The topological dynamics of continuous and noncontinuous dynamical systems are investigated. Various definitions of chaos are studied, as well as notions of stability. Results are obtained on asymptotically stable sets and the perturbation stability of such sets. The primary focus is on the traditional point sets of topological dynamics, including the chain recurrent set, omega-limit sets and attractors. The basic setting is that of a continuous function on a compact metric space, sometimes with additional properties on the space. The investigation includes results on the dynamical properties of typical continuous functions in the sense of Baire category. Results are also developed concerning dynamical systems involving quasi-continuous functions. An invariance property for the omega-limit sets of such functions is given. Omega-limit sets are characterized for Riemann integrable derivatives and derivatiyes which are continuous almost everywhere. Techniques used in the investigation and formulation of results include finding theorems which relate the rather disparate notions of dynamical properties and generalized continuity. In addition to dynamical systems, numerous other applications of generalized continuity are imoestigated. Techniques used include application of the Baire Category Theorem and the notion of semi-closure. For example, results are formulated concerning functions determined by dense sets, including separately continuous functions, thus generalizing the classical result for continuous functions on dense subsets of the domain. The uniform boundedness theorem is extended to functions which are not necessarily continuous, including various derivatives. The closed graph theorem is strictly generalized in two separate ways, and applications are presented using these generalizations. An invariance property of separately continuous functions is given. Cluster sets are studied in connection with separate continuity, and various results are presented concerning locally bounded functions.

This paper presents a Lagrangian formulation for studying the dynamics and control of the proposed Space Station based Mobile Servicing System (MSS) for a particular case of in plane libration and maneuvers. The simplified case is purposely considered to help focus on the effects of structural and joint flexibility parameters of the MSS on the complex interactions between the station and manipulator dynamics during slewing and translational maneuvers. The response results suggest that under critical combinations of parameters, the system can become unstable. During maneuvers, the deflection of the MSS can become excessive, leading to positioning error of the payload. At the same time the libration error can also be significant. A linear quadratic regulator is designed to control the deflection of the manipulator and maintain the station at its operating configuration. / Applied Science, Faculty of / Mechanical Engineering, Department of / Graduate

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New upwind kinetic-difference schemes have been developed for flows with nonequilibrium thermodynamics and chemistry. These schemes are derived from the Boltzmann equation with the resulting Euler schemes developed as moments of the discretized Boltzmann scheme with a locally Maxwellian velocity distribution. Application of a directionally-split Courant-Isaacson-Rees (CIR) scheme at the Boltzmann level results in a flux-vector splitting scheme at the Euler level and is called Kinetic Flux-Vector Splitting (KFVS). Extension to flows with finite-rate chemistry and vibrational relaxation is accomplished utilizing non-equilibrium kinetic theory. Computational examples are presented comparing KFVS with the schemes of Van-Leer and Roe for quasi-one-dimensional flow through a supersonic diffuser, inviscid flow through two-dimensional inlet, 'viscous flow over a cone at zero angle-of-attack, and shock-induced combustion/detonation in a premixed hydrogen-air mixture. Calculations are also shown for the transonic flow over a bump in a channel and the transonic flow over an NACA 0012 airfoil. The results show that even though the KFVS scheme is a Riemann solver at the kinetic level, its behavior at the Euler level is more similar to the the existing flux-vector splitting algorithms than to the flux-difference splitting scheme of Roe. A new approach toward the development of a genuinely multi-dimensional Riemann solver is also presented. The scheme is based on the same kinetic theory considerations used in the development of the KF VS scheme. The work has been motivated by the recent progress on multi-dimensional upwind schemes by the groups at the University of Michigan and the Von Karman Institute. These researchers have developed effective upwind schemes for the multi-dimensional linear advection equation using a cell-vertex fluctuation-splitting approach on unstructured grids of triangles or tetrahedra. They have made preliminary applications to the Euler equations using several wave decomposition models of the flux derivative. The issue of the appropriate wave model does not appear to be adequately resolved. The approach taken in the present work is to apply these new multi-dimensional upwind schemes for the scalar advection equation at the Boltzmann level. The resulting Euler schemes are obtained as moments of the fluctuations in the Maxwellian distribution function. The development is significantly more complicated than standard (dimensionally-split) kinetic schemes in that the Boltzmann discretization depends upon the direction of the molecular velocities which must be accounted for in the limits of integration in velocity space. The theoretical issues have been solved through analytic quadrature and Euler schemes have been developed. For this formulation it was not necessary to prescribe any explicit wave decomposition model. Encouraging preliminary results have been obtained for perfect gases on uniform Cartesian meshes with first-order spatial accuracy. Results are presented for a 29° shock reflection, a 45° shear discontinuity, and Mach 3 flow over a step. Finally, methods for obtaining accurate gas-dynamic simulations in the continuum transition regime are considered. In particular, large departures from translational equilibrium are modeled using algorithms based on the Burnett equations instead of the Navier-Stokes equations. Here, the same continuum formulation of the governing equations is retained, but new constitutive relations based on higher-order Chapman-Enskog theory are introduced. Both a rotational relaxation model and a bulk-viscosity model have been considered for simulating rotational non-equilibrium. Results are presented for hypersonic normal shock calculations in argon and diatomic nitrogen and comparisons are made with Direct Simulation Monte Carlo (DSMC) results. The present work closely follows that of the group at Stanford, however, the use of upwind schemes and the bulk-viscosity model represent new contributions. / Ph. D.

