This dissertation presents efficient, on-line, convergent methods to find defense strategies against attacks in dynamic multi-stage attacker-defender games including adaptive learning. This effort culminated in four papers submitted to high quality journals and a book and they are partially published. The first paper presents a novel fictitious play approach to describe the interactions between the attackers and network administrator along a dynamic game. Multi-objective optimization methodology is used to predict the attacker's best actions at each decision node. The administrator also keeps track of the attacker's actions and updates his knowledge on the attacker's behavior and objectives after each detected attack, and uses this information to update the prediction of the attacker's future actions to find its best response strategies. The second paper proposes a Dynamic game tree based Fictitious Play (DFP) approach to describe the repeated interactive decision processes of the players. Each player considers all possibilities in future interactions with their uncertainties, which are based on learning the opponent's decision process (including risk attitude, objectives). Instead of searching the entire game tree, appropriate future time horizons are dynamically selected for both players. The administrator keeps tracking the opponent's actions, predicts the probabilities of future possible attacks, and then chooses its best moves. The third paper introduces an optimization model to maximize the deterministic equivalent of the random payoff function of a computer network administrator in defending the system against random attacks. By introducing new variables the transformed objective function becomes concave. A special optimization algorithm is developed which requires the computation of the unique solution of a single variable monotonic equation. The fourth paper, which is an invited book chapter, proposes a discrete-time stochastic control model to capture the process of finding the best current move of the defender. The defender's payoffs at each stage of the game depend on the attacker's and the defender's accumulative efforts and are considered random variables due to their uncertainty. Their certain equivalents can be approximated based on their first and second moments which is chosen as the cost functions of the dynamic system. An on-line, convergent, Scenarios based Proactive Defense (SPD) algorithm is developed based on Differential Dynamic Programming (DDP) to solve the associated optimal control problem.

Nonrigid image registration has received much attention in the medical imaging and computer vision research communities, because it enables a wide variety of applications. Feature tracking, segmentation, classification, temporal image differencing, tumour growth estimation, and pharmacokinetic modeling are examples of the many tasks that are enhanced by the use of aligned imagery. Over the years, the medical imaging and computer vision communties have developed and refined image registration techniques in parallel, often based on similar assumptions or underlying paradigms. This thesis focuses on variational registration, which comprises a subset of nonrigid image registration. It is divided into chapters that are based on fundamental aspects of the variational registration problem: image dissimilarity measures, changing overlap regions, regularizers, and computational solution strategies. Key contributions include the development of local versions of standard dissimilarity measures, the handling of changing overlap regions in a manner that is insensitive to the amount of non-interesting background information, the combination of two standard taxonomies of regularizers, and the generalization of solution techniques based on Fourier methods and the Demons algorithm for use with many regularizers. To illustrate and validate the various contributions, two sets of example imagery are used: 3D CT, MR, and PET images of the brain as well as 3D CT images of lung cancer patients.

In the spontaneous curvature model for two-phase lipid bilayer membranes the shape of vesicles is governed by a combination of an elastic bending energy and an interface energy that penalises the size of phase boundaries. Each lipid phase induces a preferred curvature to the membrane surface, and these curvatures as well as phase boundaries may lead to the development of kinks. In a rotationally symmetric setting we introduce a family of energies for smooth surfaces and phase fields for the lipid components and study convergence to a sharp-interface limit, which depends on the choice of the bending parameters of the phase field model. We prove that, if kinks are excluded, our energies $Gamma$-converge to the commonly used sharp-interface spontaneous curvature energy with the additional assumption of $C^1$-regularity across interfaces. For a choice of parameters such that kinks may appear, we obtain a limit that coincides with the $Gamma$-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to kinks. We illustrate the theoretical result by some numerical examples.

