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Tavares, Dina dos Santos. "Fractional calculus of variations." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22184.
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Doutoramento em Matemática e AplicaçõesO cálculo de ordem não inteira, mais conhecido por cálculo fracionário, consiste numa generalização do cálculo integral e diferencial de ordem inteira. Esta tese é dedicada ao estudo de operadores fracionários com ordem variável e problemas variacionais específicos, envolvendo também operadores de ordem variável. Apresentamos uma nova ferramenta numérica para resolver equações diferenciais envolvendo derivadas de Caputo de ordem fracionária variável. Consideram- -se três operadores fracionários do tipo Caputo, e para cada um deles é apresentada uma aproximação dependendo apenas de derivadas de ordem inteira. São ainda apresentadas estimativas para os erros de cada aproximação. Além disso, consideramos alguns problemas variacionais, sujeitos ou não a uma ou mais restrições, onde o funcional depende da derivada combinada de Caputo de ordem fracionária variável. Em particular, obtemos condições de otimalidade necessárias de Euler–Lagrange e sendo o ponto terminal do integral, bem como o seu correspondente valor, livres, foram ainda obtidas as condições de transversalidade para o problema fracionário.
The calculus of non–integer order, usual known as fractional calculus, consists in a generalization of integral and differential integer-order calculus. This thesis is devoted to the study of fractional operators with variable order and specific variational problems involving also variable order operators. We present a new numerical tool to solve differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. Furthermore, we consider variational problems subject or not to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, we establish necessary optimality conditions of Euler–Lagrange. As the terminal point in the cost integral, as well the terminal state, are free, thus transversality conditions are obtained.
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Ferreira, Rui Alexandre Cardoso. "Calculus of variations on time scales and discrete fractional calculus." Doctoral thesis, Universidade de Aveiro, 2010. http://hdl.handle.net/10773/2921.
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Doutoramento em MatemáticaEstudamos problemas do cálculo das variações e controlo óptimo no contexto das escalas temporais. Especificamente, obtemos condições necessárias de optimalidade do tipo de Euler–Lagrange tanto para lagrangianos dependendo de derivadas delta de ordem superior como para problemas isoperimétricos. Desenvolvemos também alguns métodos directos que permitem resolver determinadas classes de problemas variacionais através de desigualdades em escalas temporais. No último capítulo apresentamos operadores de diferença fraccionários e propomos um novo cálculo das variações fraccionário em tempo discreto. Obtemos as correspondentes condições necessárias de Euler– Lagrange e Legendre, ilustrando depois a teoria com alguns exemplos.
We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler–Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems. We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities. In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations. Corresponding Euler–Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided.
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Zhang, Chengdian. "Calculus of variations with multiple integration." Bonn : [s.n.], 1989. http://catalog.hathitrust.org/api/volumes/oclc/20436929.html.
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Capet, SteÌphane. "Calculus of variations in quantum mechanics." Thesis, University of Warwick, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444831.
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Santos, Simão Pedro Silva. "Calculus of variations of Herglotz type." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22503.
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Doutoramento em MatemáticaWe consider several problems based on Herglotz’s generalized variational problem. We dedicate two chapters to extensions on Herglotz’s generalized variational problem to higher-order and first-order problems with time delay, using a variational approach. In the last four chapters, we rewrite Herglotz's type problems in the optimal control form and use an optimal control approach. We prove generalized higher- order Euler-Lagrange equations, first without and then with time delay; higher-order natural boundary conditions; Noether's first theorem for the first-order problem of Herglotz with time delay; Noether's first theorem for higher-order problems of Herglotz without and with time delay; and existence of Noether currents as a version of Noether's second theorem of optimal control.
