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Autonomous underwater vehicles (AUVs) have the potential to dramatically improve safety, quality of life and general scientific knowledge. Our coasts, lakes and rivers are filled with various forms of marine infrastructure including dams, bridges, ship hulls, communication lines, and oil rigs. Each of these structures requires regular inspection, and current methods utilize divers, which is dangerous, expensive, and time consuming. AUVs have the potential to alleviate these difficulties and enable more regular inspection of these structures. Furthermore, there are significant scientific discoveries in the fields of geology, marine biology and medicine that AUV exploration of our oceans will enable. Since field trials of AUVs can be both expensive and high-risk, making a simulation method to generate data for algorithm development is a necessity. For this purpose, we present HoloOcean, an open-source, fully-featured, underwater robotics simulator. Built upon Unreal Engine 4 (UE4), HoloOcean comes equipped with multi-agent communications, common underwater sensors, high-fidelity graphics, and a novel sonar simulation method. Our novel sonar simulation framework is built upon an octree structure, allowing for rapid data generation and flexible usage to simulate a variety of sonars. Further, we have augmented this simulation to incorporate various probabilistic models to account for the heavy noise found in sonar imagery. Simulation enables development of many algorithms such as mapping, localization, structure from motion, controls, and many others. Localization is one essential algorithm for AUV navigation. Recent developments in the utilization of Lie Groups for robotic localization have lead to dramatic performance improvements in convergence and uncertainty characterization. One such method, the Invariant Extended Kalman Filter (InEKF), leverages that invariant error dynamics defined on matrix Lie Groups satisfy a log-linear differential equation. We lay out the various practical decisions required for the InEKF, and show that the primary sensors used in underwater robotics with minor modifications can be used in the InEKF. We show the convergence improvements of the InEKF over the quaternion-based extended Kalman filter (QEKF) on HoloOcean data, both in low and high uncertainty scenarios.

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We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

This thesis presents the formulation and solution of a new problem in systems and control theory, called the Network Realization Problem. Its relationship to other problems, such as State Realization and Structural Identifiability, is shown. The motivation for this work is the desire to completely quantify the conditions for transitioning between different mathematical representations of linear time-invariant systems. The solution to this problem is useful for theorists because it lays a foundation for quantifying the information cost of identifying a system's complete network structure from the transfer function.

In recent years a new interesting research avenue has emerged in non-equilibrium statistical physics, namely studies of collective responses in spatially inhomogeneous systems. Whereas substantial progress has been made in understanding the origins and the often universal nature of cooperative behavior in systems far from equilibrium, it is still unclear whether it is possible to control their global collective stochastic dynamics through local manipulations. Therefore, a comprehensive characterization of spatially inhomogeneous non-equilibrium systems is required. In the first system, we explore a variant of the Katz–Lebowitz–Spohn (KLS) driven lattice gas in two dimensions, where the lattice is split into two regions that are coupled to heat baths with distinct temperatures T > T_c and T_c respectively, where T_c indicates the critical temperature for phase ordering. The geometry was arranged such that the temperature boundaries are oriented perpendicular or parallel to the external particle drive and resulting net current. For perpendicular orientation of the temperature boundaries, in the hotter region, the system behaves like the (totally) asymmetric exclusion processes (TASEP), and experiences particle blockage in front of the interface to the critical region. This blockage is induced by extended particle clusters, growing logarithmically with system size, in the critical region. We observe the density profiles in both high- and low-temperature subsystems to be similar to the well-characterized coexistence and maximal-current phases in (T)ASEP models with open boundary conditions, which are respectively governed by hyperbolic and trigonometric tangent functions. Yet if the lower temperature is set to T_c, we detect marked fluctuation corrections to the mean-field density profiles, e.g., the corresponding critical KLS power-law density decay near the interfaces into the cooler region. For parallel orientation of the temperature boundaries, we have explored the changes in the dynamical behavior of the hybrid KLS model that are induced by our choice of the hopping rates across the temperature boundaries. If these hopping rates at the interfaces satisfy particle-hole symmetry, the current difference across them generates a vector flow diagram akin to an infinite flat vortex sheet. We have studied the finite-size scaling of the particle density fluctuations in both temperature regions, and observed that it is controlled by the respective temperature values. If the colder subsystem is maintained at the KLS critical temperature, while the hotter subsystem's temperature is set much higher, the interface current greatly suppresses particle exchange between the two regions. As a result of the ensuing effective subsystem decoupling, strong fluctuations persist in the critical region, whence the particle density fluctuations scale with the KLS critical exponents. However, if both temperatures are set well above the critical temperature, the particle density fluctuations scale according to the totally asymmetric exclusion process. We have also measured the entropy production rate in both subsystems; it displays intriguing algebraic decay in the critical region, while it saturates quickly at a small but non-zero level in the hotter region. The second system is a lattice gas that simulates the spread of COVID-19 epidemics using the paradigmatic stochastic Susceptible-Infectious-Recovered (SIR) model. In our effort to control the spread of the infection of a lattice, we robustly find that the intensity and spatial spread on the epidemic recurrence wave can be limited to a manageable extent provided release of these restrictions is delayed sufficiently (for a duration of at least thrice the time until the peak of the unmitigated outbreak). / Doctor of Philosophy / In recent years a new interesting research avenue has emerged in far-from-equilibrium statistical physics, namely studies of collective behavior in spatially non-uniform systems. Whereas substantial progress has been made in understanding the origins and the often universal nature of cooperative behavior in systems far from equilibrium, it is still unclear whether it is possible to control their global collective and randomly determined dynamics through local manipulations. Therefore, a comprehensive characterization of spatially non-uniform systems out of equilibrium is required. In the first system, we explore a variant of the two-dimensional lattice gas with completely biased diffusion in one direction and attractive particle interactions. By lattice gas we mean a lattice filled with particles that can hop on nearest-neighbor empty sites. The system we are considering is a lattice that is split into two regions, which in turn are maintained at distinct temperatures T > T_c and T_c, respectively, with T_c indicating the critical temperature for the second-order phase transition. The geometry of the lattice was arranged such that the temperature boundaries are oriented perpendicular or parallel to the external particle drive that is responsible for a completely biased diffusion. When the temperature boundaries are oriented perpendicular to the drive, in the hotter region with temperature T > T_c, the system evolves as if there are no attractive interactions between the particles, and experiences particle blockage in front of the temperature boundary from the hotter region held at T>T_c to the critical region held at T_c. This accumulation of particles at the temperature boundary is induced by elongated collections of particle, i.e., particle clusters in the critical region. We observe the particle density profiles (ρ(x) vs x plots) in both high-and low-temperature subsystems to be similar to the density profiles found for other well-characterized (T)ASEP models with open boundary conditions, which are in the coexistence and maximal-current phases, and which are respectively governed by hyperbolic and trigonometric tangent functions. Yet if the lower temperature is set to T_c, we detect marked corrections to the hyperbolic and trigonometric tangent-like density profiles due to fluctuations, e.g., we observe the algebraic power-law decay of the density near the interfaces into the cooler region with the critical KLS exponent. For a parallel orientation of the temperature boundaries, we have explored the changes in the particle dynamics of the two-temperature KLS model that are induced by our choice of the particle hopping rates across the temperature boundaries. If these particle hopping rates at the temperature interfaces satisfy particle-hole symmetry (i.e. remain unchanged when particles are replaced with holes and vice versa), the particle current difference across them generates a current vector flow diagram akin to an infinite flat vortex sheet. We have studied how the particle density fluctuations in both temperature regions scale with the system size, and observed that the scaling is controlled by the respective temperature values. If the colder subsystem is maintained at the KLS critical temperature T_cold = T_c, while the hotter subsystem's temperature is set much higher T_hot >> T_c, the particle currents at the interface greatly suppresses particle exchange between the two temperature regions. As a result of the ensuing effective subsystem separation from each other, strong fluctuations persist in the critical region, whence the particle density fluctuations scale with the KLS critical exponents. However, if both temperatures are set well above the critical temperature, the particle density fluctuations scale with different scaling exponents, that fall into the totally asymmetric exclusion process (TASEP) universality class. We have also measured the rate of the entropy production in both subsystems; it displays intriguing algebraic decay in the critical region, while it reaches quickly a small but non-zero value in the hotter region. The second system is a lattice filled with particles of different types that hop around the lattice and are subjected to different sorts of reactions. That process simulates the spread of the COVID-19 epidemic using the paradigmatic random-process-based Susceptible-Infectious-Recovered (SIR) model. In our effort to control the spread of the infection of a lattice, we robustly find that the intensity and spatial spread of the epidemic second wave can be limited to a manageable extent provided release of these restrictions is delayed sufficiently (for a duration of at least thrice the time until the peak of the unmitigated outbreak).

<p> In his monograph [Ne], Nekovar studies cohomological invariants of big Galois representations and looks at the variations of Selmer groups attached to intermediate number fields in a commutative p-adic Lie extension. In view of the formulation of the "main conjecture" for noncommutative extensions, it seems natural to extend the theory to a noncommutative p-adic Lie extension. This thesis will serve as a first step in an extension of this theory, namely, we will develop duality theorems over a noncommutative p-adic Lie extension which are extensions of Tate local duality, Poitou-Tate global duality and Grothendieck duality. </p> / Thesis / Doctor of Philosophy (PhD)

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24165号 / 理博第4856号 / 新制||理||1695(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 入谷 寛, 教授 藤原 耕二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Solid Oxide Fuel Cells (SOFCs) are considered suitable for alternative energy solutions due to advantages such as high efficiency, fuel flexibility, tolerance to impurities, and potential for combined cycle operations. One of the main operating constraints of SOFCs is fuel starvation, which can occur under fluctuating power demands. It leads to voltage loss and detrimental effects on cell integrity and longevity. In addition, reformer based SOFCs require sufficient steam for fuel reforming to avoid carbon deposition and catalyst degradation. Steam to carbon ratio (STCR) is an index indicating availability of the steam in the reformer. This work takes a holistic approach to address the aforementioned concerns in SOFCs, in an attempt to enhance applicability and adaptability of such systems. To this end, we revisit prior investigation on the invariant properties of SOFC systems, that led to prediction of fuel utilization U and STCR in the absence of intrusive and expensive sensing. This work provides further insight into the reasons behind certain SOFC variables being invariant with respect to operating conditions. The work extends the idea of invariant properties to different fuel and reformer types. In SOFCs, transient control is essential for U, especially if the fuel cell is to be operated in a dynamic load-following mode at high fuel utilization. In this research, we formulate a generalized abstraction of this transient control problem. We show that a multi-variable systems approach can be adopted to address this issue in both time and frequency domains, which leads to input shaping. Simulations show the effectiveness of the approach through good disturbance rejection. The work further integrates the aforementioned transient control research with system level control design for SOFC systems hybridized with storage elements. As opposed to earlier works where centralized robust controllers were of interest, here, separate controllers for the fuel cell and storage have been the primary emphasis. Thus, the proposed approach acts as a bridge between existing centralized controls for single fuel cells to decentralized control for power networks consisting of multiple elements. As a first attempt, decentralized control is demonstrated in a SOFC ultra-capacitor hybrid system. The challenge of this approach lies in the absence of direct and explicit communication between individual controllers. The controllers are designed based on a simple, yet effective principle of conservation of energy. Simulations as well as experimental results are presented to demonstrate the validity of these designs.