Some control aspects in mathematical biology and deep learning

Abstract

In this thesis, we present different contributions on constrained controllability problems in reactiondiffusion equations, qualitative properties of optimizers, namely a fragmentation phenomena in mathematical ecology, and connections between control theory and deep learning. In the first part of the thesis, we address the main paradigms for the control of reaction-diffusion equations with constraints. It is a continuation and consolidation of some existing ideas to obtain a more complete picture of the control phenomenology. We settle the discussion on three paradigmatic target states that will serve us for explaining most of the relevant phenomena. We discuss controllability and noncontrollability, construction of paths of steady states, a characterization of the controllable initial data to a specific model target, and we observe how, in certain circumstances, the controllability time cannot be uniform with respect to the initial data. Later, we focus on the influence of spatial heterogeneity in the constrained control properties of reaction-diffusion equations. The motivation has been the so-called gene-flow models. Recently, some strategies for the control of plagues concern the release of genetically modified individuals. The question is whether the strategy’s success depends on the spatial structure of the population. We assume that the population lives in a domain to which we do not have direct access. Mathematically this can be translated to study the boundary control problem. We will show that the spatial structure of the population living in this domain plays a crucial role in determining if the controllability of the equation. A dual motivation of the problem above is treated in the second part. In this case, the objective is to design an environment to maximize the population living in it. More specifically, it concerns the qualitative properties of optimal controls. The model employed is a Fisher-KPP equation in the static regime. The problem consists on the optimal allocation of resources, which appear in a bilinear form in the equation. We prove that, in the limit when the diffusivity tends to zero, the BV norm of the optimal resources distribution should blow up. From a calculus of variations (or optimal control) perspective, the main innovation is the qualitative analysis of a non-energetic optimization problem. In the third part, we change our attention to the understanding of some mathematical properties of specific methods in Machine Learning, namely Neural Differential Equations. We will cast the problems of classification and approximation as simultaneous control problems. The simultaneous control property is rare in mechanical systems; however, the properties of the activation function allow the construction of explicit controls to achieve such a purpose. Furthermore, the fact that we are controlling every initial data with the same controls implies that we can control the characteristics of a transport equation with the same control. In particular, this property allows us to understand the classification and approximation paradigm through transport equations and giving controllability results for it