The thesis deals with various problems arising in deterministic control, jumping processes and control for locomotion in fluids. It is divided in three parts. The first part is focused on some optimal control problems on network and stratified domains with junctions, where each edge/hyper-plane has its own controlled dynamics and cost. We consider some possible approximations for such a problems given by the use of a switching rule of delayed-relay type and study the passage to the limit when the parameter of the approximation goes to zero. First, we take into account some problems on network: a twofold junction problem, a threefold junction one and an extension of the last one. For each of these problems we characterize the limit functions as viscosity solution and maximal subsolution of a suitable Hamilton-Jacobi problem. Secondly, we consider a bi-dimensional multi-domain problem and as done for the problems on network we characterize the limit function as viscosity solution of a suitable Hamilton-Jacobi problem. The second part studies a decentralized routing problem over a network, using the paradigm of mean-field games with large number of players. Building on a state-space extension technique, we turn the problem into an optimal control one for each single player. The main contribution is an explicit expression of the optimal decentralized control which guarantees the convergence both to local and to global equilibrium points. Furthermore, we study the stability of the system also in the presence of a delay which we model using an hysteresis operator. As a result of the hysteresis, we prove existence of multiple equilibrium points and analyze convergence conditions. In the last part of the thesis we investigate different strategies to overcome the so-called scallop paradox concerning periodic locomotion in fluid. We show how to obtain a net motion exploiting the fluid's type change during a periodic deformation. We consider two different models: in the first one that change is linked to the magnitude of the opening and closing velocity of the scallop's valves. Instead, in the second one it is related to the sign of the above velocity. In both cases we prove that the mechanical system is controllable, i.e. the scallop is able to move both forward and backward using cyclical deformations.

This thesis is concerned with the study of nonstandard models in measure theory and in functional analysis. In measure theory, we define elementary numerosities, that are additive measures that take on values in a non-archimedean field and for which the measure of every singleton is 1. We have shown that, by taking the ratio with a suitable unit of measurement, from a numerosity it can be defined a non-atomic real-valued measure, and that every non-atomic measure can be obtained from a numerosity by this procedure. We then used numerosities to develop a model for the probability of infinite sequences of coin tosses coherent with the original ideas of Laplace. In functional analysis, we introduce a space of functions of nonstandard analysis with a formally finite domain, that extends both the space of distributions and the space of Young measures. Among the applications of this space of functions, we develop a continuous-in-time, discrete-in-space nonstandard formulation for a class of ill-posed forward-backward parabolic equations, and on the study of the regularity and asymptotic behaviour of its nonstandard solutions. This approach proved to be a viable alternative to the study of the vanishing viscosity limit of the solution of a pseudoparabolic regularization of the original problem.

This thesis summarizes my work in systems biology as a PhD student at The Microsoft Research - University of Trento Centre for Computational and Systems Biology (COSBI) and at the University of Trento, department of Mathematics. Systems biology is an interdisciplinary field that aims at integrating biology with computational and mathematical methods to gain a better understanding of biological phenomena [5, 6]. Among these methods, mathematical and dy- namical modeling have driven the discovery of mechanistic insights from the static representations of phenomena, that is, data. As a result, mathematical and dynamical models have now become standard tools to support new discoveries in biology and in public health issues. For example, models assist governments in determining the policies to contain the spreading of the diseases and in decisions such as vaccine purchases [7]. Similarly, complex and accurate models of the cardio-vascular systems guide surgeons during many procedures on pa- tients [8]. Furthermore, dynamical models of signaling cascades help researchers in identifying new potential drug targets and therapies for many diseases [9]. We used these modeling techniques to address biological questions related to diabetes and insulin resistance. Within this framework, this thesis contains two articles I contributed to, that focus on diabetes. These works are published in the journal of Nature Scientific Reports and are included in Chapters 3 and 4. A significant contribution to the development of these models, and models in general, is given by optimization. Optimization is often used in modeling to determine certain unknown values or factors in a way that allow the model to optimally reproduce the experimental data. Moreover, the parameters of a model that correctly describe the undergoing dynamics may be used as diagnostic tools [10–13]. To this end, this thesis contains a methodological appendix that includes a review of optimization algorithms that has been submitted to the journal of Frontiers in Applied Mathematics and Statistics, special topic Optimization. The content of this article is reported in Appendix A.

