Dissertations / Theses on the topic 'Calculus of variations'
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Tavares, Dina dos Santos. "Fractional calculus of variations." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22184.
Full textO 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.
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.
Full textEstudamos 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.
Zhang, Chengdian. "Calculus of variations with multiple integration." Bonn : [s.n.], 1989. http://catalog.hathitrust.org/api/volumes/oclc/20436929.html.
Full textCapet, SteÌphane. "Calculus of variations in quantum mechanics." Thesis, University of Warwick, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444831.
Full textSantos, Simão Pedro Silva. "Calculus of variations of Herglotz type." Doctoral thesis, Universidade de Aveiro, 2017. http://hdl.handle.net/10773/22503.
Full textWe 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.
Gratwick, Richard. "Singular minimizers in the calculus of variations." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/47653/.
Full textTaheri, Ali. "Local minimizers in the calculus of variations." Thesis, Heriot-Watt University, 1997. http://hdl.handle.net/10399/656.
Full textPerrotta, Stefania. "Some Problems in the Calculus of Variations." Doctoral thesis, SISSA, 1996. http://hdl.handle.net/20.500.11767/4452.
Full textBedford, 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.
Full textChan, 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.
Full textArora, Raman. "Analysis of Economic Models Through Calculus of Variations." TopSCHOLAR®, 2005. http://digitalcommons.wku.edu/theses/453.
Full textCampos, 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.
Full textChen, 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.
Full textZagatti, Sandro. "Some Problems in the Calculus of the Variations." Doctoral thesis, SISSA, 1992. http://hdl.handle.net/20.500.11767/4226.
Full textOnofrei, Daniel T. "Homogenization of an elastic-plastic problem." Link to electronic thesis, 2003. http://www.wpi.edu/Pubs/ETD/Available/etd-0430103-121632.
Full textCrasta, Graziano. "Nonconvex problems in Control Theory and Calculus of Variations." Doctoral thesis, SISSA, 1995. http://hdl.handle.net/20.500.11767/4503.
Full textLuria, Gianvittorio. "Constrained Calculus of Variations and Geometric Optimal Control Theory." Doctoral thesis, Università degli studi di Trento, 2010. https://hdl.handle.net/11572/368036.
Full textLuria, 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.
Full textBellettini, Giovanni. "Geometric problems involving curvatures in the calculus of variations." Doctoral thesis, SISSA, 1993. http://hdl.handle.net/20.500.11767/4064.
Full textChow, 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.
Full textChow, 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.
Full textKabisch, Sandra. "On established and new semiconvexities in the calculus of variations." Thesis, University of Surrey, 2016. http://epubs.surrey.ac.uk/812076/.
Full textDodd, Thomas James. "Partial regularity of local minimisers in the calculus of variations." Thesis, Heriot-Watt University, 2010. http://hdl.handle.net/10399/2404.
Full textDryl, Monika. "Calculus of variations on time scales and applications to economics." Doctoral thesis, Universidade de Aveiro, 2014. http://hdl.handle.net/10773/12869.
Full textWe 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.
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.
Full textVarvaruca, 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.
Full textCelada, Pietro. "Some Scalar and Vectorial Problems in the Calculus of Variations." Doctoral thesis, SISSA, 1997. http://hdl.handle.net/20.500.11767/4349.
Full textBONFANTI, 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.
Full textHopper, 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.
Full textIqbal, Zamin. "Variational methods in solid mechanics." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.301901.
Full textMuller, Stefan. "Variational problems in mechanics and analysis." Thesis, Heriot-Watt University, 1989. http://hdl.handle.net/10399/925.
Full textBandeira, 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.
Full textMcMahon, Chris. "Calculus of Variations on Time Scales and Its Applications to Economics." TopSCHOLAR®, 2008. http://digitalcommons.wku.edu/theses/370.
Full textBevan, 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.
Full textPooseh, Shakoor. "Computational methods in the fractional calculus of variations and optimal control." Doctoral thesis, Universidade de Aveiro, 2013. http://hdl.handle.net/10773/11510.
Full textThe 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.
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.
Full textFanzon, 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/.
Full textMorrison, 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/.
Full textChristopher, 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.
Full textIncludes 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.
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.
Full textFordred, Gordon Ian. "An application of the Malliavin calculus in finance." Diss., Pretoria : [s.n.], 2009. http://upetd.up.ac/thesis/available/etd-07062009-123751.
Full textChen, 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.
Full text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.
Full textLauteri, 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.
Full textWong, 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.
Full textMorris, 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/.
Full textKhodadadi, 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.
Full textTurski, 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.
Full textBourdin, 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.
Full textThis 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
Botelho, Fabio Silva. "Variational Convex Analysis." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/28351.
Full textPh. D.