Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with advection terms and spatially-varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H^1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.Comment: Submitted to IEEE Transactions on Automatic Contro

State estimation for coupled PDE systems using Modulation Functions

This master thesis is devoted to the state estimation of a particular form of PDE systems, coupled parabolic PDEs with spatial dependent coefficients. This form of PDEs represent some dynamic systems such as Tubular Reactors, Diffusion in lithium-ion cells and Diffusive Gradient in Thin Films sensor. Other methods for this problem use "Backstepping" observers, in which the estimation error system is transformed into another system that is stable, reducing the problem to calculate the Kernel functions making the transformation possible. In some cases this calculation is not simple, also the simulation in real time of the observer system, that is also a PDE, can be difficult. The method presented in this thesis uses the properties of the so-called Modulating Functions in order to estimate the states. The procedure con- sists of generating an orthonormal basis of functions that can represent the state as a combination of them. Then auxiliary systems are formed from the original systems with boundary conditions that help in the simplification of the problem. Resolving these auxiliary systems, result in the calculation of the Modulating kernels. All of these steps can be made offline and do not have to be repeated. The functions are used together with the orthonormal basis in the online part, that consists of an inte- gration of a combination of the kernel functions, inputs and outputs of the system in a time window. Finally, with a matrix multiplication the coefficients for the ba- sis expansion of the state can be obtained, resulting in the desired state estimation. The present method is tested in systems that resemble the forms of the dynamics of Tubular Reactors and the performance is compared to other methods.Diese Masterarbeit widmet sich der Zustandsschätzung einer bestimmten Art von Systemen, gekoppelten partiellen Differenzialgleichungen mit raumabhängigen Ko- effizienten. Diese besondere Form von PDEs repräsentiert einige dynamische Sys- teme wie Röhrenreaktoren, Diffusion in Lithium-Ionen-Batterien und Gradienten in dünnen Schichten. Andere Methoden für dieses Problem benutzen "Backstep- ping" Beobachter, bei denen das Schätzfehlersystem in ein anderes stabiles System transformiert wird, wodurch das Problem reduziert wird, um die Kernfunktionen zu berechnen, die die Transformation ermöglichen. In manchen Fällen ist diese Berech- nung nicht einfach. Auch die Simulation in Echtzeit des Beobachters System, das auch eine PDE ist, kann sehr schwierig sein. Die in dieser Arbeit vorgestellte Meth- ode verwendet die Eigenschaften der sogenannten Modulationsfunktionen, um die Zustände zu schätzen. Das Verfahren besteht darin, eine Orthonormalbasis von Funktionen zu erzeugen können, die den Zustand als Kombination von ihnen repräsen- tieren, dann werden Hilfssysteme gebildet von dem ursprünglichen Systemen mit Randbedingungen, die bei der Vereinfachung helfen, von dem Problem. Das Au- flösen dieser Hilfssysteme ergibt die Berechnung der modulierende Kerne. Alle diese Schritte können offline durchgeführt und müssen nicht wiederholt werden. Die Funktionen werden zusammen mit der Orthonormalbasis im Online-Teil ver- wendet. Dieser Teil besteht aus einer Integration einer Kombination der Kernfunk- tionen, Eingaben und Ausgaben des Systems in einem Zeitfenster. Schließlich kön- nen die Koeffizienten zur Basiserweiterung mit einer Matrixmultiplikation berech- net werden, was zu der gewünschte Zustandsschätzung führt. Das Verfahren wird am Beispiel der Dynamik eines Rohreaktors getestet und die Ergebnisse werden mit anderen Methoden verglichen

Parameter domains for Turing and stationary flow-distributed waves: I. The influence of nonlinearity

new type of instability in coupled reaction-diffusion-advection systems is analysed in a one-dimensional domain. This instability, arising due to the combined action of flow and diffusion, creates spatially periodic stationary waves termed flow and diffusion-distributed structures (FDS). Here we show, via linear stability analysis, that FDS are predicted in a considerably wider domain and are more robust (in the parameter domain) than the classical Turing instability patterns. FDS also represent a natural extension of the recently discovered flow-distributed oscillations (FDO). Nonlinear bifurcation analysis and numerical simulations in one-dimensional spatial domains show that FDS also have much richer solution behaviour than Turing structures. In the framework presented here Turing structures can be viewed as a particular instance of FDS. We conclude that FDS should be more easily obtainable in chemical systems than Turing (and FDO) structures and that they may play a potentially important role in biological pattern formation

Hydrodynamic dispersion within porous biofilms

Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behavior by controlling nutrient supply, evacuation of waste products, and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilm-scale in the case where the width of the channels is significantly smaller than the thickness of the biofilm. We show that solute transport may be described via two coupled partial differential equations or telegrapher's equations for the averaged concentrations. These models are particularly relevant for chemicals, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterized by a second-order tensor whose components depend on (1) the topology of the channels' network; (2) the solute's diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion dominated, this analysis shows that hydrodynamic dispersion effects may significantly contribute to transport

State estimation for coupled reaction-diffusion PDE systems using modulating functions

Many systems with distributed dynamics are described by partial differential equations (PDEs). Coupled reaction-diffusion equations are a particular type of these systems. The measurement of the state over the entire spatial domain is usually required for their control. However, it is often impossible to obtain full state information with physical sensors only. For this problem, observers are developed to estimate the state based on boundary measurements. The method presented applies the so-called modulating function method, relying on an orthonormal function basis representation. Auxiliary systems are generated from the original system by applying modulating functions and formulating annihilation conditions. It is extended by a decoupling matrix step. The calculated kernels are utilized for modulating the input and output signals over a receding time window to obtain the coefficients for the basis expansion for the desired state estimation. The developed algorithm and its real-time functionality are verified via simulation of an example system related to the dynamics of chemical tubular reactors and compared to the conventional backstepping observer. The method achieves a successful state reconstruction of the system while mitigating white noise induced by the sensor. Ultimately, the modulating function approach represents a solution for the distributed state estimation problem without solving a PDE online

Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes

Spatial variation in population densities across a landscape is a feature of many ecological systems, from self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of populations. However the ways in which abiotic and biotic factors interact to determine the existence and nature of spatial patterns in population density remain poorly understood. Here we present a new approach to studying this question by analysing a predator–prey patch-model in a heterogenous landscape. We use analytical and numerical methods originally developed for studying nearest- neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a rich and highly complex array of coexisting stable patterns, located within an enormous number of unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable basins of attraction, making them significant in applications. We are able to identify mechanisms for these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby landscape heterogeneity can modulate the spatial scales at which these processes operate to structure the populations

Turing conditions for pattern forming systems on evolving manifolds

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach