The turnpike property in finite-dimensional nonlinear optimal control

Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption, and for very general terminal conditions. Not only the optimal trajectory is shown to remain exponentially close to a steady-state, but also the corresponding adjoint vector of the Pontryagin maximum principle. The exponential closedness is quantified with the use of appropriate normal forms of Riccati equations. We show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method, or a shooting method. In particular, we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector, not at the initial time, but at the middle of the trajectory

Optimal Control of the Thermistor Problem in Three Spatial Dimensions

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Pr\"uss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results

Technology for large space systems: A bibliography with indexes (supplement 10)

The bibliography lists 408 reports, articles and other documents introduced into the NASA scientific and technical information system to provide helpful information to the researcher, manager, and designer in technology development and mission design in the area of large space system technology. Subject matter is grouped according to systems, interactive analysis and design, structural and thermal analysis and design, structural concepts and control systems, electronics, advanced materials, assembly concepts, propulsion, and solar power satellite systems

Solving time-dependent parametric PDEs by multiclass classification-based reduced order model

In this paper, we propose a network model, the multiclass classification-based ROM (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based on this fact, we classify the dataset according to the magnitude of the solutions, and construct corresponding subnets dependent on different types of data. Then we train a classifier to integrate different subnets together to obtain the MC-ROM. When subsets have the same architecture, we can use transfer learning technology to accelerate the offline training. Numerical experiments show that the MC-ROM improves the generalization ability of the DL-ROM both for diffusion- and convection-dominant problems, and maintains the advantage of DL-ROM. We also compare the approximation accuracy and computational efficiency of the proper orthogonal decomposition (POD) which is not suitable for convection-dominant problems. For diffusion-dominant problems, the MC-ROM can save about 100 times online computational cost than the POD with a slightly better approximation in the reduced space of the same dimension.Comment: 19 pages, 15 figure

A Precision Experiment for the Deformation and Fracture of Blast Loaded Plates

The purpose of this thesis is to present a novel laboratory scale precision test [1] apparatus, developed specifically to study of the deformation and fracture of blast loaded plates. A review of published laboratory scale blast testing showed that classical ballistic pendulum techniques satisfy several precision testing criteria. However, specific aspects required improvement. Therefore, an instrumented ballistic pendulum has been developed, incorporating a central support, in the form of a Hopkinson bar, for use with a novel peripherally clamped centrally supported (PCCS) plate specimen, which allows the reaction force history at the inner boundary to be captured. Hence, in addition to the traditional impulse vs deflection data, this approach allows details such as the total deformation duration or the exact timing and magnitude of fracture processes to be captured. Furthermore, when used with a peripherally clamped annular (PCA) plate specimen, this approach allows the blast load pressure history to be captured. A literature review found several classical closed form solutions for the large deflection of impulsively loaded peripherally clamped solid (PCS) circular plates. However, several crucial aspects of the solutions were contradictory and/or incomplete. To address this a generalized energy method for modelling impulsively loaded axisymmetric plates was developed which subsumes previous solutions and facilitated novel analytical solutions that provide a theoretical framework for interpreting the experimental data. Extensive experimental results reported in this thesis provide a rich set of novel data for code validation. While the PCCS and PCA plate configurations require a refined definition of the observed failure modes, the transitions between the modes and the fracture behaviour are studied in great detail. The data is analysed using dimensionless impulse expressions obtained from the analytical solutions in this thesis. In addition to accurate deflection predictions, these solutions also provide improved deflection duration predictions using a novel two phase solutions that can also accommodate finite load duration. It is concluded that the instrumented ballistic pendulum, incorporating a central support in the form of a Hopkinson bar, in conjunction with the novel PCCS and PCA plate configurations and analytical solutions, satisfies all the requirement of a precision test

The term structure of commodity prices : some applications of stochastic calculus

Imperial Users onl

Analysis and design of space vehicle flight control systems. Volume 16 - Abort

Manned spacecraft control system requirements for abort problem

Genesis : the search for origins : the curation and contamination control of returned solar wind samples

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2005.Includes bibliographical references (leaves 61-62).The purpose of the studies carried out in this thesis was to aid in the curation of samples of solar wind returned to earth on the Genesis spacecraft. An experimental study was carried out to aid development of a set of protocols for the laser scribing and subdivision of the Genesis silicon collector array materials. Optimisation of the scribing speed and the positioning of the focal point of the laser were carried out. It was found that scribe width was independent of both factors. Slower scribing speeds were found to produce deeper scribes, while heating effects were minimised with faster speeds. Vertical movement of the stage of 5 pm/pass was found to optimise the focal point of the laser, and minimise heating effects. A procedure to measure the flexural modulus of samples was proposed to quantify the success of the optimisation of the scribing parameters. A theoretical study was carried out to develop a predictive kinetic model for the oxidation of the silicon collector arrays during flight. The mechanism proposed for the increase in oxide thickness over that present pre-flight was the formation of a less-dense suboxide at the SiO₂/Si interface. The driving force is the elevated temperature of the collectors during collection, in the vacuum of space.(cont.) A kinetic model was developed and growth rate expressions derived for two limiting kinetic cases. It was not possible to apply these expressions to the Genesis sample conditions, as the measurement of several experimental parameters was beyond the time limits of this study. A second model was developed alongside literature models to reduce further the number of unknown variables. Finally, the maximum possible thickness of oxide that could grow on the silicon surface was calculated. This was found to be 25 A, representing a 47 % increase over the original 17A of SiO₂ present pre-flight. It was noted that there was a non-linear increase of total oxide thickness with increase in suboxide thickness to due to density differences between Si and SiO, and SiO₂ and SiO.by Benjamin K. Jackson.S.M

Topological and Variational Methods for Differential Equations

These notes contain the extended abstracts of the talks presented at the workshop. The range of topics includes nonlinear Schrödinger equations, singularly perturbed equations, symmetry and nodal properties of solutions, long-time dynamics for parabolic equations, Morse theory