344 research outputs found
A survey on fractional variational calculus
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 compute approximated solutions
based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form is in
'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De
Gruyter. Submitted 29-March-2018; accepted, after a revision, 13-June-201
Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels
In this work we study necessary and sufficient optimality conditions for variational
problems dealing with a new fractional derivative. This fractional derivative combines two known
operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a
fractional integration by parts formula, we obtain the Euler–Lagrange equation and natural boundary
conditions for the fundamental variational problem. Also, fractional variational problems with integral
and holonomic constraints are considered. We end with some examples to exemplify our results.publishe
Herglotz variational problems involving distributed-order fractional derivatives with arbitrary smooth kernels
In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examplepublishe
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
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
An optimal polynomial approximation of Brownian motion
In this paper, we will present a strong (or pathwise) approximation of
standard Brownian motion by a class of orthogonal polynomials. The coefficients
that are obtained from the expansion of Brownian motion in this polynomial
basis are independent Gaussian random variables. Therefore it is practical
(requires independent Gaussian coefficients) to generate an approximate
sample path of Brownian motion that respects integration of polynomials with
degree less than . Moreover, since these orthogonal polynomials appear
naturally as eigenfunctions of an integral operator defined by the Brownian
bridge covariance function, the proposed approximation is optimal in a certain
weighted sense. In addition, discretizing Brownian paths as
piecewise parabolas gives a locally higher order numerical method for
stochastic differential equations (SDEs) when compared to the standard
piecewise linear approach. We shall demonstrate these ideas by simulating
Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will
also illustrate the deficiencies of the piecewise parabola approximation when
compared to a new version of the asymptotically efficient log-ODE (or
Castell-Gaines) method.Comment: 27 pages, 8 figure
Functional Calculus
The aim of this book is to present a broad overview of the theory and applications related to functional calculus. The book is based on two main subject areas: matrix calculus and applications of Hilbert spaces. Determinantal representations of the core inverse and its generalizations, new series formulas for matrix exponential series, results on fixed point theory, and chaotic graph operations and their fundamental group are contained under the umbrella of matrix calculus. In addition, numerical analysis of boundary value problems of fractional differential equations are also considered here. In addition, reproducing kernel Hilbert spaces, spectral theory as an application of Hilbert spaces, and an analysis of PM10 fluctuations and optimal control are all contained in the applications of Hilbert spaces. The concept of this book covers topics that will be of interest not only for students but also for researchers and professors in this field of mathematics. The authors of each chapter convey a strong emphasis on theoretical foundations in this book
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Sparse representations and quadratic approximations in path integral techniques for stochastic response analysis of diverse systems/structures
Uncertainty propagation in engineering mechanics and dynamics is a highly challenging problem that requires development of analytical/numerical techniques for determining the stochastic response of complex engineering systems. In this regard, although Monte Carlo simulation (MCS) has been the most versatile technique for addressing the above problem, it can become computationally daunting when faced with high-dimensional systems or with computing very low probability events. Thus, there is a demand for pursuing more computationally efficient methodologies.
Recently, a Wiener path integral (WPI) technique, whose origins can be found in theoretical physics, has been developed in the field of engineering dynamics for determining the response transition probability density function (PDF) of nonlinear oscillators subject to non-white, non-Gaussian and non-stationary excitation processes. In the present work, the Wiener path integral technique is enhanced, extended and generalized with respect to three main aspects; namely, versatility, computational efficiency and accuracy.
Specifically, the need for increasingly sophisticated modeling of excitations has led recently to the utilization of fractional calculus, which can be construed as a generalization of classical calculus. Motivated by the above developments, the WPI technique is extended herein to account for stochastic excitations modeled via fractional-order filters. To this aim, relying on a variational formulation and on the most probable path approximation yields a deterministic fractional boundary value problem to be solved numerically for obtaining the oscillator joint response PDF.
Further, appropriate multi-dimensional bases are constructed for approximating, in a computationally efficient manner, the non-stationary joint response PDF. In this regard, two distinct approaches are pursued. The first employs expansions based on Kronecker products of bases (e.g., wavelets), while the second utilizes representations based on positive definite functions. Next, the localization capabilities of the WPI technique are exploited for determining PDF points in the joint space-time domain to be used for evaluating the expansion coefficients at a relatively low computational cost.
Subsequently, compressive sampling procedures are employed in conjunction with group sparsity concepts and appropriate optimization algorithms for decreasing even further the associated computational cost. It is shown that the herein developed enhancement renders the technique capable of treating readily relatively high-dimensional stochastic systems. More importantly, it is shown that this enhancement in computational efficiency becomes more prevalent as the number of stochastic dimensions increases; thus, rendering the herein proposed sparse representation approach indispensable, especially for high-dimensional systems.
Next, a quadratic approximation of the WPI is developed for enhancing the accuracy degree of the technique. Concisely, following a functional series expansion, higher-order terms are accounted for, which is equivalent to considering not only the most probable path but also fluctuations around it. These fluctuations are incorporated into a state-dependent factor by which the exponential part of each PDF value is multiplied. This localization of the state-dependent factor yields superior accuracy as compared to the standard most probable path WPI approximation where the factor is constant and state-invariant. An additional advantage relates to efficient structural reliability assessment, and in particular, to direct estimation of low probability events (e.g., failure probabilities), without possessing the complete transition PDF.
Overall, the developments in this thesis render the WPI technique a potent tool for determining, in a reliable manner and with a minimal computational cost, the stochastic response of nonlinear oscillators subject to an extended range of excitation processes. Several numerical examples, pertaining to both nonlinear dynamical systems subject to external excitations and to a special class of engineering mechanics problems with stochastic media properties, are considered for demonstrating the reliability of the developed techniques. In all cases, the degree of accuracy and the computational efficiency exhibited are assessed by comparisons with pertinent MCS data
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