An optimal control problem governed by the heat equation with nonlinear boundary conditions is considered. The objective functional consists of a quadratic terminal part and a quadratic regularization term. It is known, that an SQP method converges quadratically to the optimal solution of the problem. To handle the quadratic optimal control subproblems with high precision, very large scale mathematical programming problems have to be treated. The constrained problem is reduced to an unconstrained one by a method due to Bertsekas. A multigrid approach developed by Hackbusch is applied to solve the unconstrained problems. Some numerical examples illustrate the behaviour of the method.

Low-frequency, long-range sound propagation over a seasurface has been calculated using a wide-angel Cranck-NicholsonParabolic Equation method. The model is developed toinvestigate noise from off-shore wind turbines. Thecalculations are made using normal meteorological conditions ofthe Baltic Sea. Special consideration has been made to a windphenomenon called low level jet with strong winds on rather lowaltitude. The effects of water waves on sound propagation have beenincorporated in the ground boundary condition using a bossmodel. This way of including roughness in sound propagationmodels is valid for water wave heights that are small comparedto the wave length of the sound. Nevertheless, since only lowfrequency sound is considered, waves up to the mean wave heightof the Baltic Sea can be included in this manner. The calculation model has been tested against benchmarkcases and agrees well with measurements. The calculations showthat channelling of sound occurs at downwind conditions andthat the sound propagation tends towards cylindrical spreading.The effects of the water waves are found to be fairlysmall. Keywords:wind turbine noise, off-shore wind power,long-range sound propagation, parabolic equation, scattering,water waves / QC 20110617

The field of mathematical modelling for pedestrian dynamics has attracted significant scientific attention, with various models proposed from perspectives such as kinetic theory, statistical mechanics, game theory and partial differential equations. Often such investigations are seen as being a part of a new branch of study in the domain of applied physics, called sociophysics. Our study proposes three models that are tailored to specific scenarios of crowd dynamics. Our research focuses on two primary issues. The first issue is centred around pedestrians navigating through a partially dark corridor that impedes visibility, requiring the calculation of the time taken for evacuation using a Markov chain model. The second issue is posed to analyse how pedestrians move through a T-shaped junction. Such a scenario is motivated by the 2022 crowd-crush disaster took place in the Itaewon district of Seoul, Korea. We propose a lattice-gas-type model that simulates pedestrians’ movement through the grid by obeying a set of rules as well as a parabolic equation with special boundary conditions. By the means of numerical simulations, we investigate a couple of evacuation scenarios by evaluating the mean velocity of pedestrians through the dark corridor, varying both the length of the obscure region and the amount of uncertainty induced by the darkness. Additionally, we propose an agent-based-modelling and cellular automata inspired model that simulates the movement of pedestrians through a T-shaped grid, varying the initial number of pedestrians. We measure the final density and time taken to reach a steady pedestrian traffic state. Finally, we propose a parabolic equation with special boundary conditions that mimic the dynamic of the pedestrian populations in a T-junction. We solve the parabolic equation using a random walk numerical scheme and compare it with a finite difference approximation. Furthermore, we prove rigorously the convergence of the random walk scheme to a corresponding finite difference scheme approximation of the solution.

Radars are used for range estimation of distant objects. They operate on the principle of sending electromagnetic pulses that are reflected off a target. This leads to the propagation of electromagnetic waves over large distances. As the waves propagate, they are affected by several aspects that decrease the performance of the radar system. In this master thesis, we derive a mathematical model that describes electromagnetic propagation in the troposphere. The model developed is based on a parabolic equation and uses the split-step Fourier method for its numerical solution. Using the model, we estimate the influence of a varying, complex, refractive index of the atmosphere, different lossy materials in the ground, terrain, and oceans. The terrain is described using a piecewise linear shift map method. The modelling of the ocean is done using a novel model which is a combination of terrain for large swells and Miller surface roughness for smaller waves, both based on a Pierson-Moskowitz sea spectrum. The model is validated and found to agree very well, with results found in the literature.

A class of Lagrange-Newton-SQP methods is investigated for optimal control problems governed by semilinear parabolic initial- boundary value problems. Distributed and boundary controls are given, restricted by pointwise upper and lower bounds. The convergence of the method is discussed in appropriate Banach spaces. Based on a weak second order sufficient optimality condition for the reference solution, local quadratic convergence is proved. The proof is based on the theory of Newton methods for generalized equations in Banach spaces.

