Publications 2019

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56.  The variable order fractional calculus of variations

Almeida, Ricardo and Tavares, Dina and Torres, Delfim F. M.

Springer

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators

ria.ua.pt | doi

Book Chapters


55.  Non-invasive control of the fractional Hegselmann-Krause type model

Almeida, Ricardo and Malinowska, A.B. and Odzijewicz, T.

Non-Integer Order Calculus and its Applications. Lecture Notes in Electrical Engineering

Springer

In this paper, the fractional order Hegselmann–Krause type model with leadership is studied. We seek an optimal control strategy for the system to reach a consensus in such a way that the control mechanism is included in the leader dynamics. Necessary optimality conditions are obtained by the use of a fractional counterpart of Pontryagin Maximum Principle. The effectiveness of the proposed control strategy is illustrated by numerical examples.

ria.ua.pt | doi

54.  A survey on fractional variational calculus

Almeida, Ricardo and Torres, Delfim F. M.

Handbook of Fractional Calculus with Applications

De Gruyter

Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and we compute approximate solutions based on truncated Grünwald-Letnikov approximations of the Caputo derivatives.

ria.ua.pt | doi | Peer Reviewed

53.  Analysis of fractional integro-differential equations of thermistor type

Sidi Ammi, Moulay Rchid and Torres, Delfim F. M.

Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory

De Gruyter

We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results as regards fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: problems concerned with Riemann–Liouville, Caputo, and time-scale fractional operators. The existence and uniqueness of positive solutions are obtained through suitable fixed-point theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence.

ria.ua.pt | doi | Peer Reviewed

52.  Variable order Mittag–Leffler fractional operators on isolated time scales and application to the calculus of variations

Abdeljawad, Thabet and Mert, Raziye and Torres, Delfim F. M.

Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control

Springer

We introduce new fractional operators of variable order in isolated time scales with Mittag–Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main results give fractional integration by parts formulas and necessary optimality conditions of Euler–Lagrange type.

ria.ua.pt | doi | Peer Reviewed

51.  Time-fractional optimal control of initial value problems on time scales

Bahaa, Gaber M. and Torres, Delfim F. M.

Nonlinear Analysis and Boundary Value Problems. NABVP 2018. Springer Proceedings in Mathematics & Statistics

Springer

We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann–Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler–Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann–Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.

ria.ua.pt | doi | Peer Reviewed

50.  O Aeroporto Francisco Sá Carneiro e a sua relação com o turismo na Região Norte de Portugal

Barros, Helena de and Alonso, Hugo

26th APDR Congress

Associação Portuguesa para o Desenvolvimento Regional

O Aeroporto Francisco Sá Carneiro e o turismo na região Norte de Portugal têm crescido de forma muito significativa nos últimos anos. Este artigo apresenta um estudo da relação entre o número de passageiros que circulam no aeroporto e as dormidas na região. O estudo é baseado numa análise de regressão linear simples. Como resultado, propõe-se uma forma de prever as dormidas, conhecido o número de passageiros.

ria.ua.pt | Peer Reviewed

49.  New inequalities for η-quasiconvex function

Nwaeze, Eze R. and Torres, Delfim F. M.

Frontiers in Functional Equations and Analytic Inequalities

Springer

The class of η-quasiconvex functions was introduced in 2016. Here we establish novel inequalities of Ostrowski type for functions whose second derivative, in absolute value raised to the power q ≥ 1, is η-quasiconvex. Several interesting inequalities are deduced as special cases. Furthermore, we apply our results to the arithmetic, geometric, Harmonic, logarithmic, generalized log and identric means, getting new relations amongst them.

ria.ua.pt | doi | Peer Reviewed

48.  Harmonic and trace inequalities in Lipschitz domains

Touhami, Soumia and Chaira, Abdellatif and Torres, Delfim F. M.

