Orientador: Geraldo Nunes Silva / Banca: Vilma Alves de Oliveira / Banca: Masayoshi Tsuchida / Resumo: Estamos interessados em estudar uma generalização do Princípio do Máximo de Pontryagin para problema de controle ótimo com restrições mistas envolvendo funções nãodiferenciáveis, pois este princípio não se aplica para todos os tipos de problemas. O principal objetivo deste trabalho é apresentar as condições necessárias de otimalidade na forma do princípio do máximo que serão aplicadas para o problema de controle ótimo com restrições mistas envolvendo funções não-diferenciáveis. Para alcançar este objetivo apresentamos estudos sobre cones normais e cones tangentes os quais são utilizados no desenvolvimento da teoria de subdiferenciais. Após esse embasamento formulamos o problema de controle ótimo envolvendo funções não-diferenciáveis, e apresentamos as condições necessárias de otimalidade. / Abstract: We are interested in study a generalization of the Pontryagin Maximum Principle for optimal control problems with mixed constraints involving nondi erentiable functions, because this principle can not be applied for all the types of problems. The main objective of this work is to present the necessary conditions of optimality in the form of the maximum principle that will be applied for the optimal control problem with mixed constraints involving nondi erentiable functions. To achieve this objective we present studies above normal cones and tangent cones which are used in the development of the theory of subdi erentials. After this foundation we formulate the optimal control problem involving nondi erentiable functions, and we present the necessary conditions of optimality. / Mestre

Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. This dissertation investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid methods to reduce accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. Parabolic initial-boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution. For those problems, specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution. This dissertation investigates characteristics of the θ-method and introduces a variable-θ method as a synergistic combination of the Crank-Nicolson (CN) method and the implicit method. It suppresses spurious oscillations, by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data. An effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations (the wobble set); the resulting variable-θ method is analyzed for its accuracy and stability. After a theory of morphogenesis in chemical cells was introduced in 1950s, much attention had been devoted to the numerical solution of reaction-diffusion (RD) equations. This dissertation studies a nonoscillatory second-order time-stepping procedure for RD equations incorporating with variable-θ method, as a perturbation of the CN method. We also perform a sensitivity analysis for the numerical solution of RD systems to conclude that it is much more sensitive to the spatial mesh resolution than the temporal one. Moreover, to enhance the spatial approximation of RD equations, this dissertation investigates the averaging scheme, that is, an interpolation of the standard and skewed discrete Laplacian operator and introduce the simple optimizing strategy to minimize the leading truncation error of the scheme.

The objective of this dissertation is to present a modeling and simulation methodology for MEMS devices and identify and understand the associated nonlinearities due to large deflections, electric actuation, impacts, and friction. In the first part of the dissertation, we introduce a reduced-order model of flexible microplates under electric excitation. The model utilizes the von Karman plate equations to account for geometric nonlinearities due to large plate deflections. The Galerkin approach is employed to reduce the partial-differential equations of motion and associated boundary conditions into a finite dimensional system of nonlinearly coupled ordinary-differential equations. We use the reduced-order model to analyze the mechanical behavior of a simply supported microplate and a fully clamped microplate. Effect of various design parameters on both the static and dynamic characteristics of microplates is studied. The second part of the dissertation presents comprehensive modeling and simulation tools for impact microactuators. Nonsmooth dynamics due to impacts and friction are studied, combining various approaches, including direct numerical integration, root-finding technique for periodic motions, continuation of grazing periodic orbits, and local analysis of the near grazing dynamics. The transition between nonimpacting and impacting long term motions, referred to as grazing bifurcations, indicates the transition between on and off states of an impact microactuator. Three different on-off switching mechanisms are identified for the Mita microactuator. These mechanisms also generalize to arbitrary impacting systems with a similar nonlinearity. A local map based on the concept of discontinuity mapping provides an effcient and accurate tool for the grazing bifurcation analysis. Nonlinear impacting dynamics of the microactuator are studied in detail to identify various bifurcations and parameter ranges corresponding to chaotic motions. We find that the frequency-response curves of the impacting dynamics are significantly different from those of the nonimpacting dynamics. / Ph. D.