While stochastic optimal control, together with associate formulations like Reinforcement Learning, provides a formal approach to, amongst other, motor control, it remains computationally challenging for most practical problems. This thesis is concerned with the study of relations between stochastic optimal control and probabilistic inference. Such dualities { exempli ed by the classical Kalman Duality between the Linear-Quadratic-Gaussian control problem and the filtering problem in Linear-Gaussian dynamical systems { make it possible to exploit advances made within the separate fields. In this context, the emphasis in this work lies with utilisation of approximate inference methods for the control problem. Rather then concentrating on special cases which yield analytical inference problems, we propose a novel interpretation of stochastic optimal control in the general case in terms of minimisation of certain Kullback-Leibler divergences. Although these minimisations remain analytically intractable, we show that natural relaxations of the exact dual lead to new practical approaches. We introduce two particular general iterative methods ψ-Learning, which has global convergence guarantees and provides a unifying perspective on several previously proposed algorithms, and Posterior Policy Iteration, which allows direct application of inference methods. From these, practical algorithms for Reinforcement Learning, based on a Monte Carlo approximation to ψ-Learning, and model based stochastic optimal control, using a variational approximation of posterior policy iteration, are derived. In order to overcome the inherent limitations of parametric variational approximations, we furthermore introduce a new approach for none parametric approximate stochastic optimal control based on a reproducing kernel Hilbert space embedding of the control problem. Finally, we address the general problem of temporal optimisation, i.e., joint optimisation of controls and temporal aspects, e.g., duration, of the task. Specifically, we introduce a formulation of temporal optimisation based on a generalised form of the finite horizon problem. Importantly, we show that the generalised problem has a dual finite horizon problem of the standard form, thus bringing temporal optimisation within the reach of most commonly used algorithms. Throughout, problems from the area of motor control of robotic systems are used to evaluate the proposed methods and demonstrate their practical utility.

In this thesis improvements, implementation and evaluation have been done on a motion planning algorithm for a full-sized reversing truck and trailer system. The motion planner is based on a motion planning algorithm called Closed-Loop Rapidly-exploring Random Tree (CL-RRT). An important property for a certain class of systems, stating that by selecting the input signals in a certain way the same result as reversing the time can be archived, is also presented. For motion planning this means that the problem of reversing from position A to position B can also be solved by driving forward from B to A and then reverse the solution. The use of this result in the motion planner has been evaluated and has shown to be very useful. The main improvements made on the CL-RRT algorithm are a faster collision detection method, a more efficient way to draw samples and a more correct heuristic cost-to-go function. A post optimizing or smoothing method that brings the system to the exact desired configuration, based on numerical optimal control, has also been developed and implemented with successful results. The motion planner has been implemented and evaluated on a full-scale truck with a dolly steered trailer prepared for autonomous operation with promising results.

This thesis is concerned with investigating new methods for the analysis of large-scale dynamical systems using convex optimization. The proposed methodology is based on composite Lyapunov theory and is computationally implemented using polynomial programming techniques. The main result of this work is the development of a system decomposition framework that makes it possible to analyze systems that are of such a scale that traditional methods cannot cope with. We begin by addressing the problem of model invalidation. A barrier certificate method for invalidating models in the presence of uncertain data is presented for both continuous and discrete time models. It is shown how a re-parameterization of the time dependent variables can improve the numerical conditioning of the underlying optimization problem. The main contribution of this thesis is the development of an automated dynamical system decomposition framework that permits us to verify the stability of systems that typically have a state dimension large enough to render traditional computational methods intractable. The underlying idea is to decompose a system into a set of lower order subsystems connected in feedback in such a manner that composite methods for stability verification may be employed. What is unique about the algorithm presented is that it takes into account both dynamics and the topology of the interconnection graph. In the first instance we illustrate the methodology with an ecological network and primal Internet congestion control scheme. The versatility of the decomposition framework is also highlighted when it is shown that when applied to a model of the EGF-MAPK signaling pathway it is capable of identifying biologically relevant subsystems in addition to stability verification. Finally we introduce stability metrics for interconnected dynamical systems based on the theory of dissipativity. We conclude by outlining a clustering based decomposition algorithm that explicitly takes into account the input and output dynamics when determining the system decomposition.