Frontiers in Functional Equations and Analytic Inequalities

Springer

We prove boundary inequalities in arbitrary bounded Lipschitz domains on the trace space of Sobolev spaces. For that, we make use of the trace operator, its Moore–Penrose inverse, and of a special inner product. We show that our trace inequalities are particularly useful to prove harmonic inequalities, which serve as powerful tools to characterize the harmonic functions on Sobolev spaces of non-integer order.

ria.ua.pt | doi | Peer Reviewed

Articles


47.  Periodic state-space representations of periodic convolutional codes

Napp, Diego and Pereira, Ricardo and Pinto, Raquel and Rocha, Paula

Cryptography and Communications

Springer Verlag

In this paper we study the representation of periodically time-varying convolutional codes by means of periodic input-state-output models. In particular, we focus on period two and investigate under which conditions a given two-periodic convolutional code (obtained by alternating two time-invariant encoders) can be represented by a periodic input-state-output system. We first show that one cannot expect, in general, to obtain a periodic input-state-output representation of a periodic convolutional code by means of the individual realizations of each of the associated time-invariant codes. We, however, provide sufficient conditions for this to hold in terms of the column degrees of the associated column reduced generator matrices. Moreover, we derive a sufficient condition to obtain a periodic state-space realization that is minimal. Finally, examples to illustrate the results are presented.

ria.ua.pt | doi | Peer Reviewed

46.  Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems

Salati, Abubakar Bello and Shamsi, Mostafa and Torres, Delfim F. M.

Communications in Nonlinear Science and Numerical Simulation

Elsevier

This paper presents three direct methods based on Grünwald–Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald–Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCPs are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.

ria.ua.pt | doi | Peer Reviewed

45.  Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales

Mozyrska, Dorota and Torres, Delfim F. M. and Wyrwas, Malgorzata

Nonlinear Analysis: Hybrid Systems

Elsevier

Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly continuous and partly discrete, i.e., hybrid time scales, one gets new fractional operators. We concentrate on the behavior of solutions to initial value problems with the Caputo-Fabrizio fractional delta derivative on an arbitrary time scale. In particular, the exponential stability of linear systems is studied. A necessary and sufficient condition for the exponential stability of linear systems with the Caputo-Fabrizio fractional delta derivative on time scales is presented. By considering a suitable fractional dynamic equation and the Laplace transform on time scales, we also propose a proper definition of Caputo-Fabrizio fractional integral on time scales. Finally, by using the Banach fixed point theorem, we prove existence and uniqueness of solution to a nonlinear initial value problem with the Caputo-Fabrizio fractional delta derivative on time scales.

ria.ua.pt | doi | Peer Reviewed

44.  The spread of a financial virus through Europe and beyond

Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.

AIMS Mathematics

AIMS Press

We analyse the importance of international relations between countries on the financial stability. The contagion effect in the network is tested by implementing an epidemiological model, comprising a number of European countries and using bilateral data on foreign claims between them. Banking statistics of consolidated foreign claims on ultimate risk bases, obtained from the Banks of International Settlements, allow us to measure the exposure of contagion spreading from a particular country to the other national banking systems. We show that the financial system of some countries, experiencing the debt crisis, is a source of global systemic risk because they threaten the stability of a larger system, being a global threat to the intoxication of the world economy and resulting in what we call a `financial virus'. Illustrative simulations were done in the NetLogo multi-agent programmable modelling environment and in MATLAB.

ria.ua.pt | doi | Peer Reviewed

43.  Column distances of convolutional codes over Z_p^r

Napp, Diego and Pinto, Raquel and Toste, Marisa

IEEE Transactions on Information Theory

Institute of Electrical and Electronics Engineers

Maximum distance profile codes over finite nonbinary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over Zpr is presented.

ria.ua.pt | doi | Peer Reviewed

42.  Existence of solution to a nonlocal conformable fractional thermistor problem

Moulay Rchid Sidi Ammi and Torres, Delfim F. M.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

Faculty of Sciences University of Ankara

We study a nonlocal thermistor problem for fractional derivatives in the conformable sense. Classical Schauder's fixed point theorem is used to derive the existence of a tube solution.

ria.ua.pt | doi | Peer Reviewed

41.  A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

Mushtaq Salh Ali and Mostafa Shamsi and Hassan Khosravian-Arab and Torres, Delfim F. M. and Farid Bozorgnia

Journal of Vibration and Control

SAGE Publications

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi–Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer–order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre–Gauss–Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low central processing unit time.

ria.ua.pt | doi | Peer Reviewed

40.  Fractional order version of the HJB equation

Razminia, Abolhassan and AsadiZadehShiraz, Mehdi and Torres, Delfim F. M.