Many real-world problems can be modeled by generalized equations. The solution of the latter can be a challenging task, and typically requires the use of some efficient numerical procedures, whose convergence analysis often relies on stability properties of a solution in question, and on a suitable over-estimate for the distance of a given point to the solution set of the problem, called error bound. With this thesis, we aim at a unified approach to first- and second-order conditions for stability properties and error bounds for generalized equations. To this end, we study existing and develop new concepts for generalized first-order derivatives of set-valued mappings, and use them to formulate criteria for Lipschitzian stability properties and Lipschitzian error bound conditions. These criteria can all be regarded as the property that a suitable generalized least singular value of a generalized derivative is nonzero. By considering generalized least singular values as an extended real-valued function that depends on arguments of an underlying mapping, we will be able to obtain second-order conditions arising from generalized derivatives of this function to guarantee non-Lipschitzian stability properties and non-Lipschitzian error bound conditions. This allows us to extend the territory covered by some seminal monographs dealing with stability properties and error bounds for generalized equations under first-order conditions. Furthermore, we discuss some specializations of our findings, and work out relations to existing results. Finally, we also investigate correlations between stability properties and error bounds with respect to different problem-formulations of one and the same generalized equation.

Discontinuous dynamical systems and multiagent systems are encountered in numerous engineering applications. This dissertation develops stability and dissipativity of nonlinear dynamical systems with discontinuous right-hand sides, optimality of discontinuous feed-back controllers for Filippov dynamical systems, almost consensus protocols for multiagent systems with innaccurate sensor measurements, and adaptive estimation algorithms using multiagent network identifiers. In particular, we present stability results for discontinuous dynamical systems using nonsmooth Lyapunov theory. Then, we develop a constructive feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function de fined in the sense of generalized Clarke gradients and set-valued Lie derivatives. Furthermore, we develop dissipativity notions and extended Kalman-Yakubovich-Popov conditions and apply these results to develop feedback interconnection stability results for discontinuous systems. In addition, we derive guaranteed gain, sector, and disk margins for nonlinear optimal and inverse optimal discontinuous feedback regulators that minimize a nonlinear-nonquadratic performance functional for Filippov dynamical systems. Then, we provide connections between dissipativity and optimality of nonlinear discontinuous controllers for Filippov dynamical systems. Furthermore, we address the consensus problem for a group of agent robots with uncertain interagent measurement data, and show that the agents reach an almost consensus state and converge to a set centered at the centroid of agents initial locations. Finally, we develop an adaptive estimation framework predicated on multiagent network identifiers with undirected and directed graph topologies that identifies the system state and plant parameters online.

In this thesis, we study the Filippov moments solution for differential equations with discontinuous right-hand side. In particular, our aim is to define a suitable Filippov sliding vector field on a co-dimension $2$ manifold $\Sigma$, intersection of two co-dimension $1$ manifolds with linearly independent normals, and then to study the dynamics provided by this selection. More specifically, we devote Chapter 1 to motivate our interest in this subject, presenting several problems from control theory, non-smooth dynamics, vehicle motion and neural networks. We then introduce the co-dimension $1$ case and basic notations, from which we set up, in the most general context, our specific problem. In Chapter 2 we propose and compare several approaches in selecting a Filippov sliding vector field for the particular case of $\Sigma$ nodally attractive: amongst these proposals, in Chapter 3 we focus on what we called \emph{moments solution}, that is the main and novel mathematical object presented and studied in this thesis. There, we extend the validity of the moments solution to $\Sigma$ attractive under general sliding conditions, proving interesting results about the smoothness of the Filippov sliding vector field on $\Sigma$, tangential exit at first-order exit points, uniqueness at potential exit points among all other admissible solutions. In Chapter 4 we propose a completely new and different perspective from which one can look at the problem: we study minimum variation solutions for Filippov sliding vector fields in $\R^{3}$, taking advantage of the relatively easy form of the Euler-Lagrange equation provided by the analysis, and of the orbital equivalence that we have in the eventuality $\Sigma$ does not have any equilibrium points on it; we further remove this assumption and extend our results. In Chapter 5 examples and numerical implementations are given, with which we corroborate our theoretical results and show that selecting a Filippov sliding vector field on $\Sigma$ without the required properties of smoothness and exit at first-order exit points ends up dynamics that make no sense, developing undesirable singularities. Finally, Chapter 6 presents an extension of the moments method to co-dimension $3$ and higher: this is the first result which provides a unique admissible solution for this problem.

En optimisation les conditions d’optimalité jouent un rôle primordial pour détecter les solutions optimales et leur étude occupe une place significative dans la recherche actuelle. Afin d’exprimer adéquatement des conditions d’optimalité les chercheurs ont introduit diverses notions de dérivées généralisées non seulement pour des fonctions non lisses, mais aussi pour des fonctions à valeurs ensemblistes, dites applications multivoques ou multifonctions. Cette thèse porte sur l’application des deux nouveaux concepts de dérivées généralisées: les ensembles variationnels de Khanh-Tuan et les approximations de Jourani-Thibault, aux problèmes d’optimisation multiobjectif et aux problèmes d’équilibre vectoriel. L’enjeu principal est d’obtenir des conditions d’optimalité du premier et du second ordre pour les problèmes ayant des données multivoques ou univoques non lisses et pas forcément continues, et des conditions assurant l’unicité des solutions dans les problèmes d’équilibre vectoriel. / Optimality conditions for nonsmooth optimization have become one of the most important topics in the study of optimization-related problems. Various notions of generalized derivatives have been introduced to establish optimality conditions. Besides establishing optimality conditions, generalized derivatives also is an important tool for studying the local uniqueness of solutions. During the last three decades, these topics have been being developed, generalized and applied to many elds of mathematics by many authors all over the world. The purpose of this thesis is to investigate the above topics. It consists of ve chapters. In Chapter 1, we develop elements of calculus of variational sets for set-valued mappings, which were recently introduced in Khanh and Tuan (2008). Most of the usual calculus rules, from chain and sum rules to rules for unions, intersections, products and other operations on mappings, are established. As applications we provide a direct employment of sum rules to establishing an explicit formula for a variational set of the solution map to a parametrized variational inequality in terms of variational sets of the data. Furthermore, chain rules and sum or product rules are also used to prove optimality conditions for weak solutions of some vector optimization problems. In Chapter 2, we propose notions of higher-order outer and inner radial derivatives of set-valued maps and obtain main calculus rules. Some direct applications of these rules in proving optimality conditions for particular optimization problems are provided. Then, we establish higher-order optimality necessary conditions and sufficient ones for a general set-valued vector optimization problem with inequality constraints. Chapter 3 is devoted to using first and second-order approximations, which were introduced by Jourani and Thibault (1993) and Allali and Amaroq (1997), as generalized derivatives, to establish both necessary and sufficient optimality conditions for various kinds of solutions to nonsmooth vector equilibrium problems with functional constraints. Our rst-order conditions are shown to be applicable in many cases, where existing ones cannot be applied. The second-order conditions are new. In Chapter 4, we consider nonsmooth multi-objective fractional programming on normed spaces. Using rst and second-order approximations as generalized derivatives, rst and second-order optimality conditions are established. For sufficient conditions no convexity is needed. Our results can be applied even in innite dimensional cases involving innitely discontinuousmaps. In Chapter 5, we establish sufficient conditions for the local uniqueness of solutions to nonsmooth strong and weak vector equilibrium problems. Also by using approximations, our results are valid even in cases where the maps involved in the problems suffer innite discontinuity at the considered point.

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Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-09-12Bitstream added on 2014-06-13T19:47:37Z : No. of bitstreams: 1 izelli_rc_me_sjrp.pdf: 916240 bytes, checksum: 24bbf9996f6955ca38766b92b37822c8 (MD5) / Estamos interessados em estudar uma generalização do Princípio do Máximo de Pontryagin para problema de controle ótimo com restrições mistas envolvendo funções nãodiferenciáveis, pois este princípio não se aplica para todos os tipos de problemas. O principal objetivo deste trabalho é apresentar as condições necessárias de otimalidade na forma do princípio do máximo que serão aplicadas para o problema de controle ótimo com restrições mistas envolvendo funções não-diferenciáveis. Para alcançar este objetivo apresentamos estudos sobre cones normais e cones tangentes os quais são utilizados no desenvolvimento da teoria de subdiferenciais. Após esse embasamento formulamos o problema de controle ótimo envolvendo funções não-diferenciáveis, e apresentamos as condições necessárias de otimalidade. / We are interested in study a generalization of the Pontryagin Maximum Principle for optimal control problems with mixed constraints involving nondi erentiable functions, because this principle can not be applied for all the types of problems. The main objective of this work is to present the necessary conditions of optimality in the form of the maximum principle that will be applied for the optimal control problem with mixed constraints involving nondi erentiable functions. To achieve this objective we present studies above normal cones and tangent cones which are used in the development of the theory of subdi erentials. After this foundation we formulate the optimal control problem involving nondi erentiable functions, and we present the necessary conditions of optimality.