This thesis addresses two problems from the theory of periodic homogenization and the related notion of two-scale convergence. Its main focus rests on the derivation of equivalent transmission conditions for the interaction of two adjacent bodies which are connected by a thin layer of interface material being perforated by identically shaped voids. Herein, the voids recur periodically in interface direction and shall in size be of the same order as the interface thickness. Moreover, the constitutive properties of the material occupying the bodies adjacent to the interface are assumed to be described by some convex energy densities of quadratic growth. In contrast, the interface material is supposed to show extremal" constitutive behavior. More precisely

In this thesis we investigate variational problems involving 1-dimensional sets (e.g., curves, networks) and variational inequalities related to obstacle-type dynamics from a twofold prospective. On one side, we provide variational approximations and convex relaxations of the relevant energies and dynamics, moving mainly within the framework of Gamma-convergence and of convex analysis. On the other side, we thoroughly investigate the numerical optimization of the corresponding approximating energies, both to recover optimal 1-dimensional structures and to accurately simulate the actual dynamics.

Evaluation of automatic speech recognition (ASR) systems is difficult and costly, since it requires manual transcriptions. This evaluation is usually done by computing word error rate (WER) that is the most popular metric in ASR community. Such computation is doable only if the manual references are available, whereas in the real-life applications, it is a too rigid condition. A reference-free metric to evaluate the ASR performance is \textit{confidence measure} which is provided by the ASR decoder. However, the confidence measure is not always available, especially in commercial ASR usages. Even if available, this measure is usually biased towards the decoder. From this perspective, the confidence measure is not suitable for comparison purposes, for example between two ASR systems. These issues motivate the necessity of an automatic quality estimation system for ASR outputs. This thesis explores ASR quality estimation (ASR QE) from different perspectives including: feature engineering, learning algorithms and applications. From feature engineering perspective, a wide range of features extractable from input signal and output transcription are studied. These features represent the quality of the recognition from different aspects and they are divided into four groups: signal, textual, hybrid and word-based features. From learning point of view, we address two main approaches: i) QE via regression, suitable for single hypothesis scenario; ii) QE via machine-learned ranking (MLR), suitable for multiple hypotheses scenario. In the former, a regression model is used to predict the WER score of each single hypothesis that is created through a single automatic transcription channel. In the latter, a ranking model is used to predict the order of multiple hypotheses with respect to their quality. Multiple hypotheses are mainly generated by several ASR systems or several recording microphones. From application point of view, we introduce two applications in which ASR QE makes salient improvement in terms of WER: i) QE-informed data selection for acoustic model adaptation; ii) QE-informed system combination. In the former, we exploit single hypothesis ASR QE methods in order to select the best adaptation data for upgrading the acoustic model. In the latter, we exploit multiple hypotheses ASR QE methods to rank and combine the automatic transcriptions in a supervised manner. The experiments are mostly conducted on CHiME-3 English dataset. CHiME-3 consists of Wall Street Journal utterances, recorded by multiple far distant microphones in noisy environments. The results show that QE-informed acoustic model adaptation leads to 1.8\% absolute WER reduction and QE-informed system combination leads to 1.7% absolute WER reduction in CHiME-3 task. The outcomes of this thesis are packed in the frame of an open source toolkit named TranscRater -transcription rating toolkit- (https://github.com/hlt-mt/TranscRater) which has been developed based on the aforementioned studies. TranscRater can be used to extract informative features, train the QE models and predict the quality of the reference-less recognitions in a variety of ASR tasks.

In this Ph.D. thesis we discuss results concerning variational convergences for functionals and differential operators on Lipschitz continuous vector fields. The convergences taken into account are gamma-convergence (for functionals) and H-convergence (for differential operators).

Electromagnetism and optimal control stand out as a topics that feature impactful applications in modern engineering, as well as challenging theoretical aspects of mathematical analysis. Within this context, a major role is played by the search of necessary and sufficient conditions characterizing optimal solutions, as they are functional to numerical algorithms aiming to approximate such solutions. In this thesis, three standalone topics in optimization sharing the underlying framework of Maxwell-related PDEs are discussed. First, I present an optimal control problem driven by a quasi-linear magneto-static obstacle problem featuring first-order differential state constraints. The non-linearity allows to suitably model electromagnetic waves in the presence of ferromagnetic materials, while the first-order obstacle is relevant for applications in the field of magnetic shielding. Existence theory and the derivation of an optimality system are addressed with an approximation technique based on a relaxation-penalization of the variational inequality. Second, I analyze an eddy current problem controlled through a dipole type source, i.e. a Dirac mass with fixed position and variable intensity: well-posedness of the state equation through a fundamental solution (of a curl curl - Id operator) approach and first order conditions are dealt with. To conclude, I discuss the computation of the topological derivative for shape functionals constrained to low-frequency electromagnetic problems (closely related to the eddy current model), with respect to the inclusion/removal of conducting material; the results are obtained using a Lagrangian approach and in particular the so-called averaged adjoint method. This approach requires the study of the asymptotic behavior of the solutions of some problems defined in the whole space, and the introduction and consequent analysis of appropriate function spaces.