Journal of Computational and Nonlinear Dynamics

American Society of Mechanical Engineers

We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

ria.ua.pt | doi | Peer Reviewed

39.  Structural derivatives on time scales

Bayour, Benaoumeur and Torres, Delfim F. M.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

Faculty of Sciences University of Ankara

We introduce the notion of structural derivative on time scales. The new operator of differentiation unifies the concepts of fractal and fractional order derivative and is motivated by lack of classical differentiability of some self-similar functions. Some properties of the new operator are proved and illustrated with examples.

ria.ua.pt | doi | Peer Reviewed

38.  Functional characterizations of trace spaces in Lipschitz domains

Touhami, Soumia and Chaira, Abdellatif and Torres, Delfim F. M.

Banach Journal of Mathematical Analysis

Duke University Press

Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces Hs (∂Ω) involving a family of positive selfadjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary ∂Ω of a bounded Lipschitz domain Ω ⊂ R d and applying Moore–Penrose pseudoinverse properties together with a special inner product on H1 (Ω). We also establish generalized results of the Moore– Penrose pseudoinverse.

ria.ua.pt | doi | Peer Reviewed

37.  Exact solution to a dynamic SIR model

Bohner, Martin and Streipert, Sabrina and Torres, Delfim F. M.

Nonlinear Analysis: Hybrid Systems

Elsevier

We investigate an epidemic model based on Bailey's continuous differential system. In the continuous time domain, we extend the classical model to time-dependent coefficients and present an alternative solution method to Gleissner's approach. If the coefficients are constant, both solution methods yield the same result. After a brief introduction to time scales, we formulate the SIR (susceptible–infected–removed) model in the general time domain and derive its solution. In the discrete case, this provides the solution to a new discrete epidemic system, which exhibits the same behavior as the continuous model. The last part is dedicated to the analysis of the limiting behavior of susceptible, infected, and removed, which contains biological relevance.

ria.ua.pt | doi | Peer Reviewed

36.  A sufficient optimality condition for delayed state-linear optimal control problems

Lemos-Paião, Ana Pedro and Silva, Cristiana J. and Torres, Delfim F. M.

Discrete and Continuous Dynamical Systems - Series B

American Institute of Mathematical Sciences

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.

ria.ua.pt | doi | Peer Reviewed

35.  On duals and parity-checks of convolutional codes over Z p r

El Oued, Mohamed and Napp, Diego and Pinto, Raquel and Toste, Marisa

Finite fields and their applications

Elsevier

A convolutional code C over Z_{p^r}((D)) is a Z_{p^r}((D))-submodule of Z_{p^r}^n((D)) that admits a polynomial set of generators, where Z_{p^r}((D)) stands for the ring of (semi-infinity) Laurent series. In this paper we study several structural properties of its dual C^{perp} . We use these results to provide a constructive algorithm to build an explicit generator matrix of C^{perp}. Moreover, we show that the transpose of such a matrix is a parity-check matrix (also called syndrome former) of C.

ria.ua.pt | doi | Peer Reviewed

34.  Analysis and numerical approximation of tempered fractional calculus of variations problems

Almeida, Ricardo and Morgado, M. Luísa

Journal of Computational and Applied Mathematics

Elsevier

In this paper, we study variational problems where the cost functional involves the tempered Caputo fractional derivative. Several important optimization conditions are derived to find the optimal solution. Sufficient and necessary conditions are presented for different variational problems. For example, the cases of integral (isoperimetric problem) and holonomic constraints are considered, as well as problems with high order derivatives. A numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples

ria.ua.pt | doi | Peer Reviewed

33.  The risk of contagion spreading and its optimal control in the economy

Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.

Statistics, Optimization and Information Computing

International Academic Press

The global crisis of 2008 provoked a heightened interest among scientists to study the phenomenon, its propagation and negative consequences. The process of modelling the spread of a virus is commonly used in epidemiology. Conceptually, the spread of a disease among a population is similar to the contagion process in economy. This similarity allows considering the contagion in the world financial system using the same mathematical model of infection spread that is often used in epidemiology. Our research focuses on the dynamic behaviour of contagion spreading in the global financial network. The effect of infection by a systemic spread of risks in the network of national banking systems of countries is tested. An optimal control problem is then formulated to simulate a control that may avoid significant financial losses. The results show that the proposed approach describes well the reality of the world economy, and emphasizes the importance of international relations between countries on the financial stability.

ria.ua.pt | Peer Reviewed

32.  Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model

Duarte, Jorge and Martins, Nuno and Rogovchenko, Svitlana and Rogovchenko, Yuriy and Januário, Cristina

