Initial Value Problem in General Relativity

This article, written to appear as a chapter in "The Springer Handbook of Spacetime", is a review of the initial value problem for Einstein's gravitational field theory in general relativity. Designed to be accessible to graduate students who have taken a first course in general relativity, the article first discusses how to reformulate the spacetime fields and spacetime covariant field equations of Einstein's theory in terms of fields and field equations compatible with a 3+1 foliation of spacetime with spacelike hypersurfaces. It proceeds to discuss the arguments which show that the initial value problem for Einstein's theory is well-posed, in the sense that for any given set of initial data satisfying the Einstein constraint equations, there is a (maximal) spacetime solution of the full set of Einstein equations, compatible with the given set of data. The article then describes how to generate initial data sets which satisfy the Einstein constraints, using the conformal (and conformal thin sandwich) method, and using gluing techniques. The article concludes with comments regarding stability and long term behavior of solutions of Einstein's equations generated via the initial value problem.Comment: To appear as a chapter in "The Springer Handbook of Spacetime," edited by A. Ashtekar and V. Petkov. (Springer-Verlag, at Press

Optimal modelling and experimentation for the improved sustainability of microfluidic chemical technology design

Optimization of the dynamics and control of chemical processes holds the promise of improved sustainability for chemical technology by minimizing resource wastage. Anecdotally, chemical plant may be substantially over designed, say by 35-50%, due to designers taking account of uncertainties by providing greater flexibility. Once the plant is commissioned, techniques of nonlinear dynamics analysis can be used by process systems engineers to recoup some of this overdesign by optimization of the plant operation through tighter control. At the design stage, coupling the experimentation with data assimilation into the model, whilst using the partially informed, semi-empirical model to predict from parametric sensitivity studies which experiments to run should optimally improve the model. This approach has been demonstrated for optimal experimentation, but limited to a differential algebraic model of the process. Typically, such models for online monitoring have been limited to low dimensions. Recently it has been demonstrated that inverse methods such as data assimilation can be applied to PDE systems with algebraic constraints, a substantially more complicated parameter estimation using finite element multiphysics modelling. Parametric sensitivity can be used from such semi-empirical models to predict the optimum placement of sensors to be used to collect data that optimally informs the model for a microfluidic sensor system. This coupled optimum modelling and experiment procedure is ambitious in the scale of the modelling problem, as well as in the scale of the application - a microfluidic device. In general, microfluidic devices are sufficiently easy to fabricate, control, and monitor that they form an ideal platform for developing high dimensional spatio-temporal models for simultaneously coupling with experimentation. As chemical microreactors already promise low raw materials wastage through tight control of reagent contacting, improved design techniques should be able to augment optimal control systems to achieve very low resource wastage. In this paper, we discuss how the paradigm for optimal modelling and experimentation should be developed and foreshadow the exploitation of this methodology for the development of chemical microreactors and microfluidic sensors for online monitoring of chemical processes. Improvement in both of these areas bodes to improve the sustainability of chemical processes through innovative technology. (C) 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved

Feedback control of the acoustic pressure in ultrasonic wave propagation

Classical models for the propagation of ultrasound waves are the Westervelt equation, the Kuznetsov and the Khokhlov-Zabolotskaya-Kuznetsov equations. The Jordan-Moore-Gibson-Thompson equation is a prominent example of a Partial Differential Equation (PDE) model which describes the acoustic velocity potential in ultrasound wave propagation, where the paradox of infinite speed of propagation of thermal signals is eliminated; the use of the constitutive Cattaneo law for the heat flux, in place of the Fourier law, accounts for its being of third order in time. Aiming at the understanding of the fully quasilinear PDE, a great deal of attention has been recently devoted to its linearization -- referred to in the literature as the Moore-Gibson-Thompson equation -- whose mathematical analysis is also of independent interest, posing already several questions and challenges. In this work we consider and solve a quadratic control problem associated with the linear equation, formulated consistently with the goal of keeping the acoustic pressure close to a reference pressure during ultrasound excitation, as required in medical and industrial applications. While optimal control problems with smooth controls have been considered in the recent literature, we aim at relying on controls which are just $L^2$ in time; this leads to a singular control problem and to non-standard Riccati equations. In spite of the unfavourable combination of the semigroup describing the free dynamics that is not analytic, with the challenging pattern displayed by the dynamics subject to boundary control, a feedback synthesis of the optimal control as well as well-posedness of operator Riccati equations are established.Comment: 39 pages; submitte