \in C^{\infty} (\partial Ω) .

The function V is supposed to be sufficiently smooth and to have the only minimum in the domain

Ω

. This minimum can degenerate. The potential V has no other stationary points in

Ω

and its normal derivative at the boundary is non-zero. Such a problem arises in studying Brownian motion governed by overdamped Langevin dynamics in the presence of a single attracting point. It describes the distribution of the points at the boundary

\partial Ω

, at which the trajectories of the Brownian particle hit the boundary for the first time. Our main result is a complete asymptotic expansion for

u_{ε}

as

ε \to + 0

. This asymptotic is a sum of a term

K_{ε} Ψ_{ε}

and a boundary layer, where

Ψ_{ε}

is the eigenfunction associated with the lowest eigenvalue of the considered problem and

K_{ε}

is some constant. We provide complete asymptotic expansions for both

K_{ε}

and

Ψ_{ε}

; the boundary layer is also an infinite asymptotic series power in

ε

. The error term in the asymptotics for

u_{ε}

is estimated in various norms. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

18 pages, 1143 KiB

Open AccessArticle

Random Homogenization in a Domain with Light Concentrated Masses

by Gregory A. Chechkin and Tatiana P. Chechkina

Mathematics 2020, 8(5), 788; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050788 - 13 May 2020

Cited by 3 | Viewed by 1528

Abstract

In the paper, we consider an elliptic problem in a domain with singular stochastic perturbation of the density located near the boundary, depending on a small parameter. Using the boundary homogenization methods, we prove the compactness theorem and study the behavior of eigenelements [...] Read more.

In the paper, we consider an elliptic problem in a domain with singular stochastic perturbation of the density located near the boundary, depending on a small parameter. Using the boundary homogenization methods, we prove the compactness theorem and study the behavior of eigenelements to the initial problem as the small parameter tends to zero. Full article

(This article belongs to the Special Issue Asymptotic Analysis and Homogenization of PDEs)

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