Journal of Mathematical Biology

Springer

Despite numerous studies of epidemiological systems, the role of seasonality in the recurrent epidemics is not entirely understood. During certain periods of the year incidence rates of a number of endemic infectious diseases may fluctuate dramatically. This influences the dynamics of mathematical models describing the spread of infection and often leads to chaotic oscillations. In this paper, we are concerned with a generalization of a classical Susceptible-Infected-Recovered epidemic model which accounts for seasonal effects. Combining numerical and analytic techniques, we gain new insights into the complex dynamics of a recurrent disease influenced by the seasonality. Computation of the Lyapunov spectrum allows us to identify different chaotic regimes, determine the fractal dimension and estimate the predictability of the appearance of attractors in the system. Applying the homotopy analysis method, we obtain series solutions to the original nonautonomous SIR model with a high level of accuracy and use these approximations to analyze the dynamics of the system. The efficiency of the method is guaranteed by the optimal choice of an auxiliary control parameter which ensures the rapid convergence of the series to the exact solution of the forced SIR epidemic model.

ria.ua.pt | doi | Peer Reviewed

31.  An epidemiological MSEIR model described by the Caputo fractional derivative

Almeida, Ricardo and Cruz, Artur Miguel and Martins, Natália and Monteiro, Maria Teresa

International Journal of Dynamics and Control

Springer

A fractional MSEIR model is presented, involving the Caputo fractional derivative. The equilibrium points and the basic reproduction number are computed. An analysis of the local asymptotic stability at the disease free equilibrium is given. Finally a numerical simulation, using Matlab based on optimization techniques, of the varicella outbreak among Shenzhen school children, China, is carried out.

ria.ua.pt | doi | Peer Reviewed

30.  A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati, Somayeh and Lima, Pedro M. and Torres, Delfim F. M.

Communications in Nonlinear Science and Numerical Simulation

Elsevier

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.

ria.ua.pt | doi | Peer Reviewed

29.  A collocation method of lines for two-sided space-fractional advection-diffusion equations with variable coefficients

Almoaeet, Mohammed K. and Shamsi, Mostafa and Khosravian-Arab, Hassan and Torres, Delfim F. M.

Mathematical Methods in the Applied Sciences

Wiley

We present the method of lines (MOL), which is based on the spectral collocation method, to solve space-fractional advection-diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left- and right-sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method

ria.ua.pt | doi | Peer Reviewed

28.  A numerical study of fractional relaxation–oscillation equations involving ψ-Caputo fractional derivative

Almeida, Ricardo and Jleli, Mohamed and Samet, Bessem

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

Springer

We provide a numerical method to solve a certain class of fractional differential equations involving ψ -Caputo fractional derivative. The considered class includes as particular case fractional relaxation–oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce ψ -shifted Legendre polynomial basis, and we derive an explicit formula for the ψ -fractional integral of ψ -shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.

ria.ua.pt | doi | Peer Reviewed

27.  Stability of a fractional HIV/AIDS model

Silva, Cristiana J. and Torres, Delfim F. M.

Mathematics and Computers in Simulation

Elsevier

We propose a fractional order model for HIV/AIDS transmission. Local and uniform stability of the fractional order model is studied. The theoretical results are illustrated through numerical simulations.

ria.ua.pt | doi | Peer Reviewed

26.  A sufficient optimality condition for non-linear delayed optimal control problems

Lemos-Paião, Ana P. and Silva, Cristiana J. and Torres, Delfim F. M.

Pure and Applied Functional Analysis

Yokohama Publishers

We prove a sufficient optimality condition for non-linear optimal control problems with delays in both state and control variables. Our result requires the verification of a Hamilton-Jacobi partial differential equation and is obtained through a transformation that allow us to rewrite a delayed optimal control problem as an equivalent non-delayed one.

ria.ua.pt | Peer Reviewed

25.  Fractional differential equations with mixed boundary conditions

Almeida, Ricardo Miguel

Bulletin of the Malaysian Mathematical Sciences Society

Springer

In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order α ∈ (2, 3), involving a general form of fractional derivative. First, we prove an equivalence between the Cauchy problem and the Volterra equation. Then, two results on the existence of solutions are proven, and we end with some illustrative examples.

ria.ua.pt | doi | Peer Reviewed

24.  Some inequalities for interval-valued functions on time scales

Zhao, Dafang and Ye, Guoju and Liu, Wei and Torres, Delfim F. M.

