Source Localization Problem for Parabolic Systems

Two problems for the linear distributed parameter systems of parabolic type motivated by environmental monitoring are discussed: (1) Nonlinear localization problem: recover the location of an unknown single source on the basis of available observations. In general, the solution of this problem is set-valued and disconnected. (2) Identifiability problem: what types of observations are able to ensure enough information to restore the location point? An approach is given, based on the introduction of a suitable space of test-functions: in order to determine the unknown location, one has to analyze a proper system of algebraic equations. The latter can be constructed in advance. Sufficient conditions for identifiability are derived and the duality relations between the above nonlinear problems and the problems of open loop control and controllability for an associated adjoint linear system are established

Quadratic Mean Field Games

Mean field games were introduced independently by J-M. Lasry and P-L. Lions, and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new approach to optimization problems with a large number of interacting agents. The description of such models split in two parts, one describing the evolution of the density of players in some parameter space, the other the value of a cost functional each player tries to minimize for himself, anticipating on the rational behavior of the others. Quadratic Mean Field Games form a particular class among these systems, in which the dynamics of each player is governed by a controlled Langevin equation with an associated cost functional quadratic in the control parameter. In such cases, there exists a deep relationship with the non-linear Schr\"odinger equation in imaginary time, connexion which lead to effective approximation schemes as well as a better understanding of the behavior of Mean Field Games. The aim of this paper is to serve as an introduction to Quadratic Mean Field Games and their connexion with the non-linear Schr\"odinger equation, providing to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure

Quasinormal modes of black holes and black branes

Quasinormal modes are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra require solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field

A novel method to design water spray cooling system to protect floating roof atmospheric storage tanks against fires

Hydrocarbon bulk storage tank fires are not very common, but their protection is essential due to severe consequences of such fires. Water spray cooling system is one of the most effective ways to reduce damages to a tank from a fire. Many codes and standards set requirements and recommendations to maximize the efficiency of water spray cooling systems, but these are widely different and still various interpretations and methods are employed to design such systems. This article provides a brief introduction to some possible design methods of cooling systems for protection of storage tanks against external non-contacting fires and introduces a new method namely “Linear Density Method” and compares the results from this method to the “Average Method” which is currently in common practice. The average Method determines the flow rate for each spray nozzle by dividing the total water demand by the number of spray nozzles while the Linear Density Method determines the nozzle flow rate based on the actual flow over the surface to be protected. The configuration of the system includes a one million barrel crude oil floating roof tank to be protected and which is placed one half tank diameter from a similar adjacent tank with a full surface fire. Thermal radiation and hydraulics are modeled using DNV PHAST Version 6.53 and Sunrise PIPENET Version 1.5.0.2722 software respectively. Spray nozzles used in design are manufactured by Angus Fire and PNR Nozzles companies. Schedule 40 carbon steel pipe is used for piping. The results show that the cooling system using the Linear Density Method consumes 3.55% more water than the design using the average method assuming a uniform application rate of 4.1 liters per minute. Despite higher water consumption the design based on Linear Density Method alleviates the problems associated with the Average Method and provides better protection

C-SIDE: The control-structure interaction demonstration experiment

The Control-Structure Interaction Demonstration Experiment (C-SIDE) is sponsored by the Electro-Optics and Cryogenics Division of Ball Aerospace Systems Group. Our objective is to demonstrate methods of solution to structure control problems utilizing currently available hardware in a system that is an extension of our corporate experience. The larger space structures with which Ball has been associated are the SEASAT radar antenna, Shuttle Imaging Radar (SIR) -A, -B and -C antennas and the Radarsat spacecraft. The motivation for the C-SIDE configuration is to show that integration of active figure control in the radar's system-level design can relieve antenna mechanical design constraints. This presentation is primarily an introduction to the C-SIDE testbed. Its physical and functional layouts, and major components are described. The sensor is of special interest as it enables direct surface figure measurements from a remote location. The Remote Attitude Measurement System (RAMS) makes high-rate, unobtrusive measurements of many locations, several of which may be collocated easily with actuators. The control processor is a 386/25 executing a reduced order model-based algorithm with provision for residual mode filters to compensate for structure interaction. The actuators for the ground demonstration are non-contacting, linear force devices. Results presented illustrate some basic characteristics of control-structure interaction with this hardware. The testbed will be used for evaluation of current technologies and for research in several areas. A brief indication of the evolution of the C-SIDE is given at the conclusion