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The analysis, finite element approximation, and numerical simulation of some control problems associated with fluid flows are considered. First, we consider a coupled solid/fluid temperature control problem. This optimization problem is motivated by the desire to remove temperature peaks, i.e., "hot spots", along the bounding surface of containers of fluid flows. The heat equation of the solid container is coupled to the energy equation for the fluid. Control is effected by adjustments to the temperature of the fluid at the inflow boundary. We give a precise statement of the mathematical model, prove the existence and uniqueness of optimal solutions, and derive an optimality system. We study a finite element approximation and provide rigorous error estimates for the error in the approximate solution of the optimality system. We then develop and implement an iterative algorithm to compute the approximate solution. Second, a computational study of the feedback control of the magnitude of the lift in flow around a cylinder is presented. The uncontrolled flow exhibits an unsymmetric Karman vortex street and a periodic lift coefficient. The size of the oscillations in the lift is reduced through an active feedback control system. The control used is the injection and suction of fluid through orifices on the cylinder; the amount of fluid injected or sucked is determined, through a simple feedback law, from pressure measurements at stations along the surface of the cylinder. The results of some computational experiments are given; these indicate that the simple feedback law used is effective in reducing the size of the oscillations in the lift. Finally, some boundary value problems which arise from a feedback control problem are considered. We give a precise statement of the mathematical problems and then prove the existence and uniqueness of solutions to the boundary value problems for the Laplace and Stokes equations by studying the boundary integral equation method. / Ph. D.

The vibration control of composite beams and plates subjected to a travelling load is studied in this dissertation. By comparing the controlled as well as uncontrolled responses of classical and refined structural models, the influence of several important composite structure properties which are not included in the classical structural model is revealed. The modal control approach is employed to suppress the structural vibration. In modal control, the control is effected by controlling the modes of the system. The control law is obtained by using the optimal control theory. Comparison of two variants of the modal control approach, the coupled modal control (CMC) and independent modal-space control (IMSC), is made. The results are found to be in agreement with those obtained by previous investigators. The differences between the controlled responses as well as actuator outputs that are predicted by the classical and the refined structural models are outlined in this work. In conclusion, it is found that, when performing the structural analysis and control system design for a composite structure, the classical structural models (such as the Euler-Bernoulli beam and Kirchhoff plate) yield erroneous conclusions concerning the performance of the actual structural system. Furthermore, transverse shear deformation, anisotropy, damping, and the parameters associated with the travelling load are shown to have great influence on the controlled as well as uncontrolled responses of the composite structure. / Ph. D.