Soft Computing

Springer

We introduce the interval Darboux delta integral (shortly, the IDΔ -integral) and the interval Riemann delta integral (shortly, the IR Δ -integral) for interval-valued functions on time scales. Fundamental properties of ID and IR Δ -integrals and examples are given. Finally, we prove Jensen’s, Hölder’s and Minkowski’s inequalities for the IR Δ -integral. Also, some examples are given to illustrate our theorems

ria.ua.pt | doi | Peer Reviewed

23.  Optimal leader–follower control for the fractional opinion formation model

Almeida, Ricardo and Malinowska, Agnieszka B. and Odzijewicz, Tatiana

Journal of Optimization Theory and Applications

Springer

This paper deals with an opinion formation model, that obeys a nonlinear system of fractional-order differential equations. We introduce a virtual leader in order to attain a consensus. Sufficient conditions are established to ensure that the opinions of all agents globally asymptotically approach the opinion of the leader. We also address the problem of designing optimal control strategies for the leader so that the followers tend to consensus in the most efficient way. A variational integrator scheme is applied to solve the leader–follower optimal control problem. Finally, in order to verify the theoretical analysis, several particular examples are presented.

ria.ua.pt | doi | Peer Reviewed

22.  Caputo fractional differential equation with state dependent delay and practical stability

Agarwal, Ravi and Almeida, Ricardo and Hristova, Snezhana and O'Regan, Donal

Dynamic Systems and Applications

Dynamic Publishers, Inc

Practical stability properties of Caputo fractional delay differential equations is studied and, in particular, the case with state dependent delays is considered. These type of delays is a generalization of several types of delays such as constant delays, time variable delays, or distributed delays. In connection with the presence of a delay in a fractional differential equation and the application of the fractional generalization of the Razumikhin method, we give a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. Three types of derivatives for Lyapunov functions, the Caputo fractional derivative, the Dini fractional derivative, and the Caputo fractional Dini derivative, are applied to obtain several sufficient conditions for practical stability. An appropriate Razumikhin condition is applied. These derivatives allow the application of non-quadratic Lyapunov function for studying stability properties. We illustrate our theory on several nonlinear Caputo fractional differential equations with different types of delays

ria.ua.pt | doi | Peer Reviewed

21.  Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives

Ammi, Moulay Rchid Sidi and Torres, Delfim F. M.

Computers and Mathematics with Applications

Elsevier

We study an optimal control problem associated to a fractional nonlocal thermistor problem involving the ABC (Atangana-Baleanu-Caputo) fractional time derivative. We first prove the existence and uniqueness of solution. Then, we show that an optimal control exists. Moreover, we obtain the optimality system that characterizes the control.

ria.ua.pt | doi | Peer Reviewed

20.  A finite element approximation for a class of Caputo time-fractional diffusion equations

Ammi, Moulay Rchid Sidi and Jamiai, Ismail and Torres, Delfim F. M.

Computers and Mathematics with Applications

Elsevier

We develop a fully discrete scheme for time-fractional diffusion equations by using a finite difference method in time and a finite element method in space. The fractional derivatives are used in Caputo sense. Stability and error estimates are derived. The accuracy and efficiency of the presented method is shown by conducting two numerical examples.

ria.ua.pt | doi | Peer Reviewed

19.  Leader-following consensus for fractional multi-agent systems

Almeida, Ricardo and Girejko, Ewa and Hristova, Snezhana and Malinowska, Agnieszka B.

Advances in Difference Equations

SpringerOpen

A leader-following consensus for Caputo fractional multi-agent systems with nonlinear intrinsic dynamics is investigated. The second Lyapunov method is used to design a control protocol ensuring a consensus for two types of multi-agent systems. Contrary to the previous studies on leader-following consensus, the investigation covers systems with bounded and unbounded time-dependent Lipschitz coefficients in the intrinsic dynamics. Moreover, coupling strength describing the interactions between agents is considered to be a function of time.

ria.ua.pt | doi | Peer Reviewed

18.  The Portuguese Meeting in Biomathematics

Torres, Delfim F. M. and Area, Iván and Silva, César and Silva, Cristiana J.