Multipath remains a dominant error source in Global Positioning System (GPS) applications that require high accuracy. With the use of differential techniques it is possible to remove many of the common-mode error sources, but the error effects of multipath have proven much more difficult to mitigate. The research aim of this work is to enhance the understanding of multipath propagation and its effects in GPS terrestrial applications, through the modelling of signal propagation behaviour and the resultant error effects. Multipath propagation occurs when environmental features cause combinations of reflected and/or diffracted replica signals to arrive at the receiving antenna. These signals, in combination with the original line-of-sight (LOS) signal, can cause distortion of the receiver correlation function and ultimately the discrimination function and hence errors in range estimation. To date, a completely satisfactory mitigation strategy has yet to be developed. In the search for such a mitigation strategy, it is imperative that a comprehensive understanding of the multipath propagation environment and the resultant error effects exists. The work presented here, provides a comprehensive understanding through the use of new modelling and simulation techniques specific to GPS multipath. This dissertation unites the existing theory of radio frequency propagation for the GPS L1 signal into a coherent treatment of GPS propagation in the terrestrial environment. To further enhance the understanding of the multipath propagation environment and the resultant error effects, this dissertation also describes the design and development of a new parabolic equation (PE) based propagation model for analysis of GPS multipath propagation behaviour. The propagation model improves on previous PE-based models by incorporating terrain features, including boundary impedance properties, backscatter and time-domain decomposition of the field into a multipath impulse response. The results provide visualisation as well as the defining parameters necessary to fully describe the multipath propagation behaviour. These resultant parameters provide the input for a correlation and discrimination model for visualisation and the generation of resultant receiver error measurements. Results for a variety of propagation environments are presented and the technique is shown to provide a deterministic methodology against real GPS data. The unique and novel combined modelling of multipath propagation and reception, presented in this dissertation, provides an effective set of tools that have enhanced the understanding of the behaviour and effect of multipath in GPS applications, and ultimately should aid in providing a solution to the GPS multipath mitigation problem.

Localisation, which is a mechanism for discovering the spatial relationship between objects, is an area that has received considerable research and development in recent times. A common name given to localisation operations based on the absolute reference frame of Earth is Geolocation. One important example of geolocation research is E-911, where wireless carriers in the United States must provide the location of 911 callers. The operation of E-911 can be based on either a network configuration, or the Global Positioning System (GPS). With the importance of localisation being acknowledged, a review concerning the vulnerability of the Global Navigation Satellite System (GNSS) is provided as background and motivation for this research. With the current vulnerability of GNSS, this dissertation presents the results of a research program undertaken with the objective of developing an electromagnetic localisation technique that can determine the relative position of GPS Radio Frequency Interference (RFI) sources. Intended for operation in a hostile environment, blind and passive localisation methodologies must be incorporated into the developed model. In performing localisation research, a background of current techniques is provided in addition to a review of current electromagnetic propagation models. From the review of propagation models, the Parabolic Equation Model (PEM) was chosen for investigation concerning localisation. The selection of PEM is due to model properties that are required for blind/passive localisation. The localisation system developed in this research program is based on the integration of inverse diffraction propagation (IDP) within the parabolic equation model. The title chosen for the localisation method is Inverse Diffraction Parabolic Equation Localisation System (IDPELS). This thesis presents the simulation and field trial results of IDPELS. Under simulation, the terrain or obstacle profiles were not based on any geodetic datum. Any estimate provided by IDPELS under simulation is therefore a "Localisation" solution. In the field trials however, IDPELS operation is referred to as "Geolocation" as geodetic datum's where used to determine the receiver's position. Under simulation analysis, IDPELS operation was considered to provide good promise as it could simultaneously perform localisation on multiple transmission sources. In each investigated simulation scenario, a display of signals amplitude (dB units) is displayed over the entire region. By determining the field convergence regions, a localisation estimate of IDPELS is provided. By defining the convergence regions as areas having the greatest signal amplitude values (i.e. ≥ 99%), elliptical areas as low as 3.2m² were considered to indicate an excellent localisation capability. With the theoretical validity of IDPELS operation in electromagnetics having been established under simulation, further investigation into the practical feasibility of the IDPELS was performed. The field trials positioned a continuous-wave (CW) transmission source at a known location. By measuring signal phasors along a straight section of road, the geodetic spatial-phase profile was used as the input signal for IDPELS. Road sections used were cross-wise to the transmitter's boresight. Many data sets were recorded, each being made over a sixty second time period. Different regions and ranges where used to continuously measure the spatial-phase profile of the signal with fixed antennas in a moving vehicle. Such a measurement process introduced an analogy with Synthetic Aperture Radar (SAR) processes. In quantitating the accuracy of the IDPELS geolocation estimate in field trials, the linear error of range and cross-range components was analysed. A free-space PEM model was chosen for development of IDPELS and hence, data sets demonstrating properties of a free-space environment were able to be considered suitable for testing of the geolocation method. Data sets demonstrating free-space propagation characteristics were measured at the base of the Mt Lofty ranges in South Australia, where the range and cross-range error are respectively 3.14m, and 0.15m. Such low error values clearly demonstrate the practical feasibility of IDPELS geolocation. With the practical feasibility of IDPELS having been established in this research program, a novel contribution to electromagnetic geolocation methodologies is provided. An important characteristic of any geolocation technique concerns its robustness to operate in a wide variety of possible environments. With continued development of IDPELS, the robustness of this passive/blind geolocation technique can be enhanced. Further assistance with geolocation of multiple transmission sources is also indicated to be available by IDPELS, as shown in the simulation analysis.