The application of techniques from nonlinear analysis to materials science has seen great developments in the recent years and it has really been a driving force for substantial mathematical research in the area of partial differential equations and the multi-dimensional calculus of variations. This thesis has been motivated by two recent and remarkable experimental observations of H. Seiner in shape-memory alloys which we attempt to interpret mathematically. Much of the work is original and has given rise to deep problems in the calculus of variations. Firstly, we study the formation of non-classical austenite-martensite interfaces. Ball & Carstensen (1997, 1999) theoretically investigated the possibility of the occurrence of such interfaces and studied the cubic-to-tetragonal case extensively. In this thesis, we present an analysis of non-classical austenite-martensite interfaces recently observed by Seiner et al.~in a single crystal of a CuAlNi shape-memory alloy, undergoing a cubic-to-orthorhombic transition. We show that these can be described by the general nonlinear elasticity model and we make some predictions regarding the admissible volume fractions of the martensitic variants involved, as well as the habit plane normals. Interestingly, in the above experimental observations, the interface between the austenite and the martensitic configuration is never exactly planar, but rather slightly curved, resulting from the pattern of martensite not being exactly homogeneous. However, it is not clear how one can reconstruct the inhomogeneous configuration as a stress-free microstructure and, instead, a theoretical approach is followed. In this approach, a general method is provided for the construction of a compatible curved austenite-martensite interface and, by exploiting the structure of quasiconvex hulls, the existence of curved interfaces is shown in two and three dimensions. As far as the author is aware of, this is the first construction of such a curved austenite-martensite interface. Secondly, we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy consisting of a single variant of stabilized 2H martensite. The nucleation process is induced by localized heating and it is observed that, regardless of where the localized heating is applied, the nucleation points are always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant throughout the specimen, provided this is suitably cut. The proofs of quasiconvexity are based on a rigidity argument and are specific to the change of symmetry in the phase transformation. To the best of the author's knowledge, quasiconvexity conditions at edges and corners have not been considered before.

Throughout the course of this thesis, we give an introduction to optimal control theory and its necessary conditions, prove Pontryagin's Maximum Principle, and present the life-cycle saving under uncertain lifetime optimal control problem. We present a very involved sensitivity analysis that determines how a change in the initial wealth, discount factor, or relative risk aversion coefficient may affect the model the terminal depletion of wealth time, optimal consumption path, and optimal accumulation of wealth path. Through simulation of the life-cycle saving under uncertain lifetime model, we are not only able to present the model dynamics through time, but also to demonstrate the feasibility of the model.

This thesis concerns the regularity of holonomic minimisers of variational integrals in the context of direct methods in the calculus of variations. Specifically, we consider Sobolev mappings from a bounded domain into a connected compact Riemannian manifold without boundary, to which such mappings are said to be holonomically constrained. For a general class of strictly quasiconvex integral functionals, we give a direct proof of local C^1,α-Hölder continuity, for some 0 < α < 1, of holonomic minimisers off a relatively closed 'singular set' of Lebesgue measure zero. Crucially, the proof constructs comparison maps using the universal covering of the target manifold, the lifting of Sobolev mappings to the covering space and the connectedness of the covering space. A certain tangential A-harmonic approximation lemma obtained directly using a Lipschitz approximation argument is also given. In the context of holonomic minimisers of regular variational integrals, we also provide bounds on the Hausdorff dimension of the singular set by generalising a variational difference quotient method to the holonomically constrained case with critical growth. The results are analogous to energy-minimising harmonic maps into compact manifolds, however in this case the proof does not use a monotonicity formula. We discuss several applications to variational problems in condensed matter physics, in particular those concerning the superfluidity of liquid helium-3 and nematic liquid crystals. In these problems, the class of mappings are constrained to an orbit of 'broken symmetries' or 'manifold of internal states', which correspond to a sub-group of residual symmetries.

In this work we introduce a novel queueing model with two classes of users in which, instead of accessing a resource, users wait in the system to match with a candidate from the other class. The users are selective and the matchings occur probabilistically. This new model is useful for analysing the traffic in web portals that match people who provide a service with people who demand the same service, e.g. employment portals, matrimonial and dating sites and rental portals. We first provide a Markov chain model for these systems and derive the probability distribution of the number of matches up to some finite time given the number of arrivals. We then prove that if no control mechanism is employed these systems are unstable for any set of parameters. We suggest four different classes of control policies to assure stability and conduct analysis on performance measures under the control policies. Contrary to the intuition that the rejection rate should decrease as the users become more likely to be matched, we show that for certain control policies the rejection rate is insensitive to the matching probability. Even more surprisingly, we show that for reasonable policies the rejection rate may be an increasing function of the matching probability. We also prove insensitivity results related to the average queue lengths and waiting times. Further, to gain more insight into the behaviour of probabilistic matching systems, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyse the basic properties of these approximations and show that some performance measures are insensitive to the matching probability agreeing with the results found by the exact analysis. Finally we study the optimal control and revenue management for the systems with the objective of profit maximization. We formulate mathematical models for both unobservable and observable systems. For an unobservable system we suggest a deterministic optimal control, while for an observable system we develop an optimal myopic state dependent pricing.