Consideramos vários problemas com base no problema variacional generalizado de Herglotz. Dois capítulos são dedicados à extensão do problema variacional generalizado de Herglotz para ordem superior e para problemas de primeira ordem com atraso no tempo, utilizando uma abordagem variacional. Nos últimos quatro capítulos, reescrevemos os problemas de Herglotz na forma do controlo ótimo e usamos essa abordagem. Demonstramos equações generalizadas de Euler-Lagrange de ordem superior, inicialmente sem e depois com atraso no tempo; condições de fronteira de ordem superior; o primeiro teorema de Noether para o problema de Herglotz de primeira ordem com atraso no tempo; o primeiro teorema de Noether para problemas de ordem superior de Herglotz sem e com atraso no tempo; e a existência de leis de conservação de Noether numa versão do segundo teorema de Noether do controlo ótimo.
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Gratwick, Richard. "Singular minimizers in the calculus of variations." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/47653/.
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This thesis examines the possible failure of regularity for minimizers of onedimensional variational problems. The direct method of the calculus of variations gives rigorous assurance that minimizers exist, but necessarily admits the possibility that minimizers might not be smooth. Regularity theory seeks to assert some extra smoothness of minimizers. Tonelli's partial regularity theorem states that any absolutely continuous minimizer has a (possibly infinite) classical derivative everywhere, and this derivative is continuous as a function into the extended real line. We examine the limits of this theorem. We find an example of a reasonable problem where partial regularity fails, and examples where partial regularity holds, but the infinite derivatives of minimizers permitted by the theorem occur very often, in precise senses. We construct continuous Lagrangians, strictly convex and superlinear in the third variable, such that the associated variational problems have minimizers nondifferentiable on dense second category sets. Thus mere continuity is an insufficient smoothness assumption for Tonelli's partial regularity theorem. Davie showed that any compact null set can occur as the singular set of a minimizer to a problem given via a smooth Lagrangian with quadratic growth. The proof relies on enforcing the occurrence of the Lavrentiev phenomenon. We give a new proof of the result, but constructing also a Lagrangian with arbitrary superlinear growth, and in which the Lavrentiev phenomenon does not occur in the problem. Universal singular sets record how often a given Lagrangian can have minimizers with infinite derivative. Despite being negligible in terms of both topology and category, they can have dimension two: any compact purely unrectifiable set can lie inside the universal singular set of a Lagrangian with arbitrary superlinearity. We show this also to be true of Fσ purely unrectifiable sets, suggesting a possible characterization of universal singular sets.
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Taheri, Ali. "Local minimizers in the calculus of variations." Thesis, Heriot-Watt University, 1997. http://hdl.handle.net/10399/656.
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Perrotta, Stefania. "Some Problems in the Calculus of Variations." Doctoral thesis, SISSA, 1996. http://hdl.handle.net/20.500.11767/4452.
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Bedford, Stephen James. "Calculus of variations and its application to liquid crystals." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:a2004679-5644-485c-bd35-544448f53f6a.
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The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function n ε W1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state n to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV2 (Ω,S2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.
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Chan, Ka-bo, and 陳家寶. "On Griffiths' formalism of the calculus of variations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B30456630.
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Arora, Raman. "Analysis of Economic Models Through Calculus of Variations." TopSCHOLAR®, 2005. http://digitalcommons.wku.edu/theses/453.
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This thesis is a combination of two science fields: Mathematics and Economics. Mathematics is often used to formulate a clear and concise solution to economic problems. In my observation calculus of variation has often been used in various macroeconomic problems. This mathematical method deals with maximizing or minimizing of various objective functions given a set of constraints. This topic brings out one of the best ways to show the relationship between mathematics and economics. My thesis consists of three parts: The first chapter contains a review of the calculus of variations. Basic definitions and important conditions have been stated. The aim of this chapter was to set the groundwork for understanding calculus of variations so that it can be used in solving various economics models. In the second chapter we study an economic model from which calculus of variations has been used to solve it. The macroeconomic model deals with optimizing the social welfare function. The entire working of the model has been discussed and documented in the thesis report. The third chapter deals with the analysis of the Lucas model which concentrated on how the accumulation of human capital impacts the growth rate of the economy. Lucas assumes that the growth rate of the human capital is linearly related to its level. If we abandon this assumption, will the optimal value of the time devoted to education in the steady state exist? If it exists, will it be same or different? So we introduced a new model in which the only modification we made to the Lucas model was in the equation that describes the process of human accumulation by introducing a nonlinear component. On investigation of this new model we have shown that it is possible that optimal behavior for an individual can be not to educate himself.
