Analytical foundations of a class of hybrid models with applications to collective dynamics

The thesis as a whole focus on mathematical models describing collective dynamics of agents in different contexts. In the first part of this work, we study a distinct type of systems of differential equations, arising from mathematical models that combine discrete and continuum approaches, known as hybrid or discrete-continuum models. In the last years, these kind of models has gained interest in those biological phenomena involving cell-cell interactions and cell-matrix interaction, specifically chemotactic ones. The terminology hybrid thus refers to the presence of different scales within the same model: while cells are treated as discrete units, the chemotactic signal influencing their dynamics is represented as a continuum. The great variety of applications of these kind of models does not correspond to a relevant literature published elsewhere concerning theoretical fundaments. From a mathematical point of view, the general structure of the investigated models combines ordinary and partial differential equations. There are at least two main aspects that differentiate our formulation from those available in the current state of the art concerning analytical results for coupled systems. Part I of the thesis is devoted to analytical results concerning existence and uniqueness of the solution of hybrid systems. In a first published work, we propose well-posedness theorems for the solutions in RN assuming, between others, the source term g and the initial data f continuous functions. This contribution has been extended in a paper, where we investigated the case of a source term with less regularity properties. In fact, the present literature exhibits models with discontinuous source term, in order to differentiate regions from which a signal arises from the others. In an ongoing work, we further generalize the above structure of hybrid system, assuming in particular f ? H1(RN ) and g ? Lip(RN×n; L2(RN )). In this work we are going to introduce a different technique, based on a preliminary study concerning well-posedness of a pseudo-parabolic approximation problem, in order to prove existence and uniqueness of the solution. Another crucial investigated aspect concerns the asymptotic behavior of the solution. In a more applied and numerical oriented work, we consider an alignment and chemotaxis mechanism, whose evolution in time is modeled by a parabolic equation with constant coefficient, acting on a system of interacting particles, and initial null concentration of chemical. The proposed model is studied both from an analytical and a numerical point of view. From the analytic point of view we proved existence and uniqueness of the solution. Then, the asymptotic behavior of a linearised version of the system is investigated. We proved that the migrating aggregate exponentially converges to a state in which all the particles have a same position with zero velocity. Theoretical results were compared with numerical simulations, based on finite difference schemes, concerning the behavior of the full nonlinear system. Part II of the thesis introduces a different approach to collective dynamics of agents. The research started with a published book chapter focused on robustness and control of distributed systems. In order to provide evidence of the robustness of distributed biological systems, we considered a case study describing chemotaxis processes for a colony of E. Coli bacteria. Afterwards, distributed systems of agents have been investigated in a decision making perspective. The research in this area has led to two contributions. In a first work, we aimed at finding effective distributed algorithms to solve the Sparse Analytic Hierarchy Process problem, where a set of networked agents (e.g., wireless sensors, mobile robots or IoT devices) need to be ranked based on their utility/importance. However, instead of knowing their absolute importance, the agents know their relative utility/importance with respect to their neighbors. Moreover, such a relative information is perturbed due to errors, subjective biases or incorrect information. The aim of this paper is to provide a numerical comparison of the performances of four methods over networks with different characteristics. In the second contribution, we considered a scenario where a set of agents, interconnected by a network topology, aim at computing an estimate of their own utility, based on pairwise relative information having hybrid nature. In greater detail, the agents are able to measure the difference between their value and the value of some of their neighbors, or have an estimate of the ratio between their value and the value of the remaining neighbors. The "hybrid" scenario represents the novelty with respect to previous work in literature, where the two types of information are treated separately. Specifically, we developed a distributed algorithm that lets each agent asymptotically compute a utility value. To this end, we first characterized the task at hand in terms of a least-squares minimum problem, providing a necessary and sufficient condition for the existence of a unique global minimum. Moreover, we proved that the proposed algorithm asymptotically converges to the global minimum.