Orientador: Geraldo Nunes Silva / Banca: Iguer Luis Domini dos Santos / Banca: Marko Antonio Rojas Medar / Resumo: Neste trabalho caracterizaremos sistemas dinâmicos na forma dos chamados domínios estratificados. Bressan e Hong[9] foram os primeiros a definir e trabalhar em domínios estratificados. Grosso modo, estes são uma coleção de domínios disjuntos, cada um tendo sua própria dinâmica; mas não se exige que seus domínios sejam proximamente suaves e nem wedged. Estes termos foram introduzidos por P. Wolenski e R. Barnard em[10]. Primeiramente, estabeleceremos condições Hamiltonianos para caracterizar invariância fraca e forte para sistemas não Lipschitz em domínios estratificados. Depois, estudamos condições Hamiltonianas para sistemas fracamente e fortemente decrescentes e apresentamos condições que garantem a estabilidade assintótica global para uma dinâmica estratificada e finalmente apresentamos o problema tipo Mayer em domínios estratificados em que mostramos que a função valor e a única solução semicontínua inferior de uma generalização adequada da clássica equação Hamilton-Jacobi-Bellman, para a dinâmica estratificada / Abstract: In this work will characterize dynamical systems in the form of the so-called strati ed domain. Bressan and Hong [9] were the rst to de ne and work in strati ed domains. Roughly speaking, these are a collection of disjoint domains, each having its own dynamics; but not requiring that their domains are proximally smooth and not wedged. These terms were introduced by P. Wolenski and R. Barnard in [10]. At rst, we will establish Hamiltonian conditions to characterize weak and strong invariance for systems non-Lipschitz in strati ed domains. Secondly, we study the Hamiltonian conditions for systems weakly and strongly de- creasing and present conditions that guarantee global asymptotic stability for a strati ed dynamics and nally we present the problem Mayer type in strati ed domains where we show that the value function is the unique lower semicontinuous solution of an appropriate generalization of the classical Hamilton-Jacobi-Bellman equation for strati ed dynamics / Mestre