[pt] Os efeitos da difração na propagação de ondas de rádio sobre terreno irregular em VHF e outras bandas a ser usado por futuras aplicações da TV digital são normalmente estimados usando um dos muitos modelos clássicos. Nesta dissertação é feita uma comparação dos erros cometidos na previsão do sinal recebido por três modelos de propagação computacionalmente intensivos. Os resultados da presente comparação indicarão se os esforços computacionais envolvidos na aplicação destes métodos são capazes de diminuir o valor médio e desvio padrão das diferenças entre as medidas e predições determinadas pelos métodos clássicos. / [en] Diffraction effects on the propagation of radio wave over irregular terrain in the VHF and other bands to be used by future digital TV applications are normally estimated using one of many classical models. In this dissertation is made a comparison of the errors committed prediction of signal received by three propagation models computationally-intensive. The results of the present comparison will indicate whether the computational efforts involved on the application of these methods are capable of decreasing the mean value and the standard deviation of the difference between measurements and predictions determined by the classical methods.

Étant donné l'urgence environnementale, le développement des énergies renouvelables s'est fortement accru ces dernières années. L'implantation de champs d'éoliennes est notamment en pleine expansion dans toute l'Europe. Ces éoliennes, de structure diélectrique et métallique et de grande taille, peuvent avoir un impact significatif sur les systèmes radiofréquences. En particulier, les systèmes de radionavigation et de surveillance opérés par la DGAC (VOR, radar) doivent cohabiter avec de nouveaux champs d'éoliennes. En effet, ces dernières influent sur le champ électromagnétique des systèmes et peuvent dégrader leurs performances (multitrajets, masquages... ). Il est alors important de pouvoir quantifier ces dégradations, en particulier dans le cas du VOR où les multitrajets engendrent une erreur de relèvement. Dans ce travail de thèse, un simulateur électromagnétique appelé VERSO (VOR ERror SimulatOr) est développé. Il permet d'estimer l'impact d'objets diffractants, en particulier d'éoliennes, sur le signal VOR. Dans la littérature, différentes techniques de modélisation sont proposées pour prédire ces phénomènes. Certaines sont trop approximatives, d'autres trop coûteuses en temps. Ainsi, le choix des techniques utilisées dans ce simulateur a été guidé par le compromis entre précision et temps de calcul. L'équation parabolique est utilisée pour modéliser la propagation de la source jusqu'aux éoliennes afin de prendre en compte le relief. Ensuite, une méthode d'optique physique sur matériaux diélectriques est mise en œuvre pour calculer le champ diffracté par ces objets. Le modèle électromagnétique de l'éolienne et les hypothèses inhérentes aux méthodes utilisées par VERSO ont été validés aux fréquences VHF (VOR) par comparaison avec la méthode des moments qui fait office de référence. Une extension de VERSO pour les systèmes radars a été abordée. Par conséquent des validations similaires à ces fréquences ont été réalisées. Plus spécifiquement, un modèle de pale d'éolienne prenant en compte la présence du parafoudre est proposé aux fréquences VOR et radar. Les effets de masquage sont eux aussi quantifiés pour ces 2 domaines de fréquences. Il est notamment démontré que l'effet de masquage pour des éoliennes alignées radialement à un VOR est négligeable. Cette approximation ainsi que le modèle de pale sont ensuite utilisés dans le programme VERSO. Ce dernier est validé à l'aide de contrôles en vol sur un scénario de 9 éoliennes implantées à 5 km du VOR de Boulogne-sur-Mer. Une étude quantitative de l'impact de chaque partie des éoliennes est menée afin de discriminer la source majoritaire d'erreur VOR. On constate notamment qu'à grande distance du VOR dans le scénario d'observation considéré, le mât constitue le principal contributeur en terme de champ diffracté et d'erreur VOR. Enfin, une étude statistique sur l'erreur VOR a permis d'obtenir un simulateur de scénarios qui donne l'erreur maximale avec une confiance fixée, en minimisant le nombre de simulations à effectuer. Pour accélérer cette méthode, une expression analytique approchée de l'erreur VOR maximum a été développée en fonction de la distance d'implantation et de la hauteur du mât de l'éolienne. Cette dernière étude fournit une méthode rapide pour évaluer l'impact de la construction d'un champ d'éoliennes quelconque à proximité de systèmes de l'aviation civile. / Considering the ecological emergency, the renewable energy development has greatly increased for a decade. In particular, the windfarms implantation rapidly expands in Europe. These windturbines are large obstacles composed by dielectric and metallic materials. So their impact on electromagnetic devices is significant. The radionavigation systems for the civil aviation services are particularly concerned. However, they have to work side with new windfarms. Actually, these latter cause scattering effects on the electromagnetic signals and can degrade the performances of these equipments (multipaths, shadowing effects etc.). Thus, quantifying these degradations is crucial, particularly on the VOR devices where multipath effects cause an error on the azimuth. In this thesis work, an electromagnetic simulator called VERSO (VOR ERror SimulatOr) is developed. It can estimate the impact of scattering objects, especially windturbines, on the VOR signal. In literature, several techniques are proposed to model these phenomena: some make coarse approximations and some others are memory intensive. Thus, the choice over the methods used in VERSO is a compromise between precision and memory requirement. The parabolic equation is used to model the propagation from the source to the windturbines so as to take the relief into account. A physical optic based method is used to compute the field scattered by these objects. The electromagnetic model of the windturbine and the hypothesis due to the methods used by VERSO have been validated in the VHF (VOR) frequency by comparison with the method of moments, which is the reference. An extension of VERSO for the radar systems is introduced. Consequently similar validations have been performed at radar frequencies. In particular, a windturbine blade model taking into account the lightning protection is proposed for the VOR and the radar frequencies. The shadowing effects are also quantified in both frequency domains. Especially, a demonstration that the shadowing effects due to radially implemented windturbines can be neglected around a VOR beacon is proposed. This approximation and the blade model are used for the implementation of VERSO. This simulator is validated by comparison with measurements on 9 windturbines built 5~km far from a VORC in Boulogne-sur-Mer (France). A study is performed to quantify the influence of each part of the windturbine. The mast is shown to be the main contributor regarding to the electromagnetic field and the VOR error. Finally, parametric simulations are performed and analytic expressions are proposed to describe the evolution of the maximum VOR error with respect to the mast size and the distance VOR-windturbine. The latter study gives some key parameters that need to be considered for the elaboration of a windfarm building plan close to civil aviation systems for the project to be viable.