Statistics, Optimization and Information Computing

IAPress

The main contributions of [Stat. Optim. Inf. Comput. Vol. 7, No. 3 (2019)], consisting of 7 papers selected and revised from the 2nd Portuguese Meeting in Biomathematics (EPB’2018), are highlighted.

ria.ua.pt | doi | Peer Reviewed

17.  Analysis of a SIRI epidemic model with distributed delay and relapse

Elazzouzi, Abdelhai and Alaoui, Abdesslem Lamrani and Tilioua, Mouhcine and Torres, Delfim F. M.

Statistics, Optimization and Information Computing

IAPress

We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number R0 for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when R0 < 1. Moreover,we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when R0 > 1.

ria.ua.pt | doi | Peer Reviewed

16.  A minimal HIV-AIDS infection model with general incidence rate and application to Morocco data

Lotfi, El Mehdi and Mahrouf, Marouane and Maziane, Mehdi and Silva, Cristiana J. and Torres, Delfim F. M. and Yousfi, Noura

Statistics, Optimization and Information Computing

IAPress

We study the global dynamics of a SICA infection model with general incidence rate. The proposed model is calibrated with cumulative cases of infection by HIV–AIDS in Morocco from 1986 to 2015. We first prove that our model is biologically and mathematically well-posed. Stability analysis of different steady states is performed and threshold parameters are identified where the model exhibits clearance of infection or maintenance of a chronic infection. Furthermore, we examine the robustness of the model to some parameter values by examining the sensitivity of the basic reproduction number. Finally, using numerical simulations with real data from Morocco, we show that the model predicts well such reality.

ria.ua.pt | doi | Peer Reviewed

15.  Optimal control and sensitivity analysis of a fractional order TB model

Rosa, Silvério and Torres, Delfim F. M.

Statistics, Optimization and Information Computing

A Caputo fractional-order mathematical model for the transmission dynamics of tuberculosis (TB) was recently proposed in [Math. Model. Nat. Phenom. 13 (2018), no. 1, Art. 9]. Here, a sensitivity analysis of that model is done, showing the importance of accuracy of parameter values. A fractional optimal control (FOC) problem is then formulated and solved,with the rate of treatment as the control variable. Finally, a cost-effectiveness analysis is performed to assess the cost and the effectiveness of the control measures during the intervention, showing in which conditions FOC is useful with respect to classical (integer-order) optimal control.

ria.ua.pt | doi | Peer Reviewed

14.  Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models

Cantin, Guillaume and Silva, Cristiana J.

AIMS Mathematics

AIMS Press

In this paper, we propose an original complex network model for an epidemic problem in an heterogeneous geographical area. The complex network is constructed by coupling nonidentical instances of a HIV/AIDS epidemiological model for which a disease-free equilibrium and an endemic equilibrium can coexist. After proving the existence of a positively invariant region for the solutions of the complex network problem, we investigate the effect of the coupling on the dynamics of the network, and establish the existence of a unique disease-free equilibrium for the whole network, which is globally asymptotically stable. We prove the existence of an optimal topology that minimizes the level of infected individuals, and apply the theoretical results to the case of the Cape Verde archipelago.

ria.ua.pt | doi | Peer Reviewed

13.  Realization of 2D (2,2)-periodic encoders by means of 2D periodic separable Roesser models

Pereira, Ricardo and Napp, Diego and Pinto, Raquel and Rocha, Paula

International Journal of Applied Mathematics and Computer Science

University of Zielona Gora Press

It is well-known that convolutional codes are linear systems when they are defined over a finite field. A fundamental issue in the implementation of convolutional codes is to obtain a minimal state representation of the code. In comparison to the literature on one-dimensional (1D) time-invariant convolutional codes, there exists only relatively few results on the realization problem for the time-varying 1D convolutional codes and even fewer if the convolutional codes are two-dimensional (2D). In this paper we consider 2D periodic convolutional codes and address the minimal state space realization problem for this class of codes. This is, in general, a highly nontrivial problem. Here, we focus on separable Roesser models and show that in this case it is possible to derive, under weak conditions, concrete formulas for obtaining a 2D Roesser state space representation. Moreover, we study minimility and present necessary conditions for these representations to be minimal. Our results immediately lead to constructive algorithms to build these representations.

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12.  Towards a geometric theory for nD behaviors: conditioned invariance and detectability subspaces

Pereira, Ricardo and Rocha, Paula

IFAC-PapersOnLine

Elsevier

We introduce the definitions of observer, conditioned invariance and detectability subspaces for discrete multidimensional behavioral systems, based on our previous work for the continuous 1D case, as a step forward in the attempt to develop a geometric theory for nD behaviors.