The present thesis considers the development and analysis of arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods with time-dependent approximation spaces for conservation laws and the Hamilton-Jacobi equations. Fundamentals about conservation laws, Hamilton-Jacobi equations and discontinuous Galerkin methods are presented. In particular, issues in the development of discontinuous Galerkin (DG) methods for the Hamilton-Jacobi equations are discussed. The development of the ALE-DG methods based on the assumption that the distribution of the grid points is explicitly given for an upcoming time level. This assumption allows to construct a time-dependent local affine linear mapping to a reference cell and a time-dependent finite element test function space. In addition, a version of Reynolds’ transport theorem can be proven. For the fully-discrete ALE-DG method for nonlinear scalar conservation laws the geometric conservation law and a local maximum principle are proven. Furthermore, conditions for slope limiters are stated. These conditions ensure the total variation stability of the method. In addition, entropy stability is discussed. For the corresponding semi-discrete ALE-DG method, error estimates are proven. If a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell, the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence for monotone fuxes and the optimal $(k+1)$ convergence for an upwind flux are proven in the $\mathrm{L}^{2}$-norm. The capability of the method is shown by numerical examples for nonlinear conservation laws. Likewise, for the semi-discrete ALE-DG method for nonlinear Hamilton-Jacobi equations, error estimates are proven. In the one dimensional case the optimal $\left(k+1\right)$ convergence and in the two dimensional case the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence are proven in the $\mathrm{L}^{2}$-norm, if a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell. For the fullydiscrete method, the geometric conservation is proven and for the piecewise constant forward Euler step the convergence of the method to the unique physical relevant solution is discussed. / Die vorliegende Arbeit beschäftigt sich mit der Entwicklung und Analyse von arbitrar Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) Methoden mit zeitabhängigen Testfunktionen Räumen für Erhaltungs- und Hamilton-Jacobi Gleichungen. Grundlagen über Erhaltungsgleichungen, Hamilton-Jacobi Gleichungen und discontinuous Galerkin Methoden werden präsentiert. Insbesondere werden Probleme bei der Entwicklung von discontinuous Galerkin Methoden für die Hamilton-Jacobi Gleichungen untersucht. Die Entwicklung der ALE-DG Methode basiert auf der Annahme, dass die Verteilung der Gitterpunkte zu einem kommenden Zeitpunkt explizit gegeben ist. Diese Annahme ermöglicht die Konstruktion einer zeitabhängigen lokal affin-linearen Abbildung auf eine Referenzzelle und eines zeitabhängigen Testfunktionen Raums. Zusätzlich kann eine Version des Reynolds’schen Transportsatzes gezeigt werden. Für die vollständig diskretisierte ALE-DG Methode für nichtlineare Erhaltungsgleichungen werden der geometrischen Erhaltungssatz und ein lokales Maximumprinzip bewiesen. Des Weiteren werden Bedingungen für Limiter angegeben. Diese Bedingungen sichern die Stabilität der Methode im Sinne der totalen Variation. Zusätzlich wird die Entropie-Stabilität der Methode diskutiert. Für die zugehörige semi-diskretisierte ALE-DG Methode werden Fehlerabschätzungen gezeigt. Wenn auf der Referenzzelle ein Testfunktionen Raum, der stückweise Polynome vom Grad $k$ enthält verwendet wird, kann für einen monotonen Fluss die suboptimale Konvergenzordnung $\left(k+\frac{1}{2}\right)$ und für einen upwind Fluss die optimale Konvergenzordnung $\left(k+1\right)$ in der $\mathrm{L}^{2}$-Norm gezeigt werden. Die Leistungsfähigkeit der Methode wird anhand numerischer Beispiele für nichtlineare Erhaltungsgleichungen untersucht. Ebenso werden für die semi-diskretisierte ALE-DG Methode für nichtlineare Hamilton-Jacobi Gleichungen Fehlerabschätzungen gezeigt. Wenn auf der Referenzzelle ein Testfunktionen Raum, der stückweise Polynome vom Grad k enthält verwendet wird, kann im eindimensionalen Fall die optimale Konvergenzordnung $\left(k+1\right)$ und im zweidimensionalen Fall die suboptimale Konvergenzordnung $\left(k+\frac{1}{2}\right)$ in der $\mathrm{L}^{2}$-Norm gezeigt werden. Für die vollständig diskretisierte ALE-DG Methode werden der geometrischen Erhaltungssatz bewiesen und für die stückweise konstante explizite Euler Diskretisierung wird die Konvergenz gegen die eindeutige physikalisch relevante Lösung diskutiert.

The WET-Temp (Watershed Evaluation Tool Temperature) model is designed to take advantage of spatially explicit datasets to predict stream temperature distribution. Datasets describing vegetation cover, stream network locations, elevation and stream discharge are utilized by WET-Temp to quantify geometric relationships between the sun, stream channel and riparian areas. These relationships are used to estimate the energy gained or lost by the stream via various heat flux processes (solar and longwave radiation, evaporation, convection and advection). The sum of these processes is expressed as a differential energy balance equation applied at discrete locations across the stream network. The model describes diurnal temperature dynamics at each of these locations and thus temperature distribution across the entire network. WET-Temp is calibrated to a tributary of the South Santiam River in western Oregon, McDowell Creek. The mean differences between measured and modeled values in McDowell Creek were 0.6��C for daily maximum temperature and 1.3��C for daily minimum temperature. The model was then used to predict maximum and minimum temperatures in an adjacent tributary, Hamilton Creek. The mean differences between modeled and measured values in this paired basin were 1.8��C for daily maximum temperatures and 1.4��C for daily minimum temperatures. Influences of model parameters on modeled temperature distributions are explored in a sensitivity analysis. The ability of WET-Temp to utilize spatially explicit datasets in estimating temperature distributions across stream networks advances the state of the art in modeling stream temperature. / Graduation date: 2003

The solution of optimal control problems through the Hamilton-Jacobi-Bellman (HJB) equation offers guaranteed satisfaction of both the necessary and sufficient conditions for optimality. However, finding an exact solution to the HJB equation is a near impossible task for many optimal control problems. This thesis presents an approximation method for solving finite-horizon optimal control problems involving nonlinear dynamical systems. The method uses finite-order approximations of the partial derivatives of the cost-to-go function, and successive higher-order differentiations of the HJB equation. Natural byproducts of the proposed method provide sensitivities of the controls to changes in the initial states, which can be used to approximate the solution to neighboring optimal control problems. For highly nonlinear problems, the method is modified to calculate control sensitivities about a nominal trajectory. In this framework, the method is shown to provide accurate control sensitivities at much lower orders of approximation. Several numerical examples are presented to illustrate both applications of the approximation method.

