Thesis (PhD)--Montana State University--Bozeman, 2009. / Typescript. Chairperson, Graduate Committee: Tomas Gedeon. Includes bibliographical references (leaves 234-237).

A multiple-valued function is a "function" that assumes two or more distinct values in its range for at least one point in its domain. While these "functions" are not functions in the normal sense of being single-valued, the usage is so common that there is no way to dislodge it. This thesis is devoted to a particular class of multiple-valued functions: Q-valued functions. A Q-valued function is essentially a rule assigning Q unordered and not necessarily distinct points of R^n to each element of R^m. This object is one of the key ingredients of Almgren's 1700 pages proof that the singular set of an m-dimensional mass minimizing integral current in R^n has dimension at most m-2. We start by developing a decomposition theory and show for instance when a continuous Q-valued function can or cannot be seen as Q "glued" continuous classical functions. Then, the decomposition theory is used to prove intrinsically a Rademacher type theorem for Lipschitz Q-valued functions. A couple of Lipschitz extension theorems are also obtained for partially defined Lipschitz Q-valued functions. The second part is devoted to a Peano type result for a particular class of nonconvex-valued differential inclusions. To the best of the author's knowledge this is the first theorem, in the nonconvex case, where the existence of a continuously differentiable solution is proved under a mere continuity assumption on the corresponding multifunction. An application to a particular class of nonlinear differential equations is included. The third part is devoted to the calculus of variations in the multiple-valued framework. We define two different notions of Dirichlet nearly minimizing Q-valued functions, generalizing Dirichlet energy minimizers studied by Almgren. Hölder regularity is obtained for these nearly minimizers and we give some examples showing that the branching phenomena can be much worse in this context.

This thesis addressed non-smooth dynamics of lumped parameter systems, and was restricted to Filippov-type systems. The main objective of this thesis was twofold. Firstly, modeling aspects of Filippov-type non-smooth dynamical systems were addressed with an emphasis on the constitutive assumptions and mathematical structure behind these models. Secondly, robust algorithms were presented to obtain numerical solutions for various Filippov-type lumped parameter systems. Governing equations were written using two different mathematical approaches. The first approach was based on differential inclusions and the second approach was based on differential-algebraic equations. The differential inclusions approach is more amenable to mathematical analysis using existing mathematical tools. On the other hand, the approach based on differential-algebraic equations gives more insight into the constitutive assumptions of a chosen model and easier to obtain numerical solutions. Bingham-type models in which the force cannot be expressed in terms of kinematic variables but the kinematic variables can be expressed in terms of force were considered. Further, Coulomb friction was considered in which neither the force can be expressed in terms of kinematic variables nor the kinematic variables in terms of force. However, one can write implicit constitutive equations in terms of kinematic quantities and force. A numerical framework was set up to study such systems and the algorithm was devised. Towards the end, representative dynamical systems of practical significance were considered. The devised algorithm was implemented on these systems and the results were obtained. The results show that the setting offered by differential-algebraic equations is appropriate for studying dynamics of lumped parameter systems under implicit constitutive models.

L'objet de ce travail est l’étude des propriétés de stabilité et de robustesse des systèmes non-linéaires via des méthodes basées sur l'homogénéité. Dans un premier temps, nous rappelons le contexte usuel des systèmes homogènes ainsi que leurs caractéristiques principales. La suite du travail porte sur l'extension de l'homogénéisation des systèmes non-linéaires, déjà proposée dans le cadre de l'homogénéité à poids, au cadre plus général de l'homogénéité géométrique. Les principaux résultats d'approximation sont étendus. Nous développons ensuite un cadre théorique pour définir l'homogénéité de systèmes discontinus et/ou donnés par des inclusions différentielles. Nous montrons que les propriétés bien connues des systèmes homogènes restent vérifiées dans ce contexte. Ce travail se poursuit par l'étude de la robustesse des systèmes homogènes ou homogénéisables. Nous montrons que sous des hypothèses peu restrictives, ces systèmes sont input-to-state stable. Enfin, la dernière partie de ce travail consiste en l'étude du cas particulier du double intégrateur. Nous développons pour ce système un retour de sortie qui le stabilise en temps fini, et pour lequel nous prouvons des propriétés de robustesse par rapport à des perturbations ou à la discrétisation en exploitant les résultats développés précédemment. Des simulations viennent compléter l'étude théorique de ce système et illustrer son comportement / The purpose of this work is the study of stability and robustness properties of nonlinear systems using homogeneity-based methods. Firstly, we recall the usual context of homogeneous systems as well as their main features. The sequel of this work extends the homogenization of nonlinear systems, which was already defined in the framework of weighted homogeneity, to the more general setting of the geometric homogeneity. The main approximation results are extended. Then we develop a theoretical framework for defining homogeneity of discontinuous systems and/or systems given by a differential inclusion. We show that the well-known properties of homogeneous systems persist in this context. This work is continued by a study of the robustness properties of homogeneous or homogenizable systems. We show that under mild assumptions, these systems are input-to-state stable. Finally, the last part of this work consists in the study of the example of the double integrator system. We synthesize a finite-time stabilizing output feedback, which is shown to be robust with respect to perturbations or discretization by using techniques developed before. Simulations conclude the theoretical study of this system and illustrate its behavior