Cette thèse est dédiée a l''etude de certaines propriétés des équations d' évolutions non-autonomes $u'(t)+A(t)u(t)=f(t), u(0)=x.$ Il s'agit précisément de la propriété de la régularité maximale $L^p$: étant donnée $fin L^{p}(0,tau;H)$, montrer l'existence et unicité de la solution $u in W^{1,p}(0,tau;H)$. Ce problème a 'et'e intensivement étudie dans le cas autonome, i.e., $A(t)=A$ pour tout $t$. Dans le cas non-autonome, le problème a été considéré par J.L.Lions en 1960. Nous montrons divers résultats qui étendent tout ce qui est connu sur ce problème. On suppose ici que la famille des opérateurs $(mathcal{A}(t))_{tin [0,tau]}$ est associée à des formes quasi-coercives, non autonomes $(fra(t))_{t in [0,tau]}.$ Nous considérons également le problème de régularité maximale pour les d'ordre 2 (équations des ondes). Plusieurs exemples et applications sont considérés. / This Thesis is devoted to certain properties of non-autonomous evolution equations $u'(t)+A(t)u(t)=f(t), u(0)=x.$ More precisely, we are interested in the maximal $L^p$-regularity: given $fin L^{p}(0,tau;H),$ prove existence and uniqueness of the solution $u in W^{1,p}(0,tau;H)$. This problem was intensively studied in the autonomous cas, i.e., $A(t)=A$ for all $t.$ In the non-autonomous cas, the problem was considered by J.L.Lions in 1960. We prove serval results which extend all previously known ones on this problem. Here we assume that the familly of the operators $(mathcal{A}(t))_{tin [0,tau]}$ is associated with quasi-coercive, non-autonomous forms $(fra(t))_{t in [0,tau]}.$ We also consider the problem of maximal regularity for second order equations (the wave equation). Serval examples and applications are given in this Thesis.