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Campos, Cordero Judith. "Regularity and uniqueness in the calculus of variations." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:81e69dac-5ba2-4dc3-85bc-5d9017286f13.
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This thesis is about regularity and uniqueness of minimizers of integral functionals of the form F(u) := ∫Ω F(∇u(x)) dx; where F∈C2(RNn) is a strongly quasiconvex integrand with p-growth, Ω⊆RnRn is an open bounded domain and u∈W1,pg(Ω,RN) for some boundary datum g∈C1,α(‾Ω, RN). The first contribution of this work is a full regularity result, up to the boundary, for global minimizers of F provided that the boundary condition g satisfies that ΙΙ∇gΙΙLP < ε for some ε > 0 depending only on n;N, the parameters given by the strong quasiconvexity and p-growth conditions and, most importantly, on an arbitrary but fixed constant M > 0 for which we require that ΙΙ∇gΙΙO,α < M. Furthermore, when the domain Ω is star-shaped, we extend the regularity result to the case of W1,p-local minimizers. On the other hand, for the case of global minimizers we exploit the compactness provided by the aforementioned regularity result to establish the main contribution of this thesis: we prove that, under essentially the same smallness assumptions over the boundary condition g that we mentioned above, the minimizer of F in W1,pg is unique. This result appears in contrast to the non-uniqueness examples previously given by Spadaro [Spa09], for which the boundary conditions are required to be suitably large.
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Chen, Chuei Yee. "Quasiminimality and coercivity in the calculus of variations." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:d6daadfd-92fb-4fa6-9d5b-fd8403955079.
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Zagatti, Sandro. "Some Problems in the Calculus of the Variations." Doctoral thesis, SISSA, 1992. http://hdl.handle.net/20.500.11767/4226.
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Onofrei, Daniel T. "Homogenization of an elastic-plastic problem." Link to electronic thesis, 2003. http://www.wpi.edu/Pubs/ETD/Available/etd-0430103-121632.
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Crasta, Graziano. "Nonconvex problems in Control Theory and Calculus of Variations." Doctoral thesis, SISSA, 1995. http://hdl.handle.net/20.500.11767/4503.
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Luria, Gianvittorio. "Constrained Calculus of Variations and Geometric Optimal Control Theory." Doctoral thesis, Università degli studi di Trento, 2010. https://hdl.handle.net/11572/368036.
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The present work provides a geometric approach to the calculus of variations in the presence of non-holonomic constraints. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The usual classification of the evolutions into normal and abnormal ones is also discussed, showing the existence of a universal algorithm assigning to every admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. A gauge-invariant formulation of the variational problem, based on the introduction of the bundle of affine scalars over the configuration manifold, is then presented. The analysis includes a revisitation of Pontryagin Maximum Principle and of the Erdmann-Weierstrass corner conditions, a local interpretation of Pontryagin's equations as dynamical equations for a free (singular) Hamiltonian system and a generalization of the classical criteria of Legendre and Bliss for the characterization of the minima of the action functional to the case of piecewise-differentiable extremals with asynchronous variation of the corners.
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Luria, Gianvittorio. "Constrained Calculus of Variations and Geometric Optimal Control Theory." Doctoral thesis, University of Trento, 2010. http://eprints-phd.biblio.unitn.it/170/1/Constrained_Calculus_of_Variations_and_Geometric_Optimal_Control_Theory.pdf.
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The present work provides a geometric approach to the calculus of variations in the presence of non-holonomic constraints. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The usual classification of the evolutions into normal and abnormal ones is also discussed, showing the existence of a universal algorithm assigning to every admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. A gauge-invariant formulation of the variational problem, based on the introduction of the bundle of affine scalars over the configuration manifold, is then presented. The analysis includes a revisitation of Pontryagin Maximum Principle and of the Erdmann-Weierstrass corner conditions, a local interpretation of Pontryagin's equations as dynamical equations for a free (singular) Hamiltonian system and a generalization of the classical criteria of Legendre and Bliss for the characterization of the minima of the action functional to the case of piecewise-differentiable extremals with asynchronous variation of the corners.
