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Discrete & Continuous Dynamical Systems - S

February 2018 , Volume 11 , Issue 1

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Preface
Catherine Choquet, Natália Martins, M. Rchid Sidi Ammi, Mouhcine Tilioua and Delfim F. M. Torres
2018, 11(1): ⅰ-ⅱ doi: 10.3934/dcdss.201801i +[Abstract](200) +[HTML](117) +[PDF](102.6KB)
Abstract:
Optimality conditions for fractional variational problems with free terminal time
Ricardo Almeida
2018, 11(1): 1-19 doi: 10.3934/dcdss.2018001 +[Abstract](403) +[HTML](135) +[PDF](412.8KB)
Abstract:

This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler-Lagrange equations are established for the fundamental problem and when in presence of an integral constraint. A Legendre condition, which is a second-order necessary condition, is also obtained. Other cases, such as the infinite horizon problem, the problem with delays in the Lagrangian, and the problem with high-order derivatives, are considered. Finally, a necessary condition for the optimal fractional order to satisfy is proved.

Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem
Hassan Belhadj, Mohamed Fihri, Samir Khallouq and Nabila Nagid
2018, 11(1): 21-34 doi: 10.3934/dcdss.2018002 +[Abstract](473) +[HTML](95) +[PDF](1401.6KB)
Abstract:

The purpose of this paper is to establish a new numerical approach to solve, in two dimensions, a semilinear reaction diffusion equation combining finite volume method and Schur complement method. We applied our method for q = 2 non-overlapping subdomains and then we generalized in the case of several subdomains (q≥2). A large number of numerical test cases shows the efficiency and the good accuracy of the proposed approach in terms of the CPU time and the order of the error, when increasing the number of subdomains, without using the parallel computing. After several variations of the number of subdomains and the mesh grid, we remark two significant results. On the one hand, the increase related to the number of subdomains does not affect the order of the error, on the other hand, for each mesh grid when we augment the number of subdomains, the CPU time reaches the minimum for a specific number of subdomains. In order to have the minimum CPU time, we resorted to a statistical study between the optimal number of subdomains and the mesh grid.

Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted composite medium
Catherine Choquet, Mohammed Moumni and Mouhcine Tilioua
2018, 11(1): 35-57 doi: 10.3934/dcdss.2018003 +[Abstract](321) +[HTML](113) +[PDF](537.6KB)
Abstract:

We study the Landau-Lifshitz-Gilbert equation in a composite ferromagnetic medium made of two different materials with highly contrasted properties. Over the so-called matrix domain, the effective field, the demagnetizing field and the bulk anisotropy field are scaled with regard to a parameter $ε$ representing the size of the matrix blocks. This scaling preserves the physics of the magnetization as $ε$ tends to zero. Using homogenization theory, we derive the corresponding effective model. To this aim we use the concept of two-scale convergence together with a new homogenization procedure for handling with the nonlinear terms. More precisely, an appropriate dilation operator is applied in a embedded cells network, the network being constrained by the microscopic geometry. We prove that the less magnetic part of the medium contributes through additional memory terms in the effective field.

A necessary condition of Pontryagin type for fuzzy fractional optimal control problems
Omid S. Fard, Javad Soolaki and Delfim F. M. Torres
2018, 11(1): 59-76 doi: 10.3934/dcdss.2018004 +[Abstract](344) +[HTML](123) +[PDF](463.6KB)
Abstract:

We prove necessary optimality conditions of Pontryagin type for a class of fuzzy fractional optimal control problems with the fuzzy fractional derivative described in the Caputo sense. The new results are illustrated by computing the extremals of three fuzzy optimal control systems, which improve recent results of Najariyan and Farahi.

On consensus in the Cucker–Smale type model on isolated time scales
Ewa Girejko, Luís Machado, Agnieszka B. Malinowska and Natália Martins
2018, 11(1): 77-89 doi: 10.3934/dcdss.2018005 +[Abstract](312) +[HTML](121) +[PDF](842.7KB)
Abstract:

This article addresses a consensus phenomenon in a Cucker-Smale model where the magnitude of the step size is not necessarily a constant but it is a function of time. In the considered model the weights of mutual influences in the group of agents do not change. A sufficient condition under which the proposed model tends to a consensus is obtained. This condition strikingly demonstrates the importance of the graininess function in a consensus phenomenon. The results are illustrated by numerical simulations.

Noether currents for higher-order variational problems of Herglotz type with time delay
Simão P. S. Santos, Natália Martins and Delfim F. M. Torres
2018, 11(1): 91-102 doi: 10.3934/dcdss.2018006 +[Abstract](266) +[HTML](108) +[PDF](388.1KB)
Abstract:

We study, from an optimal control perspective, Noether currents for higher-order problems of Herglotz type with time delay. Main result provides new Noether currents for such generalized variational problems, which are particularly useful in the search of extremals. The proof is based on the idea of rewriting the higher-order delayed generalized variational problem as a first-order optimal control problem without time delays.

Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration
Moulay Rchid Sidi Ammi and Ismail Jamiai
2018, 11(1): 103-117 doi: 10.3934/dcdss.2018007 +[Abstract](392) +[HTML](161) +[PDF](638.0KB)
Abstract:

In this paper, we consider a time fractional diffusion-convection equation and its application for image processing. A time discretization scheme is introduced and a stability result and error estimates are proved. Practical experiments are then provided showing that the fractional approach is more efficient than the ordinary integer one (Perona-Malik). A fully discrete scheme is proposed by using a Legendre collocation method. The convergence of this method is obtained by proving a priori error estimates.

Modeling and optimal control of HIV/AIDS prevention through PrEP
Cristiana J. Silva and Delfim F. M. Torres
2018, 11(1): 119-141 doi: 10.3934/dcdss.2018008 +[Abstract](513) +[HTML](114) +[PDF](675.9KB)
Abstract:

Pre-exposure prophylaxis (PrEP) consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals and has recently demonstrated to be highly efficacious for HIV prevention. We propose a new epidemiological model for HIV/AIDS transmission including PrEP. Existence, uniqueness and global stability of the disease free and endemic equilibriums are proved. The model with no PrEP is calibrated with the cumulative cases of infection by HIV and AIDS reported in Cape Verde from 1987 to 2014, showing that it predicts well such reality. An optimal control problem with a mixed state control constraint is then proposed and analyzed, where the control function represents the PrEP strategy and the mixed constraint models the fact that, due to PrEP costs, epidemic context and program coverage, the number of individuals under PrEP is limited at each instant of time. The objective is to determine the PrEP strategy that satisfies the mixed state control constraint and minimizes the number of individuals with pre-AIDS HIV-infection as well as the costs associated with PrEP. The optimal control problem is studied analytically. Through numerical simulations, we demonstrate that PrEP reduces HIV transmission significantly.

Fractional Herglotz variational problems of variable order
Dina Tavares, Ricardo Almeida and Delfim F. M. Torres
2018, 11(1): 143-154 doi: 10.3934/dcdss.2018009 +[Abstract](338) +[HTML](114) +[PDF](357.8KB)
Abstract:

We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper.

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