391 research outputs found

    Solving inverse source problems using observability. Applications to the Euler-Bernoulli plate equation

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    International audienceThe aim of this paper is to provide a general framework allowing to use exact observability of infinite dimensional systems to solve a class of inverse source problems. More precisely, we show that if a system is exactly observable, then we can identify a source term in this system by knowing the corresponding intensity and appropriate observations which often correspond to the measure of some boundary traces. This abstract theory is then applied to a system governed by the Euler-Bernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation

    Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler-Bernoulli beam and heat equations

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    We are concerned with the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. We establish stability estimates of logarithmic type when the measurements are given by the operator who maps the initial condition to Neumann boundary trace of the solution of the corresponding initial-boundary value problem. We build a method combining an observability inequality together with a spectral decomposition. We also apply this method to a clamped Euler-Bernoulli beam equation. Finally, we indicate how the present approach can be adapted to a heat equation

    Recovering Coefficients of Second-Order Hyperbolic and Plate Equations via Finite Measurements on the Boundary

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    Abstract In this dissertation, we consider the inverse problem for a second-order hyperbolic equation of recovering n + 3 unknown coefficients defined on an open bounded domain with a smooth enough boundary. We also consider the inverse problem of recovering an unknown coefficient on the Euler- Bernoulli plate equation on a lower-order term again defined on an open bounded domain with a smooth enough boundary. For the second-order hyperbolic equation, we show that we can uniquely and (Lipschitz) stably recover all these coefficients from only using half of the corresponding boundary measurements of their solutions, and for the plate equation, we show that we can uniquely and stably recover the coefficient by using two measurements on the boundary. The proofs for solving both inverse problems are based on a post-Carleman estimate strategy developed by Isakov in [19], continuous observability inequalities, and regularity theory

    Determining the potential in a wave equation without a geometric condition. Extension to the heat equation

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    International audienceWe prove a logarithmic stability estimate for the inverse problem of determining the potential in a wave equation from boundary measurements obtained by varying the first component of the initial condition. The novelty of the present work is that no geometric condition is imposed to the sub-boundary where the measurements are made. Our results improve those obtained by the first and second authors in [2]. We also show how the analysis for the wave equation can be adapted to an inverse coefficient problem for the heat equatio

    Uniqueness in the determination of loads in multi-span beams and plates

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    Most of the results available on the inverse problem of determining loads acting on elastic beams or plates under transverse vibration refer to single beam or single plate. In this paper, we consider the determination of sources in multi-span systems obtained by connecting either two Euler-Bernoulli elastic beams or two rectangular Kirchhoff-Love elastic plates. The material of the structure is assumed to be homogeneous and isotropic. The transverse load is of the form g(t)f(x), where g(t) is a known function of time and f(x) is the unknown term depending on the position variable x. Under slight a priori assumptions, we prove a uniqueness result for f(x) in terms of observations of the dynamic response taken at interior points of the structure in an arbitrary small interval of time. A numerical implementation of the method is included to show the possible application of the results in the practical identification of the source term

    Estimation of Spacecraft Attitude Motion and Vibrational Modes Using Simultaneous Dual-Latitude Ground-Based Data

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    Cutting-edge Space Situational Awareness (SSA) research calls for improved methods for rapidly characterizing resident space objects. In this thesis, this will take the form of speeding up convergence of spacecraft attitude estimates, and of a non-model-based approach to the detection of vibrational modes. Because attitude observability from photometric data is angle-based, dual-site simultaneous photometric observations of a resident space object are predicted to improve the convergence speed and steady-state error of spacecraft attitude state estimation from ground-based sensor data. Additionally, it is predicted that by adding polarimetric data to the measurements, the speed of convergence and steady-state error will be reduced further. This thesis models satellite motion and measurements from ground-based sensors for dual-latitude simultaneous light curve simulation, then develops a data fusion process to combine photometric, astrometric, and polarimetric data from both sites in order to more quickly estimate the attitude of an RSO. The Fractional Fourier Transform shows promise as a non-model-based approach to the detection of input vibrational frequencies from the degree of linear polarization. The main results are that dual-site observation geometry is conducive to slight improvements of attitude filter performance, and the addition of polarimetric data to the measurements yields much improved performance over both the single-site and dual-site cases

    The taut string approach to statistical inverse problems: theory and applications

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    A novel solution approach to a class of nonlinear statistical inverse problems with finitely many observations collected over a compact interval on the real line blurred by Gaussian white noise of arbitrary intensity is presented. Exploiting the nonparametric taut string estimator, we prove the state recovery strategy is convergent to a solution of the unnoisy problem at the rate of n−1/2n^{−1/2} as the number of observations n grows to infinity. Illustrations of the method\u27s application to real-world examples from hydrology, civil & electrical engineering are given andan empirical study on the robustness of our approach is presented

    Modelling, system identification and control of a fibre optic accelerometer

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    A research report submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in partial ful lment of the requirements for the degree of Master of Science in Engineering. Johannesburg, 2015Control of systems are important in most industrial sectors, they nd applications in electronics, machine design and navigation. These control systems often use sensors to work e ectively. One such sensor is an accelerometer, which is used to measure acceleration with one or more degrees of freedom. This research report investigates the modelling, system identi cation and controller design for an accelerometer, a Fibre Optic Accelerometer (FOA). Such a device may be applied in many applications such as anti-skid control, structural failure in buildings and bridges, as well as strategic missile guidance. This report presents a model of a FOA demonstrator which crudely models an industrially developed accelerometer, the demonstrator is made of a jig consisting of a guitar string and electromagnets. Such a model needs to account for a distributed parameter beam combined with a permanent magnet and four electromagnets. The guitar string is modelled using three beam models, namely a spring/damper model, an Assumed Modes Model (ASM) and a Transfer Function Model (TFM). The parameters for these beam models are identi ed using the Nelder-Mead simplex algorithm and the least squares method. The electromagnets within the jig, are modelled using a mathematical model obtained through curve tting of experimental data. The overall FOA sensor is optimised using a lead-lag controller. Five cost functions where investigated, these cost functions are H1, Integral Square Error (ISE), Integral Absolute Error (IAE), Integral Time Square Error (ITSE) and Integral Absolute Time Error (IATE). It was found that the guitar string may be modelled using a single degree of freedom beam model. This is based on a number of reasons, such as the aperture size - through which the tip Light Emitting Diode (LED) projects, the tip mass (permanent magnet) - acting as a natural damper and the fact that Position Sensing Device (PSD) only measures the tip position. It was found that a single degree of freedom model in two orthogonal axes, with a single link beam spring/damper model was the most suitable representation of the guitar string. And the IAE lead-lag controller was found to be the most e ective in controlling a guitar string, this e ectiveness was due to least settling time.MT201
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