143 research outputs found
Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
We are interested in evolution phenomena on star-like networks composed of
several branches which vary considerably in physical properties. The initial
boundary value problem for singularly perturbed hyperbolic differential
equation on a metric graph is studied. The hyperbolic equation becomes
degenerate on a part of the graph as a small parameter goes to zero. In
addition, the rates of degeneration may differ in different edges of the graph.
Using the boundary layer method the complete asymptotic expansions of solutions
are constructed and justified.Comment: 18 pages, 3 figure
Applications of Singular Perturbation Techniques to Control Problems
Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-79-C-0424National Science Foundation / NSF ECS 82-1763
Singular Perturbations and Time-Scale Methods in Control Theory: Survey 1976-1982
Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-79-C-0424U.S. Air Force / AFOSR 78-363
Optimal trajectory tracking
This thesis investigates optimal trajectory tracking of nonlinear dynamical
systems with affine controls. The control task is to enforce the system state
to follow a prescribed desired trajectory as closely as possible. The concept
of so-called exactly realizable trajectories is proposed. For exactly
realizable desired trajectories exists a control signal which enforces the
state to exactly follow the desired trajectory. For a given affine control
system, these trajectories are characterized by the so-called constraint
equation. This approach does not only yield an explicit expression for the
control signal in terms of the desired trajectory, but also identifies a
particularly simple class of nonlinear control systems. Based on that insight,
the regularization parameter is used as the small parameter for a perturbation
expansion. This results in a reinterpretation of affine optimal control
problems with small regularization term as singularly perturbed differential
equations. The small parameter originates from the formulation of the control
problem and does not involve simplifying assumptions about the system dynamics.
Combining this approach with the linearizing assumption, approximate and partly
linear equations for the optimal trajectory tracking of arbitrary desired
trajectories are derived. For vanishing regularization parameter, the state
trajectory becomes discontinuous and the control signal diverges. On the other
hand, the analytical treatment becomes exact and the solutions are exclusively
governed by linear differential equations. Thus, the possibility of linear
structures underlying nonlinear optimal control is revealed. This fact enables
the derivation of exact analytical solutions to an entire class of nonlinear
trajectory tracking problems with affine controls. This class comprises
mechanical control systems in one spatial dimension and the FitzHugh-Nagumo
model.Comment: 240 pages, 36 figures, PhD thesi
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Dynamical problems and phase transitions
Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68
Numerical Treatment of Non-Linear singular pertubation problems
Magister Scientiae - MScThis thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.South Afric
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
São Carlos, Brasil. 3-7 february 2014
Multimodeling, Singular Perturbations and Chained Aggregation of Large Scale Systems
Coordinated Science Laboratory was formerly known as Control Systems LaboratoryDepartment of Energy / US ERDA EX-76-C-01-208
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