The gap between necessity and sufficiency for stability of sparse matrix systems: simulation studies

Sparse matrix systems (SMSs) are potentially very useful for graph analysis and topological representations of interaction and communication among elements within a system. Such systems’ stability can be determined by the Routh-Hurwitz criterion. However, simply using the Routh-Hurwitz criterion is not efficient in this kind of system. Therefore, a necessary condition can save a lot of work. The necessary condition is of importance and will be discussed in this thesis. Also, meeting the necessary condition does not mean it is safe to claim the SMS is stable. Therefore, another part of this project is to see how effective the necessary condition is by simulations. The simulation shows that approximate SMSs meeting the necessary condition are very likely to be stable. The results approach 90-95% effectiveness given enough trials

Singular Cucker-Smale Dynamics

The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared - analytically and numerically - to the porous medium equation

Mean-field optimal control as Gamma-limit of finite agent controls

This paper focuses on the role of a government of a large population of interacting agents as a mean field optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a PDE of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a $\Gamma$-convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle

Mean field control hierarchy

In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrainted by aPDE of continuity-type, governing the dynamics of the probability distributionof the agent population. We show the existence of mean field optimal controlsboth in the stochastic and deterministic setting. We derive rigorously thefirst order optimality conditions useful for numerical computation of meanfield optimal controls. We introduce a novel approximating hierarchy ofsub-optimal controls based on a Boltzmann approach, whose computation requiresa very moderate numerical complexity with respect to the one of the optimalcontrol. We provide numerical experiments for models in opinion formationcomparing the behavior of the control hierarchy