This dissertation is divided into three different parts. In the first part we analyse collected data on the occurrence of influenza-like illness (ILI) symptoms regarding the 2009 influenza A/H1N1 virus pandemic in two primary schools of Trento, Italy. These data were used to calibrate a discrete-time SIR model, which was designed to estimate the probabilities of influenza transmission within the classes, grades and schools using Markov Chain Monte Carlo (MCMC) methods. We found that the virus was mainly transmitted within class, with lower levels of transmission between students in the same grade and even lower, though not significantly so, among different grades within the schools. We estimated median values of R0 from the epidemic curves in the two schools of 1.16 and 1.40; on the other hand, we estimated the average number of students infected by the first school case to be 0.85 and 1.09 in the two schools. This discrepancy suggests that household and community transmission played an important role in sustaining the school epidemics. The high probability of infection between students in the same class confirms that targeting within-class transmission is key to controlling the spread of influenza in school settings and, as a consequence, in the general population. In the second part, by starting from a basic host-parasitoid model, we study the dynamics of a 2 hosts-1 parasitoid model assuming, for the sake of simplicity, that larval stages have a fixed duration. If each host is subjected to density-dependent mortality in its larval stage, we obtain explicit conditions for coexistence of both hosts, as long as each 1 host-parasitoid system would tend to an equilibrium point. Otherwise, if mortality is density-independent, under the same conditions host coexistence is impossible. On the other hand, if at least one of the 1 host-parasitoid systems has an oscillatory dynamics (which happens under some parameter values), we found, through numerical bifurcation, that coexistence is favoured. It is also possible that coexistence between the two hosts occurs even in the case without density-dependence. Analysis of this case has been based on methods of approximation of the dominant characteristic multipliers of the monodromy operator using a recent method introduced by Breda et al. Models of this type may be relevant for modelling control strategies for Drosophila suzukii, a recently introduced fruit fly that caused severe production losses, based on native parasitoids of indigenous fruit flies. In the third part, we present a starting point to analyse raw data collected by Stacconi et al. in the province of Trento, Italy. We present an extensions of the model presented in Part 2 where we have two hosts and two parasitoids. Since its analysis is complicated, we begin with a simpler one host-one parasitoid model to better understand the possible impact of parasitoids on a host population. We start by considering that the host population is at an equilibrium without parasitoids, which are then introduced as different percentages of initial adult hosts. We compare the times needed by parasitoids to halve host pupae and we found that the best percentage choice is 10%. Thus we decide to fix this percentage of parasitoid introduction and analyse what happens if parasitoids are introduced when the host population is not at equilibrium both by introducing always the same percentage or the same amount of parasitoids. In this case, even if the attack rate is at 1/10 of its maximum value, parasitoids would have a strong effect on host population, shifting it to an oscillatory regime. However we found that this effect would require more than 100 days but we also found that it can faster if parasitoids are introduced before the host population has reached the equilibrium without parasitoids. Thus there could be possible releases when host population is low. Last we investigate also what happens if in nature mortality rates of these species increase and we found that there is not such a big difference respect to the results obtained using laboratory data.

Firstly, I proposed a very simple SIS/SIR model for a general vector-borne disease transmission considering constant population sizes over the season, where contact between the host and the vector responsible of the transmission is assumed to occur only during the summer of each year. I discussed two different types of threshold for pathogen persistence that I explicitly computed: a "short-term threshold" and a "long-term threshold". Later, I took into account the seasonality of the populations involved in the transmission. For a single season, the model consists of system of non linear differential equations considering the various stages of the infection transmission between the vector and the host population. Assuming the overwintering in the mosquito populations, I simulated the model for several years. Finally, I studied the spatial spread of a vector-borne disease throught an impusive reaction-diffusion model and I showed some simulations.