On a brisk fall night in Detroit, after watching the San Francisco Giants celebrate the 2012 World Series championship live, baseball agent, Michael Moye, hailed a cab for the airport – the post season had ended and free agency was underway. Moye, who had years of experience managing players, knew that this off-season would be different from the rest. He was heading to Westlake, Texas, to meet his star client, outfielder Josh Hamilton, who had entered free agency after five years with the Texas Rangers.

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The method of separation of variables facilitates the integration of the Hamilton-Jacobi equation by reducing its solution to a series of quadratures in the separable coordinates. The case in which the metric tensor is diagonal in the separable coordinates, that is, orthogonal separability, is fundamental. Recent theory by Benenti has established a concise geometric (coordinate-independent) characterisation of orthogonal separability of the Hamilton-Jacobi equation on a pseudoRiemannian manifold. It generalises an approach initiated by Eisenhart and developed by Kalnins and Miller. Benenti has shown that the orthogonal separability of a system via a point transformation is equivalent to the existence of a Killing tensor with real simple eigen values and orthogonally integrable eigenvectors. Applying a moving frame formalism, we develop a method that produces the orthogonal separable coordinates for low dimensional Hamiltonian systems. The method is applied to a two dimensional Riemannian manifold of arbitrary curvature. As an illustration, we investigate Euclidean 2-space, and the two dimensional surfaces of constant curvature, recovering known results. Using our formalism, we also derive the known superseparable potentials for Euclidean 2-space. Some of the original results presented in this thesis were announced in [8, 9, 10].

This thesis puts forward the conjecture that for <i>n</i> > 3<i>k</i> with <i>k</i> > 2, the generalized Petersen graph, <i>GP</i>(<i>n,k</i>) is Hamilton-laceable if <i>n</i> is even and <i>k</i> is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case <i>n</i> = 3<i>k</i> + 1 and <i>k</i> greater than or equal to 1. We do this mainly by means of an induction which takes us from <i>GP</i>(3<i>k</i> + 1, <i>k</i>) to <i>GP</i>(3(<i>k</i> + 2) + 1, <i>k</i> + 2). The induction takes the form of mapping a Hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call <i>rotors</i> to obtain a Hamilton path in the larger graph.

This thesis puts forward the conjecture that for <i>n</i> > 3<i>k</i> with <i>k</i> > 2, the generalized Petersen graph, <i>GP</i>(<i>n,k</i>) is Hamilton-laceable if <i>n</i> is even and <i>k</i> is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case <i>n</i> = 3<i>k</i> + 1 and <i>k</i> greater than or equal to 1. We do this mainly by means of an induction which takes us from <i>GP</i>(3<i>k</i> + 1, <i>k</i>) to <i>GP</i>(3(<i>k</i> + 2) + 1, <i>k</i> + 2). The induction takes the form of mapping a Hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call <i>rotors</i> to obtain a Hamilton path in the larger graph.

The method of separation of variables facilitates the integration of the Hamilton-Jacobi equation by reducing its solution to a series of quadratures in the separable coordinates. The case in which the metric tensor is diagonal in the separable coordinates, that is, orthogonal separability, is fundamental. Recent theory by Benenti has established a concise geometric (coordinate-independent) characterisation of orthogonal separability of the Hamilton-Jacobi equation on a pseudoRiemannian manifold. It generalises an approach initiated by Eisenhart and developed by Kalnins and Miller. Benenti has shown that the orthogonal separability of a system via a point transformation is equivalent to the existence of a Killing tensor with real simple eigen values and orthogonally integrable eigenvectors. Applying a moving frame formalism, we develop a method that produces the orthogonal separable coordinates for low dimensional Hamiltonian systems. The method is applied to a two dimensional Riemannian manifold of arbitrary curvature. As an illustration, we investigate Euclidean 2-space, and the two dimensional surfaces of constant curvature, recovering known results. Using our formalism, we also derive the known superseparable potentials for Euclidean 2-space. Some of the original results presented in this thesis were announced in [8, 9, 10].