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In this thesis we study the space L = L(M,N) of all Lagrangian mappings of a fixed closed surface or 3-manifold M to respectively another surface or 3-manifold N. In most cases we are assuming both M and N oriented. We are looking for local (order 1) invariants of such generic maps, that is, for those whose increments along generic paths in L are completely determined by diffeomorphism types of the local bifurcations of the caustics in N. Such invariants are dual to trivial cycles supported on the discriminantal hypersurface ? in L. The duality here is in the sense of the increment of an invariant along a generic path γ in L is the index of intersection of γ with the cycle, and the triviality means that if γ is a loop then its index of intersection with the cycle must vanish. For surfaces, we obtain a complete description of the spaces of discriminantal cycles, possibly non-trivial. For N = R² and the subset of maps in L without corank 2 singularities, this description implies that any rational local invariant itself is a linear combinations of the numbers of various singular points of the caustics and of the Ohmoto-Aicardi linking invariant of ordinary maps between surfaces. Using our discriminantal cycles, we also prove Ohmoto's conjecture about non-contractability of a certain loop in L(S²,R²). Our surface results are now published in [16]. For oriented 3-manifolds, we prove that the space of all rational local invariants is ten-dimensional and spanned by the numbers of various isolated-type singularities of the caustics and the Euler characteristic of the critical point set. We also show that the rank of the space of the mod2 invariants has dimension 16. The results of the thesis are based on our study of generic one- and two-parameter families of caustics. In our 3-dimensional constructions, we had to analyse generic projections of the D6 and E6 caustics to the plane. Nothing anyhow close to this rather delicate analysis has been done before, and it occupies nearly half the thesis.

The aim of this thesis is threefold. Firstly, we study the stochastic evolution equations (driven by an infinite dimensional cylindrical Wiener process) in a class of Banach spaces satisfying the so-called H-condition. In particular, we deal with the questions of the existence and uniqueness of solutions for such stochastic evolution equations. Moreover, we analyse the Markov property of the solution. Secondly, we apply the abstract results obtained in the first part to the so-called Heath-Jarrow-Morton-Musiela (HJMM) equation. In particular, we prove the existence and uniqueness of solutions to the HJMM equation in a large class of function spaces, such as the weighted Lebesgue and Sobolev spaces. Thirdly, we study the ergodic properties of the solution to the HJMM equation. In particular, we analyse the Markov property of the solution and we find a sufficient condition for the existence and uniqueness of an invariant measure for the Markov semigroup associated to the HJMM equation (when the coefficients are time independent) in the weighted Lebesgue spaces.

In this dissertation, I develop a phase-field approach to minimising a geometric energy functional in the class of connected structures confined to a small container. The functional under consideration is Willmore's energy, which depends on the mean curvature and area measure of a surface and thus allows for a formulation in terms of varifold geometry. In this setting, I prove existence of a minimiser and a very low level of regularity from simple energy bounds. In the second part, I describe a phase-field approach to the minimisation problem and provide a sample implementation along with an algorithmic description to demonstrate that the technique can be applied in practice. The diffuse Willmore functional in this setting goes back to De Giorgi and the novel element of my approach is the design of a penalty term which can control a topological quantity of the varifold limit in terms of phase-field functions. Besides the design of this functional, I present new results on the convergence of phase-fields away from a lower-dimensional subset which are needed in the proof, but interesting in their own right for future applications. In particular, they give a quantitative justification for heuristically identifying the zero level set of a phase field with a sharp interface limit, along with a precise description of cases when this may be admissible only up to a small additional set. The results are optimal in the sense that no further topological quantities can be controlled in this setting, as is also demonstrated. Besides independent geometric interest, the research is motivated by an application to certain biological membranes.

In this work, we study a class of continuous generalisations of the kinematic Ponomarenko Dynamo, in an annulus with perfectly conducting boundary conditions.\par We first consider the fundamentals of dynamo theory, deriving the governing equations and a general numerical code to find the growth rates for all modes and magnetic Reynolds numbers $R$. We concentrate on three types of flow fields: (a) flows which approximate the discontinuous Ponomarenko dynamo, (b) full solutions of the Navier Stokes driven by an axial pressure gradient and moving boundaries, and (c) flows where both the axial and azimuthal velocity components are powers of the cylindrical radius. Good agreement is found between the numerical results and the known asymptotic theory for large $R$. The smallest $R$-values permitting dynamo action are found, along with the values which gives rise to the fastest growing mode.

This thesis addresses the observer design problem, for a class of linear one-dimensional parabolic Partial Differential Equations, considering the simultaneous estimation of states and parameters from boundary measurements. The design is based on the Backstepping methodology for Partial Differential Equations and extends its central idea, the Volterra transformation, to compensate for the parameters uncertainties. The design steps seek to reject time-varying parameter uncertainties setting forth a type of differential boundary value problems (Kernel-PDE/ODEs) to accomplish its objective, the solution of which is computed at every fixed sampling time and constitutes the observer gains for states and parameters. The design does not include any pre-transformation to some canonical form and/or a finite-dimensional formulation, and performs a direct parameter estimation from the original model. The observer design problem considers two cases of parameter uncertainty, at the boundary: control gain coefficient, and in-domain: diffusivity and reactivity parameters, respectively. For a Luenberger-type observer structure, the problems associated to one and two points of measurement at the boundary are studied through the application of an intuitive modification of the Volterra-type and Fredholm-type transformations. The resulting Kernel-PDE/ODEs are addressed by means of a novel methodology based on polynomial optimization and Sum-of-Squares decomposition. This approach allows recasting these coupled differential equations as convex optimization problems readily implementable resorting to semidefinite programming, with no restrictions to the spectral characteristics of some integral operators or system's coefficients. Additionally, for polynomials Kernels, uniqueness and invertibility of the Fredholm-type transformation are proved in the space of real analytic and continuous functions. The direct and inverse Kernels are approximated as the optimal polynomial solution of a Sum-of-Squares and Moment problem with theoretically arbitrary precision. Numerical simulations illustrate the effectiveness and potentialities of the methodology proposed to manage a variety of problems with different structures and objectives.

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The use of Lyapunov's direct method in obtaining regions of asymptotic stability of non-linear autonomous systems is well-known. This thesis is an investigation into the optimization of some function of these systems over different classes of Lyapunov functions. In Chapter 2 bounds on the transient response of two systems are optimized over a subset of quadratic Lyapunov functions and numerical work is carried out to compare several bounds. Zubov's equation is the subject of Chapter 3. The non-uniformity of the series-construction procedure is studied analytically and a new approach is made to the solution of the equation by finite difference methods. Chapters 4, 5 and 6 have a common theme of optimizing the RAS over a class of Lyapunov functions. Chapter 4 is restricted to optimal quadratics which are investigated analytically and numerically, two algorithms being developed. An optimal quadratic algorithm and a RAS algorithm are proposed in Chapter 5 for high order systems. Extensions are made in Chapter 6 to optimal Lyapunov functions of general degree and relay control systems and systems of Lur'e form are considered.

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We study the gradient flow model of the Landau-de Gennes energy functional for nematic liquid crystals at the isotropic-nematic transition temperature on prototype geometries. We focus on the three-dimensional droplet, the disc and the square with Dirichlet boundary conditions and different types of initial conditions, with the aim of observing interesting transient dynamics which may be of practical relevance. We use a fourth-order Landau-de Gennes bulk potential which admits isotropic and uniaxial minima at the transition temperature. For a droplet with radial boundary conditions, a large class of physically relevant initial conditions generate dynamic solutions with a well-defined isotropic-nematic front which propagates according to mean curvature for significant times. We introduce radially symmetric obstacles into the droplet and prove the existence of pulsating wave solutions of the gradient flow model in certain parameter regimes. The average velocity of the pulsating wave is determined by some critical forcing which can be verified numerically. On the unit disc, we make a distinction between planar and non-planar initial conditions and minimal and non-minimal Dirichlet boundary conditions. Planar initial conditions generate solutions with an isotropic core for all times whereas non-planar initial conditions generate solutions that escape into the third dimension. Non-minimal boundary conditions result in solutions with boundary layers. These solutions can have either a largely nematic interior profile or a largely isotropic interior profile, depending on the initial conditions. On the square, we provide an analytic description of the Well Order Reconstruction solution first reported numerically by Kralj and Majumdar in 2014. We interpret the Well Order Reconstruction solution as a critical point of a related scalar variational problem and prove that the solution is globally stable on small domains. We use the gradient flow model of the Landau-de Gennes energy to numerically study the emergence of new solution branches from the Well Order Reconstruction solution. We conclude this thesis by studying a triple phase Landau-de Gennes model with a sixth-order bulk potential which admits isotropic, uniaxial and biaxial minima at a special temperature known as the triple point temperature. For some model problems, we can use asymptotic methods to prove that isotropic-uniaxial, uniaxial-biaxial and isotropic-biaxial fronts propagate according to mean curvature and to prove an angle condition that holds when the fronts intersect at a triple junction. We corroborate our formal calculations with a numerical investigation of the full Landau-de Gennes gradient flow system.