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Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1990. / Vita. Abstract. Includes bibliographical references (leaves 119-127). Also available via the Internet.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 137-139).

In the classical calculus of variations, the Hamilton - Jacobi theory leads, under general hypotheses, to sufficient conditions for a local minimum. The optimal control problem as well has its own Hamilton -Jacobi approach to sufficient conditions for optimality. In this thesis we extend this approach to the differential inclusion problem; a general, nonconvex, nondifferentiable control problem. In particular, the familiar Hamilton - Jacobi equation is generalized and a corresponding necessary condition (chapter 2) is obtained. The sufficiency condition (chapter 3) is derived and an example is presented where it is shown how this result may lead to considerable simplification. Finally, we show (chapter 4) how the classical theory of canonical transformations may be brought to bear on certain Hamiltonian inclusions associated with the differential inclusion problem. Our main tool will be the generalized gradient, a set valued derivative for Lipschitz functions which reduces to the subdifferential of convex analysis in the convex case and the familiar derivative in the C¹ case. / Science, Faculty of / Mathematics, Department of / Graduate

Thesis (PhD (Mathematics and Applied Mathematics)) -- University of Pretoria, 2007. / Abstract in English. Includes bibliographical references.

Cette thèse est consacrée à l’étude des propriétés qualitatives de solutions d’une équation d’évolution de type Hamilton-Jacobi avec une diffusion donnée par l’opérateur p-Laplacien. On s’attache principalement à l’étude de l’effet de la diffusion non-linéaire sur le phénomène d’explosion du gradient. Les principales questions qu’on étudie portent sur l’existence locale, régularité, profil spatial d’explosion et la localisation des points d’explosion. En particulier on montre un résultat d’explosion en seul point du bord. Dans le chapitre 4, on utilise une approche de solutions de viscosité pour prolonger la solution explosive au delà des singularités et on étudie son comportement en temps grands. Dans l’avant dernier chapitre on s’intéresse au caractère borné des solutions globales du problème unidimensionnel. Dans le dernier chapitre on démontre une estimation de gradient locale en espace et on l’utilise pour obtenir un résultat de type Liouville. On s’inspire et on compare nos résultats avec les résultats connus pour le cas de la diffusion linéaire. / This thesis is devoted to the study of qualitative properties of solutions of an evolution equation of Hamilton-Jacobi type with a p-Laplacian diffusion. It is mainly concerned with the study of the effect of the non-linear diffusion on the gradient blow-up phenomenon. The main issues we are studying are: local existence and uniqueness, regularity, spatial profile of gradient blow-up and localization of the singularities. We provide examples where the gradient blow-up set is reduced to a single point. In Chapter 4, a viscosity solution approachis used to extend the blowing-up solutions beyond the singularities and an ergodic problem is also analyzed in order to study their long time behavior. In the penultimate chapter, we address the question of boundedness of global solutions to the one-dimensional problem. In the last chapter we prove a local in space, gradient estimate and we use it to obtain a Liouville-type theorem.

We consider the numerical solution of discretised Hamilton-Jacobi-Bellman (HJB) equations with applications in finance. For the discrete linear complementarity problem arising in American option pricing, we study a policy iteration method. We show, analytically and numerically, that, in standard situations, the computational cost of this approach is comparable to that of European option pricing. We also characterise the shortcomings of policy iteration, providing a lower bound for the number of steps required when having inaccurate initial data. For discretised HJB equations with a finite control set, we propose a penalty approach. The accuracy of the penalty approximation is of first order in the penalty parameter, and we present a Newton-type iterative solver terminating after finitely many steps with a solution to the penalised equation. For discretised HJB equations and discretised HJB obstacle problems with compact control sets, we also introduce penalty approximations. In both cases, the approximation accuracy is of first order in the penalty parameter. We again design Newton-type methods for the solution of the penalised equations. For the penalised HJB equation, the iterative solver has monotone global convergence. For the penalised HJB obstacle problem, the iterative solver has local quadratic convergence. We carefully benchmark all our numerical schemes against current state-of-the-art techniques, demonstrating competitiveness.

In this dissertation we prove the homogenization for two very different classes of nonlinear partial differential equations and nonlinear elliptic integro-differential equations. The first result covers the homogenization of convex and superlinear Hamilton-Jacobi equations with stationary ergodic dependence in time and space simultaneously. This corresponds to equations of the form: [mathematical equation]. The second class of equations is nonlinear integro-differential equations with periodic coefficients in space. These equations take the form, [mathematical equation]. / text

No description available.

Thesis (Ph.D.)--York University, 2006. Graduate Programme in Applied Mathematics. / Typescript. Includes bibliographical references (leaves 134-142). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR19766