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11.  Fractional differential equations and Volterra–Stieltjes integral equations of the second kind

Asanov, Avyt and Almeida, Ricardo and Malinowska, Agnieszka B.

Computational and Applied Mathematics

Springer

In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an equivalence relation between the fractional differential equation and the Volterra– Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to solve numerically the integral equation and an estimation for the error is given. Results of numerical experiments demonstrate that satisfactory and reliable results could be obtained by the proposed method.

ria.ua.pt | doi | Peer Reviewed

10.  Preface of the “4th Symposium on Modelling and Simulation in Computer Sciences and Engineering”

Miranda, Francisco and Abreu, Carlos and Miranda, Daniel

AIP Conference Proceedings

AIP Publishing

The 4th Symposium on Modelling and Simulation in Computer Sciences and Engineering was held in the 16th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2018), Rhodes, Greece, 13-18 September 2018.

ria.ua.pt | doi | Peer Reviewed

9.  Theoretical simulation of the influence of cathode formulation on lithium-ion battery performance

Miranda, D. and Miranda, F. and Almeida, A. M. and Lanceros-Méndez, S. and Costa, C. M.

AIP Conference Proceedings

AIP Publishing

Optimizing cathode electrode formulation is essential in the development and performance of lithium-ion batteries, as the cathode affects the capacity of the battery. Cathode electrode formulation is based on different materials and relative contents. In this work, the cathode performance for the LiMn2O4, LMO, active material has been obtained by theoretical simulations for different materials formulations and at various discharge rates. Further, the simulations were compared with experimental results. It is demonstrated that the optimization of the electrode formulation strongly depends on the percentage of conductive material, existing a minimum conductive filler content that optimizes the delivered capacity of the battery and being that delivery capacity independent of the conductive filler content for higher concentrations.

ria.ua.pt | doi | Peer Reviewed

8.  Carbohydrate counting: how accurate should it be to achieve glycemic control in patients on intensive insulin regimens?

Abreu, Carlos and Miranda, Francisco and Felgueiras, Paula

AIP Conference Proceedings

AIP Publishing

Carbohydrate counting is an important meal-planning tool for patients on intensive insulin regimens. Preprandial insulin bolus is adjusted taking into account the carbohydrate content of each meal and the insulin-to-carb ratio of each patient throughout the day. Evidence suggests that accurate carbohydrate counting may have positive effects not only on reducing glycosylated hemoglobin concentration but also on decreasing the incidence of hypoglycemic episodes. Nevertheless, despite its benefits, the efficacy of carbohydrate counting depends on the ability of each patient, or its caregiver, to accurately estimate the carbohydrate content of each meal. Therefore, it is of great importance to understand how accurate should carbohydrate counting be, and the impact of inaccurate carbohydrate counting on the glycemic control of each patient. Within this work, we propose an analytic method that uses the insulin-to-carb ratio and the insulin sensitivity factor, along with the glycemic targets of each patient to calculate the limits of accurate carbohydrate counting, in order to achieve better glycemic control and to reduce hypoglycemic episodes.

ria.ua.pt | doi | Peer Reviewed

7.  Home glucose meters: how accurate should they be to avoid dysglycemia in patients using carbohydrate counting?

Abreu, Carlos and Miranda, Francisco and Dabrowska, Anna and Felgueiras, Paula

AIP Conference Proceedings

AIP Publishing

Accurate self-monitoring of blood glucose is the key to an effective and safe intensive insulin therapy. Indeed, most insulin dosing decisions are made based on the blood glucose values obtained from home glucose meters, in particular for those using diet planning and carbohydrate counting. Patients on that therapeutic regimen depend not only on their ability to accurately estimate the carbohydrate content of each meal but also on the accuracy of the glucose meter being used. Therefore, in order to avoid postprandial hypoglycemia and hyperglycemia, it is of great importance to realize how important is the accuracy of blood glucose meters according to the particular characteristics of each patient. In this regard, we propose an analytic method to find the limits of the blood glucose meters accuracy according to the insulin-to-carb ratio, the insulin sensitivity factor and the ability of each patient to estimate the carbohydrate content of each meal.