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Bellettini, Giovanni. "Geometric problems involving curvatures in the calculus of variations." Doctoral thesis, SISSA, 1993. http://hdl.handle.net/20.500.11767/4064.
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Chow, Hong-Yu, and 周康宇. "Griffiths' formalism of the calculus of variations and applications toinvariants." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2005. http://hub.hku.hk/bib/B35812503.
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Chow, Hong-Yu. "Griffiths' formalism of the calculus of variations and applications to invariants." Click to view the E-thesis via HKUTO, 2005. http://sunzi.lib.hku.hk/hkuto/record/B35812503.
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Kabisch, Sandra. "On established and new semiconvexities in the calculus of variations." Thesis, University of Surrey, 2016. http://epubs.surrey.ac.uk/812076/.
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After introducing the topics that will be covered in this work we review important concepts from the calculus of variations in elasticity theory. Subsequently the following three topics are discussed: The first originates from the work of Post and Sivaloganathan [\emph{Proceedings of the Royal Society of Edinburgh, Section: A Mathematics}, 127(03):595--614, 1997] in the form of two scenarios involving the twisting of the outer boundary of an annulus $A$ around the inner. It seeks minimisers of $∫_A \frac{1}{2}|∇u|^2 \d x$ among deformations $u$ with the constraint $\det ∇u ≥0$ a.e.~as well as of $∫_A \frac{1}{2}|∇u|^2 + h(\det ∇u) \d x$ in which $h$ penalises volume compression so that $\det ∇u > 0$ a.e.~is imposed on minimisers. In the former case we find infinitely many explicit solutions for which $\det ∇u = 0$ holds on a region around the inner boundary of $A$. In the latter we expand on known results by showing similar growth properties of the solutions compared to the previous case while contrasting that $\det ∇u>0$ holds everywhere. In the second we introduce a new semiconvexity called $n$-polyconvexity that unifies poly- and rank-one convexity in the sense that for $f:ℝ^{d×D}→\bar{ℝ}$ we have that $n$-polyconvexity is equivalent to polyconvexity for $n=\min\{d,D\}=:d∧D$ and equivalent to rank-one convexity for $n=1$. For $d,D≥3$ we gain previously unknown semiconvexities in hierarchical order ($2$-polyconvexity, \dots, $(d∧D-1)$-polyconvexity, weakest to strongest). We further define functions which are `$n$-polyaffine at $F$' and find that they are not necessarily polyaffine for $n
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Dodd, Thomas James. "Partial regularity of local minimisers in the calculus of variations." Thesis, Heriot-Watt University, 2010. http://hdl.handle.net/10399/2404.
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Dryl, Monika. "Calculus of variations on time scales and applications to economics." Doctoral thesis, Universidade de Aveiro, 2014. http://hdl.handle.net/10773/12869.
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Doutoramento em MatemáticaWe consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.
Consideramos alguns problemas do cálculo das variações em escalas temporais. Primeiramente, demonstramos equações do tipo de Euler-Lagrange e condições de transversalidade para problemas de horizonte infinito generalizados. De seguida, consideramos a composição de uma certa função escalar com os integrais delta e nabla de um campo vetorial. Presta-se atenção a problemas extremais inversos para funcionais variacionais em escalas de tempo arbitrárias. Começamos por demonstrar uma condição necessária para uma equação dinâmica integro-diferencial ser uma equação de Euler-Lagrange. Resultados novos e interessantes para o cálculo discreto e quantum são obtidos como casos particulares. Além disso, usando a equação de Euler-Lagrange e a condição de Legendre fortalecida, obtemos uma forma geral para uma funcional variacional atingir um mínimo local, num determinado ponto do espaço vetorial. No final, duas aplicações interessantes em termos económicos são apresentadas. No primeiro caso, consideramos uma empresa que quer programar as suas políticas de produção e de investimento para alcançar uma determinada taxa de produção e maximizar a sua competitividade no mercado futuro. O outro problema é mais complexo e relaciona a inflação e o desemprego, que inflige uma perda social. A perda social é escrita como uma função da taxa de inflação p e a taxa de desemprego u, com diferentes pesos. Em seguida, usando as relações conhecidas entre p, u, e a taxa de inflação esperada π, reescrevemos a função de perda social como uma função de π. A resposta é obtida através da aplicação do cálculo das variações, a fim de encontrar a curva ótima π que minimiza a perda social total ao longo de um determinado intervalo de tempo.
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Crispin, Daniel John. "Brake periodic orbits and linking in the calculus of variations." Thesis, University of Bath, 2004. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.410925.
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Varvaruca, Lorina. "Singular minimizers in the calculus of variations and nonlinear elasticity." Thesis, University of Bath, 2007. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.439277.
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Celada, Pietro. "Some Scalar and Vectorial Problems in the Calculus of Variations." Doctoral thesis, SISSA, 1997. http://hdl.handle.net/20.500.11767/4349.
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BONFANTI, GIOVANNI. "Progresses on some classical problems of the calculus of variations." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2012. http://hdl.handle.net/10281/28223.
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Some theorems about the validity of the Euler-Lagrange equation are proved. In particular, Lagrangians without regularity hypothesis or without growth assumptions are considered. Moreover, a result about the non-occurrence of the Lavrentiev phenomenon in the scalar case is proved.
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Hopper, Christopher Peter. "On the regularity of holonomically constrained minimisers in the calculus of variations." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:d8bde7a2-7dae-44d2-919d-48b9f2543789.
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This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.
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Iqbal, Zamin. "Variational methods in solid mechanics." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.301901.
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Muller, Stefan. "Variational problems in mechanics and analysis." Thesis, Heriot-Watt University, 1989. http://hdl.handle.net/10399/925.
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Bandeira, Luís Miguel Zorro. "Analysis of new situations for quasiconvexity versus rank-one convexity in 2 x 2 and other dimensions." Doctoral thesis, Universidade de Évora, 2008. http://hdl.handle.net/10174/11119.
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It is well-known that quasiconvexity is a fundamental concept for vector
problems in the Calculus of Variations. Its main necessary condition is rankone
convexity. Still today it is not known whether it is also sufficient or not, when
the target space of deformations is m=2 (in the general case).
We introduce a method to find, in a systematic way, rank-one convex
polynomials. We show how it works in several examples. It can also be
applied to convexity along general cones.
An alternative proof is provided for the well-known quadratic case of
quasiconvexity, which does not use the Plancherel formula. An application
to the case of 4th degree homogeneous polynomials is shown.
We also explore an attempt to disprove the implication, from rank-one
convexity to quasiconvexity for 2x2 symmetric matrices, using the viewpoint
of laminates and homogeneous gradient Young measures. /RESUMO - É bem conhecido que a quasiconvexidade é um conceito fundamental para problemas vectoriais do Cálculo das Variações. A sua principal condição necessária é a convexidade característica-l. Ainda hoje não é conhecido se é ou não suficiente, quando o espaço alvo das deformações é m=2 (no caso geral).
Introduzimos um método para determinar, de uma forma sistemática, polinómios convexos característica-1. Mostramos como funciona em diversos exemplos. Pode também ser aplicado à convexidade ao longo de cones gerais.
Providenciamos uma demonstração alternativa para o bem conhecido caso quadrático da quasiconvexidade, que não utiliza a fórmula de Plancherel. Apresentamos uma aplicação para o caso dos polinómios homogéneos de grau 4.
Exploramos também uma tentativa para refutar a implicação da convex-idade característica-1 para a quasiconvexidade nas matrizes 2x2 simétricas, sob o ponto de vista dos laminados e das medidas de Young gradiente homogéneas.
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McMahon, Chris. "Calculus of Variations on Time Scales and Its Applications to Economics." TopSCHOLAR®, 2008. http://digitalcommons.wku.edu/theses/370.
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The goal of time scale research is to progress the development of a harmonized theory that is all encompassing of the more commonly known specialized forms. The main results of this paper is the presentation of the Ramsey model which can be written using both the A and V operators, and solved using the two separate theories of the calculus of variations on time scales. The next presentation will be of the solution of an adjustment model, for a specific form of a time scale, whose functional can only be optimized, using the existing theory, when written with the A operator. We will also develop certain elements of stochastic time scale calculus, in order to lay the groundwork necessary to develop the theory of stochastic calculus of variations on time scales.
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Bevan, Jonathan. "Polyconvexity and counterexamples to regularity in the multidimensional calculus of variations." Thesis, University of Oxford, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275363.
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Pooseh, Shakoor. "Computational methods in the fractional calculus of variations and optimal control." Doctoral thesis, Universidade de Aveiro, 2013. http://hdl.handle.net/10773/11510.
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Doutoramento em MatemáticaThe fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.
O cálculo das variações e controlo óptimo fraccionais são generalizações das correspondentes teorias clássicas, que permitem formulações e modelar problemas com derivadas e integrais de ordem arbitrária. Devido à carência de métodos analíticos para resolver tais problemas fraccionais, técnicas numéricas são desenvolvidas. Nesta tese, investigamos a aproximação de operadores fraccionais recorrendo a séries de derivadas de ordem inteira e diferenças finitas generalizadas. Obtemos majorantes para o erro das aproximações propostas e estudamos a sua eficiência. Métodos directos e indirectos para a resolução de problemas variacionais fraccionais são estudados em detalhe. Discutimos também condições de optimalidade para diferentes tipos de problemas variacionais, sem e com restrições, e para problemas de controlo óptimo fraccionais. As técnicas numéricas introduzidas são ilustradas recorrendo a exemplos.
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Saunders, D. J. "The geometry of jet bundles, with applications to the calculus of variations." Thesis, Open University, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.376035.
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Fanzon, Silvio. "Geometric patterns and microstructures in the study of material defects and composites." Thesis, University of Sussex, 2018. http://sro.sussex.ac.uk/id/eprint/72566/.
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The main focus of this PhD thesis is the study of microstructures and geometric patterns in materials, in the framework of the Calculus of Variations. My PhD research, carried out in collaboration with my supervisor Mariapia Palombaro and Marcello Ponsiglione, led to the production of three papers [21, 22, 23]. Papers [21, 22] have already been published, while [23] is currently in preparation. This thesis is divided into two main parts. In the first part we present the results obtained in [22, 23]. In these two works geometric patterns have to be understood as patterns of dislocations in crystals. The second part is devoted to [21], where suitable microgeometries are needed as a mean to produce gradients that display critical integrability properties.
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Morrison, George. "Rotationally-symmetric solutions to a nonlinear elliptic system under an incompressibility constraint and related problems." Thesis, University of Sussex, 2018. http://sro.sussex.ac.uk/id/eprint/79856/.
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Christopher, Jason W. "Using calculus of variations to optimize paths of descent through ski race courses." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/32840.
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Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2005.Includes bibliographical references (p. 69-70).
The goal of ski racing is to pass through a series of gates as quickly as possible. There are many paths from gate to gate, but there is only one path that is fastest. By knowing what the fastest path is, a racer could shave tenths of seconds off his or her time. That is a tremendous amount of time considering that races are won by hundredths of a second. This thesis attempts to calculate the fastest path through a ski race course using several simplifications such as neglecting friction. The method of attacking this problem is to modify the Brachistochrone problem. It is found that it is best if the skier places the apex of the turn at the gate, and that turning more after the gate is better than turning more above the gate. In the case of a rhythmical course, it is found that turning more below the gate is still true, but not as evident. Instead the optimal path appears more symmetric about the gate.
by Jason W. Christopher.
S.B.
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Buß, Hinderk M. "A posteriori error estimators based on duality techniques from the calculus of variations." [S.l. : s.n.], 2003. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10790752.
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Fordred, Gordon Ian. "An application of the Malliavin calculus in finance." Diss., Pretoria : [s.n.], 2009. http://upetd.up.ac/thesis/available/etd-07062009-123751.
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Chen, Pei-Tai. "Axisymmetric vibration, acoustic radiation, and the influence of eigenvalue veering phenomena in prolate spheroidal shells using variational principles." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/19407.
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黃志榮 and Chi-wing Wong. "On Cartan form and equivalence of variational problems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B31220071.
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Lauteri, Gianluca. "The Emergence of Cosserat-type Structures in Metal Plasticity." Doctoral thesis, Universitätsbibliothek Leipzig, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-225513.
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We study an energy functional able to describe low energy configurations of a two dimensional lattice with dislocations in a nonlinear elasticity regime. The main result can be described as follows: configurations of energy comparable to the lattice spacing consist of piecewise constant microrotations with small angle grain boundaries between them. Moreover, we also give bounds to the energy of particular configurations describing a small angle symmetric tilt grain boundary.
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Wong, Chi-wing. "On Cartan form and equivalence of variational problems /." Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19472602.
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Morris, Charles Graham. "Mapping problems in the calculus of variations : twists, L1-local minimisers and vectorial symmetrisation." Thesis, University of Sussex, 2017. http://sro.sussex.ac.uk/id/eprint/72567/.
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Khodadadi, Mohammad. "Exploration of variations of unrestricted blocking for description logics." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/exploration-of-variations-of-unrestricted-blocking-for-description-logics(3fc15638-1483-42d0-a827-75cb231f0737).html.
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Description logics are a family of logics that provide formalisms for representing and reasoning about knowledge, based on describing concepts, in a structured and formally well-understood way. They provide the logical foundation for the web ontology language (OWL), which increased awareness of them recently. The most popular techniques for decision procedures for description logics are tableau reasoning methods, which have a long tradition and are well established in automated reasoning. This thesis investigates the possibility of finding a general and optimised blocking mechanism for description logics with the finite model property. It suggests that, while the high branching factor for unrestricted blocking reduces its performance, suitable control of the application of the blocking rule can make the performance acceptable while preserving termination. This claim is supported by experiments that compare the performance of two sample controlled versions of unrestricted blocking. In order to show the generality and power of controlled versions of unrestricted blocking, it is shown how some of the mainstream and most successful standard blocking mechanisms can be approximated as restricted forms of unrestricted blocking. These approximations have the advantage of always being sound compared to their standard versions, which are known to be sound only for some logics. Here, a variation of unrestricted blocking which can ensure strong termination is also introduced. This is done through introducing a new rule that uses the inequality expressions introduced by the blocking rule. The weak termination property of unrestricted blocking is one of its weak points which by this variant of blocking can be addressed. The work presented in this thesis should be of value to people who are working on generalising different aspects of reasoning methods. As blocking plays a critical role in termination of tableau provers, exploration of different variations of unrestricted blocking introduced here may be also of interest for the artificial intelligence researcher.
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Turski, Jacek. "Calculus of variations for discontinous fields and its applications to selected topics in continuum mechanics." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=72804.
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Bourdin, Loïc. "Contributions au calcul des variations et au principe du maximum de Pontryagin en calculs time scale et fractionnaire." Thesis, Pau, 2013. http://www.theses.fr/2013PAUU3009/document.
