Practical stability for fractional impulsive control systems with noninstantaneous impulses on networks

This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunov-like functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our result

Comparison of Quantities of Information in the Human Memory

A mathematical model of changing the amount of information in the abstract human memory is proposed in the presence of the subsequent "external discrete" training (filling the information). Under this model, the amount of information is a solution of impulsive differential equation with fixed moments of impulsive effects and variable structure. Sufficient conditions are proposed related to the moments and magnitudes of  the impulsive effects (i.e., to the moments of discrete training and the volume of the received information), where the quantities of information in two different models of learning can be compared

Iterative learning control for impulsive multi-agent systems with varying trial lengths

In this paper, we introduce iterative learning control (ILC) schemes with varying trial lengths (VTL) to control impulsive multi-agent systems (I-MAS). We use domain alignment operator to characterize each tracking error to ensure that the error can completely update the control function during each iteration. Then we analyze the system’s uniform convergence to the target leader. Further, we use two local average operators to optimize the control function such that it can make full use of the iteration error. Finally, numerical examples are provided to verify the theoretical results

Fractional Calculus - Theory and Applications

In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications

Fractional Differential Equations, Inclusions and Inequalities with Applications

During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering

Controlo ótimo fracionário e aplicações biológicas

In this PhD thesis, we derive a Pontryagin Maximum Principle (PMP) for fractional optimal control problems and analyze a fractional mathematical model of COVID– 19 transmission dynamics. Fractional optimal control problems consist on optimizing a performance index functional subject to a fractional control system. One of the most important results in optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. First, we study properties of optimality for a dynamical system described by distributed-order non-local derivatives associated to a Lagrangian cost functional. We start by proving continuity and differentiability of solutions due to control perturbations. For smooth and unconstrained data, we obtain a weak version of Pontryagin's Maximum principle and a sufficient optimality condition under appropriate convexity. However, for controls taking values on a closed set, we use needle like variations to prove a strong version of Pontryagin's maximum principle. In the second part of the thesis, optimal control problems for fractional operators involving general analytic kernels are studied. We prove an integration by parts formula and a Gronwall inequality for fractional derivatives with a general analytic kernel. Based on these results, we show continuity and differentiability of solutions due to control perturbations leading to a weak version of the maximum principle. In addition, a wide class of combined fractional operators with general analytic kernels is considered. For this later problem, the control set is a closed convex subset of L2. Thus, using techniques from variational analysis, optimality conditions of Pontryagin type are obtained. Lastly, a fractional model for the COVID--19 pandemic, describing the realities of Portugal, Spain and Galicia, is studied. We show that the model is mathematically and biologically well posed. Then, we obtain a result on the global stability of the disease free equilibrium point. At the end we perform numerical simulations in order to illustrate the stability and convergence to the equilibrium point. For the data of Wuhan, Galicia, Spain, and Portugal, the order of the Caputo fractional derivative in consideration takes different values, characteristic of each region, which are not close to one, showing the relevance of the considered fractional models. 2020 Mathematics Subject Classification: 26A33, 49K15, 34A08, 34D23, 92D30.Nesta tese, derivamos o Princípio do Máximo de Pontryagin (PMP) para problemas de controlo ótimo fracionário e analisamos um modelo matemático fracionário para a dinâmica de transmissão da COVID-19. Os problemas de controlo ótimo fracionário consistem em otimizar uma funcional de índice de desempenho sujeita a um sistema de controlo fracionário. Um dos resultados mais importantes no controlo ótimo é o Princípio do Máximo de Pontryagin, que fornece uma condição de otimalidade necessária que toda a solução para o problema de otimização deve verificar. Primeiramente, estudamos propriedades de otimalidade para sistemas dinâmicos descritos por derivadas não-locais de ordem distribuída associadas a uma funcional de custo Lagrangiana. Começamos demonstrando a continuidade e a diferenciabilidade das soluções usando perturbações do controlo. Para dados suaves e sem restrições, obtemos uma versão fraca do princípio do Máximo de Pontryagin e uma condição de otimalidade suficiente sob convexidade apropriada. No entanto, para controlos que tomam valores num conjunto fechado, usamos variações do tipo agulha para provar uma versão forte do princípio do máximo de Pontryagin. Na segunda parte da tese, estudamos problemas de controlo ótimo para operadores fracionários envolvendo um núcleo analítico geral. Demonstramos uma fórmula de integração por partes e uma desigualdade Gronwall para derivadas fracionárias com um núcleo analítico geral. Com base nesses resultados, mostramos a continuidade e a diferenciabilidade das soluções por perturbações do controlo, levando a uma formulação de uma versão fraca do princípio do máximo de Pontryagin. Além disso, consideramos uma classe ampla de operadores fracionários combinados com núcleo analítico geral. Para este último problema, o conjunto de controlos é um subconjunto convexo fechado de L2. Assim, usando técnicas da análise variacional, obtemos condições de otimalidade do tipo de Pontryagin. Finalmente, estudamos um modelo fracionário da pandemia de COVID-19, descrevendo as realidades de Portugal, Espanha e Galiza. Mostramos que o modelo proposto é matematicamente e biologicamente bem colocado. Então, obtemos um resultado sobre a estabilidade global do ponto de equilíbrio livre de doença. No final, realizamos simulações numéricas para ilustrar a estabilidade e convergência do ponto de equilíbrio. Para os dados de Wuhan, Galiza, Espanha e Portugal, a ordem da derivada fracionária de Caputo em consideração toma valores diferentes característicos de cada região, e não próximos de um, mostrando a relevância de se considerarem modelos fracionários.Programa Doutoral em Matemática Aplicad

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia