Made available in DSpace on 2016-04-27T16:58:31Z (GMT). No. of bitstreams: 1 Desiree Frasson Balielo Picone.pdf: 820410 bytes, checksum: a1da8465596d29a290d2f4e59e068d05 (MD5) Previous issue date: 2007-10-19 / Secretaria da Educação do Estado de São Paulo / The discipline Calculus is included in the curriculum of many courses, not only in the Exact Sciences but also in other areas, as it involves concepts that permeate various scientific fields. Because of its association with high rates of failure, the teaching and learning of Calculus has been the subject of numerous researches that have sought to propose more effective teaching approaches. Considering the context of the difficulties students face during a Calculus course, and more specifically those related to the teaching and learning of the Fundamental Theory of Calculus (FTC), this work seeks to investigate the representation registers mobilised by teachers in the teaching of this theorem, considers the importance of the coordination of this registers and the ways in which visualisation is explored (or not) by means of graphical representations. The research is based on the theory of Semiotic Representation Registers of Raymond Duval, and emphasises the role of the identification of relevant visual variables, the conversion of the graphical register to the algebraic and vice-versa and the arguments presented in natural language. The study involves the conception and administration of a questionnaire divided into two stages followed by an interview with teachers of Calculus from public and private educational institutions in the state of São Paulo. Data indicated that the teachers consider that in the teaching of the FTC it is important to stress how this theorem can be used as a tool for calculating areas and to establish connections between differentiation and integration, but this connection was not explored graphically by all the teachers. As regards the inter-relationships between relevant visual variables, we verified that the articulation between different registers is not always emphasised by teachers. In general, the teachers considered important the coordination of different representations of the same mathematical object in the teaching of Calculus, with the principle registers used, algebra, graphs and natural language. To analyse a situation which explores the connection between the derivative and the integral graphically, some affirmed that, although they use similar situations, they do not perceive the ways in which these situation can contribute to the understanding of the Thereom. Others, in relation to the same situation, affirmed that they do not make use of this type of activity with their students, and in this case, they offered diverse justifications, none of which suggested the proposals were not important. We believe that the study offers contributions to the teaching and learning of the FTC, but that the results require further study including the amplification of the questionnaire and interviews and their application with different populations of subjects / A disciplina de Cálculo Diferencial e Integral consta na grade curricular de vários cursos da área de Ciências Exatas e também de outras áreas, por tratar de conceitos que permeiam vários campos de Ciência. Seu ensino e aprendizagem tem sido alvo de muitas pesquisas devido aos altos índices de desistência e retenção comprovados, a fim de propor abordagens de ensino que possam amenizar seus problemas existentes. Considerando o contexto das dificuldades enfrentadas num curso de Cálculo e mais precisamente as relacionadas ao ensino e aprendizagem do Teorema Fundamental do Cálculo (TFC), o presente trabalho busca investigar que registros de representação são mobilizados por professores no ensino desse Teorema, bem como se consideram importante a coordenação desses registros e, ainda, se exploram a visualização por meio da representação gráfica. A pesquisa fundamentou-se na teoria dos Registros de Representação Semiótica de Raymond Duval, destacando o papel da identificação das variáveis visuais pertinentes, na conversão do registro gráfico para o algébrico e vice-versa e nas argumentações da língua natural. Para atingir esse objetivo, elaboramos e aplicamos um questionário dividido em duas etapas seguido por uma entrevista com professores de Cálculo de instituições públicas e particulares do Estado de São Paulo. Constatamos que eles consideram importante no ensino do TFC enfatizar que o mesmo pode ser utilizado como uma ferramenta para o cálculo de áreas e que estabelece uma conexão entre derivação e integração, mas essa conexão não é explorada graficamente, por todos. Com relação à inter-relação entre as variáveis visuais pertinentes verificamos que nem sempre foram destacadas pelos professores, na articulação de diferentes registros. Os professores consideram importante a coordenação das diferentes representações do mesmo objeto matemático no ensino do Cálculo de modo geral, sendo os mais utilizados os registros algébrico, gráfico e língua natural. Ao analisarem uma situação que explora a conexão entre a derivada e a integral graficamente, alguns afirmaram que apesar de propor situações parecidas não percebiam de que modo essas situações poderiam contribuir para o entendimento do Teorema. Enquanto outros, ao analisarem a mesma situação, afirmaram que não costumam propor esse tipo de atividade aos seus alunos e, nesse caso, as justificativas foram diversas, porém em nenhum momento apontaram para a não importância de serem propostas. Acreditamos que este estudo apresenta contribuições ao ensino e aprendizagem do TFC, mas julgamos que ele pode ser continuado, quer com a ampliação do questionário e entrevistas, quer com a mudança ou ampliação da amostra de sujeitos da pesquisa

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results.

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results.