Made available in DSpace on 2014-07-29T16:02:20Z (GMT). No. of bitstreams: 1 ALEX NERI GUTIERREZ DISSERTACAO.pdf: 1234288 bytes, checksum: ae65a58b7c2fd793b3c15d44001d82d6 (MD5) Previous issue date: 2012-03-23 / This work has the main objective in the context of the fuzzy theory. Averaging method, differential inclusions are studied; finally this context of the fuzzy theory. / O trabalho tem como objetivo principal, o estudo de um método de averaging em problemas de valor inicial no contexto fuzzy. Com o intuito de facilitar a compreensão do trabalho, faz-se um estudo do, um método de averaging no contexto determinístico, teoria de inclusões diferencias, teoria dos conjuntos fuzzy, inclusões diferenciais fuzzy e finalmente mostra-se o um resultado da validade do método de averaging no contexto fuzzy.

We study a class of differential inclusions involving maximal monotone relations, which cover a huge class of problems in mathematical physics. For this purpose we introduce the time derivative as a continuously invertible operator in a suitable Hilbert space. It turns out that this realization is a strictly monotone operator and thus, the question on existence and uniqueness can be answered by well-known results in the theory of maximal monotone relations. Furthermore, we show that the resulting solution operator is Lipschitz-continuous and causal, which is a natural property of evolutionary processes. Finally, the results are applied to a system of partial differential equations and inclusions, which describes the diffusion of a compressible fluid through a saturated, porous, plastically deforming media, where certain hysteresis phenomena are modeled by maximal montone relations.

Network utility maximization has emerged as a powerful tool in studying flow control, resource allocation and other cross-layer optimization problems. In this work, we study a flow control problem in the optimization framework. The objective is to maximize the sum utility of the users subject to the flow constraints of the network. The utility maximization is solved in a distributed setting; the network operator does not know the user utility functions and the users know neither the rate choices of other users nor the flow constraints of the network. We build upon a popular decomposition technique proposed by Kelly [Eur. Trans. Telecommun., 8(1), 1997] to solve the utility maximization problem in the aforementioned distributed setting. The technique decomposes the utility maximization problem into a user problem, solved by each user and a network problem solved by the network. We propose an iterative algorithm based on this decomposition technique. In each iteration, the users communicate to the network their willingness to pay for the network resources. The network allocates rates in a proportionally fair manner based on the prices communicated by the users. The new feature of the proposed algorithm is that the rates allocated by the network remains feasible at all times. We show that the iterates put out by the algorithm asymptotically tracks a differential inclusion. We also show that the solution to the differential inclusion converges to the system optimal point via Lyapunov theory. We use a popular benchmark algorithm due to Kelly et al. [J. of the Oper. Res. Soc., 49(3), 1998] that involves fast user updates coupled with slow network updates in the form of additive increase and multiplicative decrease of the user flows. The proposed algorithm may be viewed as one with fast user update and fast network update that keeps the iterates feasible at all times. Simulations suggest that our proposed algorithm converges faster than the aforementioned benchmark algorithm. When the flows originate or terminate at a single node, the network problem is the maximization of a so-called d-separable objective function over the bases of a polymatroid. The solution is the lexicographically optimal base of the polymatroid. We map the problem of finding the lexicographically optimal base of a polymatroid to the geometrical problem of finding the concave cover of a set of points on a two-dimensional plane. We also describe an algorithm that finds the concave cover in linear time. Next, we consider the minimization of a more general objective function, i.e., a separable convex function, over the bases of a polymatroid with a special structure. We propose a novel decomposition algorithm and show the proof of correctness and optimality of the algorithm via the theory of polymatroids. Further, motivated by the need to handle piece-wise linear concave utility functions, we extend the decomposition algorithm to handle the case when the separable convex functions are not continuously differentiable or not strictly convex. We then provide a proof of its correctness and optimality.

We study a class of differential inclusions involving maximal monotone relations, which cover a huge class of problems in mathematical physics. For this purpose we introduce the time derivative as a continuously invertible operator in a suitable Hilbert space. It turns out that this realization is a strictly monotone operator and thus, the question on existence and uniqueness can be answered by well-known results in the theory of maximal monotone relations. Furthermore, we show that the resulting solution operator is Lipschitz-continuous and causal, which is a natural property of evolutionary processes. Finally, the results are applied to a system of partial differential equations and inclusions, which describes the diffusion of a compressible fluid through a saturated, porous, plastically deforming media, where certain hysteresis phenomena are modeled by maximal montone relations.