Analysis of 2nd order differential equations : applications to chaos synchronisation and control

In this thesis a number of open problems in the theory of ordinary differen-tial equations (ODEs) and dynamical systems are considered. The intention being to address current problems in the theory of systems control and synchronisation as well as enhance the understanding of the dynamics of those systems treated herein. More specifically, we address three central problems; the determination of exact analytical solutions of (non)linear (in)homogeneous ODEs of order 1 and 2, the de¬termination of upper/lower bounds on solutions of nonlinear ODEs and finally, the synchronisation of dynamical systems for the purposes of secure communication. With regard to the first of these problems we identify a new solvable class of Riccati equations and show that the solution may be written in closed-form. Fol¬lowing this we show how the Riccati equation solution leads us quite naturally to the identification of a new solvable class of 2nd order linear ODEs, as well as a yet more general class of Riccati equations. In addition, we demonstrate a new alter¬native method to Lagrange's variation of parameters for the solution of 2nd order linear inhomogeneous ODEs. The advantage of our approach being that a choice of solution methods is offered thereby allowing the solver to pick the simplest op¬tion. Furthermore, we solve, by means of variable transforms and identification of the first integral, an example of the Duffing-van der Pol oscillator and an associated ODE that connects the equations of Lienard and Riccati. These fundamental results are subsequently applied to the problem of solving the ODE describing a lengthen Abstract ii ing pendulum and the matter of bounded controller design for linear time-varying systems. In addressing the second of the above problems we generalise an existing GrOnwall-like integral inequality to yield several new such inequalities. Using one of the new inequalities we show that a certain class of nonlinear ODEs will always have bounded solutions and subsequently demonstrate how one can numerically evaluate the upper limits on the square of the solution of any given ODE in this class. Finally, we apply our results to an academic example and verify our conclu¬sions with numerical simulation. The third and final open problem we consider herein is concerned with the synchronisation of chaotic dynamical systems with the express intention of exploit¬ing that synchronisation for the purposes of secure transmission of information. The particular issue that we concern ourselves with is the matter of limiting the amount of distortion present in the message arriving at the receiver. Since the distortion encountered is primarily a due to the presence of noise and the message itself we meet our ends by employing an observer-based synchronisation technique incorpo¬rating a proportional-integral observer. We show how the PI observer used gives us the freedom to reduce message distortion without compromising on synchronisa¬tion quality and rate. We verify our results by applying the method to synchronise two parameter-matched Duffing oscillators operating in a chaotic regime. Simula tions clearly show the enhanced performance of the proposed method over the more traditional proportional observer-based approach under the same conditions. The structure of thesis is as follows: first of all we describe the motivation be¬hind object of study before going on to give a general introduction to the theory of ODEs and dynamical systems. This lead-in also includes a brief history of the the¬ory ODEs and dynamical systems, a general overview of the subject (as wholly as is possible without getting into the mathematical detail that is left to the appendices) and concludes with a statement of the scope of the thesis as well as the contribu¬tions to knowledge contained herein. We then go on to state and prove our main results and contributions to the solution of those problems detailed above starting with the solution of ODEs ..

Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Curriculum Guidelines for Undergraduate Programs in Data Science

The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in Data Science. The group consisted of 25 undergraduate faculty from a variety of institutions in the U.S., primarily from the disciplines of mathematics, statistics and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in Data Science