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Cette thèse est une contribution au calcul des variations et à la théorie du contrôle optimal dans les cadres discret, plus généralement time scale, et fractionnaire. Ces deux domaines ont récemment connu un développement considérable dû pour l’un à son application en informatique et pour l’autre à son essor dans des problèmes physiques de diffusion anormale. Que ce soit dans le cadre time scale ou dans le cadre fractionnaire, nos objectifs sont de : a) développer un calcul des variations et étendre quelques résultats classiques (voir plus bas); b) établir un principe du maximum de Pontryagin (PMP en abrégé) pour des problèmes de contrôle optimal. Dans ce but, nous généralisons plusieurs méthodes variationnelles usuelles, allant du simple calcul des variations au principe variationnel d’Ekeland (couplé avec la technique des variations-aiguilles), en passant par l’étude d’invariances variationnelles par des groupes de transformations. Les démonstrations des PMPs nous amènent également à employer des théorèmes de point fixe et à prendre en considération la technique des multiplicateurs de Lagrange ou encore une méthode basée sur un théorème d’inversion locale conique. Ce manuscrit est donc composé de deux parties : la Partie 1 traite de problèmes variationnels posés sur time scale et la Partie 2 est consacrée à leurs pendants fractionnaires. Dans chacune de ces deux parties, nous suivons l’organisation suivante : 1. détermination de l’équation d’Euler-Lagrange caractérisant les points critiques d’une fonctionnelle Lagrangienne ; 2. énoncé d’un théorème de type Noether assurant l’existence d’une constante de mouvement pour les équations d’Euler-Lagrange admettant une symétrie ; 3. énoncé d’un théorème de type Tonelli assurant l’existence d’un minimiseur pour une fonctionnelle Lagrangienne et donc, par la même occasion, d’une solution pour l’équation d’Euler-Lagrange associée (uniquement en Partie 2) ; 4. énoncé d’un PMP (version forte en Partie 1, version faible en Partie 2) donnant une condition nécessaire pour les trajectoires qui sont solutions de problèmes de contrôle optimal généraux non-linéaires ; 5. détermination d’une condition de type Helmholtz caractérisant les équations provenant d’un calcul des variations (uniquement en Partie 1 et uniquement dans les cas purement continu et purement discret). Des théorèmes de type Cauchy-Lipschitz nécessaires à l’étude de problèmes de contrôle optimal sont démontrés en AnnexeThis dissertation deals with the mathematical fields called calculus of variations and optimal control theory. More precisely, we develop some aspects of these two domains in discrete, more generally time scale, and fractional frameworks. Indeed, these two settings have recently experience a significant development due to its applications in computing for the first one and to its emergence in physical contexts of anomalous diffusion for the second one. In both frameworks, our goals are: a) to develop a calculus of variations and extend some classical results (see below); b) to state a Pontryagin maximum principle (denoted in short PMP) for optimal control problems. Towards these purposes, we generalize several classical variational methods, including the Ekeland’s variational principle (combined with needle-like variations) as well as variational invariances via the action of groups of transformations. Furthermore, the investigations for PMPs lead us to use fixed point theorems and to consider the Lagrange multiplier technique and a method based on a conic implicit function theorem. This manuscript is made up of two parts : Part A deals with variational problems on time scale and Part B is devoted to their fractional analogues. In each of these parts, we follow (with minor differences) the following organization: 1. obtaining of an Euler-Lagrange equation characterizing the critical points of a Lagrangian functional; 2. statement of a Noether-type theorem ensuring the existence of a constant of motion for Euler-Lagrange equations admitting a symmetry;3. statement of a Tonelli-type theorem ensuring the existence of a minimizer for a Lagrangian functional and, consequently, of a solution for the corresponding Euler-Lagrange equation (only in Part B); 4. statement of a PMP (strong version in Part A and weak version in Part B) giving a necessary condition for the solutions of general nonlinear optimal control problems; 5. obtaining of a Helmholtz condition characterizing the equations deriving from a calculus of variations (only in Part A and only in the purely continuous and purely discrete cases). Some Picard-Lindelöf type theorems necessary for the analysis of optimal control problems are obtained in Appendices
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Botelho, Fabio Silva. "Variational Convex Analysis." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/28351.
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This work develops theoretical and applied results for variational convex analysis. First we present the basic tools of analysis necessary to develop the core theory and applications.
New results concerning duality principles for systems originally modeled by non-linear differential equations are shown in chapters 9 to 17. A key aspect of this work is that although the original problems are non-linear with corresponding non-convex variational formulations, the dual formulations obtained are almost always concave and amenable to numerical computations. When the primal problem has no solution in the classical sense, the solution of dual problem is a weak limit of minimizing sequences, and the evaluation of such average behavior is important in many practical applications. Among the results we highlight the dual formulations for micro-magnetism, phase transition models, composites in elasticity and conductivity and others. To summarize, in the present work we introduce convex analysis as an interesting alternative approach for the understanding and computation of some important problems in the modern calculus of variations.Ph. D.