ria.ua.pt | doi | Peer Reviewed

6.  An adaptive mealtime bolus calculator to minimize the effects of inaccurate carbohydrate counting

Abreu, Carlos and Miranda, Francisco and Felgueiras, Paula

AIP Conference Proceedings

AIP Publishing

Evidence suggests that accurate carbohydrate counting along with self-monitoring of blood glucose is the key to a successful diabetes management, in particular for patients on intensive insulin regimens. However, despite its benefits, accurate carbohydrate counting is a complex, difficult, time-consuming, and error-prone task for most patients. Several studies show that most patients frequently estimate the carbohydrate content of meals within an error of about 10-15 g of the real value. In addition, fearing hypoglycemic events, patients frequently underestimate the carbohydrate content of meals and, consequently, they have high levels of HbA1C. Therefore, is important to avoid the consequences of incorrect carbohydrate counting in order to improve the patient’s glycemic control. To that end, this work presents an adaptive mealtime bolus calculator that uses the patient’s glycemic data to dynamically adjust the mealtime bolus and counterbalance the negative effects of inaccurate carbohydrate counting.

ria.ua.pt | doi | Peer Reviewed

5.  A fractional measles model having monotonic real statistical data for constant transmission rate of the disease

Almeida, Ricardo and Qureshi, Sania

Fractal and Fractional

MDPI

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number R0 < 1, which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model.

ria.ua.pt | doi | Peer Reviewed

4.  Optimal control of HIV treatment and immunotherapy combination with state and control delays

Silva, Cristiana J. and Maurer, Helmut

Optimal Control Applications and Methods

Wiley

In this paper, we propose and analyze an optimal control problem where human immunodeficiency virus treatment and immunotherapy are described by two control functions that are subject to time delays representing pharmacological and absorption delays, respectively. The goal is to propose effective optimal control solutions for the combination of human immunodeficiency virus treatment and immunotherapy, ensuring a functional behavior of the immune system. The incubation period is mathematically represented by a time delay in the virus load, and the local asymptotic and Hopf bifurcation analysis of the CTL-equilibrium point of the uncontrolled delayed system is studied. We obtain optimal controls of bang-singular type both for the nondelayed and delayed optimal control problem with and without state constraints.We study boundary arcs of state constraints and junction properties of the control and adjoint variables at entry and exit points of boundary arcs. Moreover, we derive an explicit formula of the multiplier associated with the state constraint.

ria.ua.pt | doi | Peer Reviewed

3.  Further properties of Osler's generalized fractional integrals and derivatives with respect to another function

Almeida, Ricardo

Rocky Mountain Journal of Mathematics

Rocky Mountain Mathematics Consortium

In this paper we discuss fractional integrals and fractional derivatives of a function with respect to another function. We present some fundamental properties for both types of fractional operators, such as Taylor’s theorem, Leibniz and semigroup rules. We also provide a numerical tool to deal with these operators, by approximating them with a sum involving integer-order derivatives.

ria.ua.pt | doi | Peer Reviewed

2.  A stochastic analysis of the impact of fluctuations in the environment on pre-exposure prophylaxis for HIV infection

Djordević, Jasmina and Silva, Cristiana J.

Soft Computing

Springer Nature

We propose a stochastic model for HIV/AIDS transmission where pre-exposure prophylaxis is considered as a prevention measure for new HIV infections. A white noise is introduced into the model, representing fluctuations in the environment that manifest themselves on the transmission coefficient rate. We prove the existence and uniqueness of a global positive solution of the stochastic model and establish conditions under which extinction and persistence in mean hold. Numerical simulations are provided which illustrate the theoretical results, and conclusions are derived on the impact of the fluctuations in the environment on the number of the susceptible individuals that are under pre-exposure prophylaxis.

ria.ua.pt | doi | Peer Reviewed

Proceedings


1.  The impact of accurate carbohydrate counting on patient’s glycemic targets

Abreu, C. and Miranda, F. and Felgueiras, P.

Diabetes Technology and Therapeutics

Mary Ann Liebert

Preprandial insulin bolus is adjusted taking into account the carbohydrate content of each meal, the patient’s glycemic targets (GHyper, GT and GHypo), the insulin sensitivity factor (ISF), and the insulin-to-carb ratio (ICR) throughout the day. Evidence suggests that accurate carbohydrate counting may have positive effects not only on reducing glycosylated hemoglobin concentration but also on decreasing the incidence of hypoglycemic episodes. Therefore, the efficacy of carbohydrate counting depends not only on the ability of each patient accurately estimate the carbohydrate content of each meal but also on each patient glycemic targets.

ria.ua.pt | doi | Peer Reviewed
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