Lieu : Université McGill, McConnell Engineering Building, salle 603
Date : le vendredi 28 février à 10h30
Directeur de recherche : Peter E. Caines

The modelling of linear quadratic Gaussian optimal control problems on large complex networks is intractable computationally. Graphon theory provides an approach to overcome these issues by defining limit objects for infinite sequences of graphs. This permits one to approximate arbitrarily large networks by infinite dimensional operators. This is extended to stochastic systems by the use of Q-noise, a generalization of Wiener processes in finite dimensional spaces to processes in function spaces. This thesis concerns the synthesis of two types of stochastic system on large graphs: linear quadratic Gaussian problems with estimation and linear quadratic field tracking games.

The optimal control and estimation of linear quadratic problems on graphon systems with Q-noise disturbances are defined here and shown to be the limit of the corresponding finite graph optimal control problem. The theory is extended to low rank systems, and a fully worked special case is presented. In addition, the worst-case long-range average and infinite horizon discounted optimal control performance with respect to Q-noise distribution are computed for a set of standard graphon limits. The convergence of finite network linear system state estimates to their graphon limit counterparts is established. Computational examples of this convergence behaviour is illustrated with a set of standard graphon examples.

In this thesis, linear quadratic games on very large dense networks are modelled with discrete time linear quadratic graphon field games with Q-noise. In such a game, the agents are interconnected via an undirected network with one agent per node. Gaussian disturbances that are correlated over nodes affect each agent. The limit of the finite-sized linear quadratic network tracking game in discrete time is formulated, and it is shown that under the proper assumptions, the game has a graphon limit system with Q-noise. Then, the optimal control of the discrete time system is found in closed-form and the Nash equilibrium behavior of the game is demonstrated numerically. The infinite time horizon discounted case is also analyzed, and a closed form feedback solution is presented in the special case where the underlying graphon is finite rank.

Bio: Alex Dunyak is a PhD student at McGill University supervised by Peter E. Caines studying the application of control theory to large networks. He received a master’s degree from the New York University Tandon School of Engineering in 2019, working with Professor Quanyan Zhu, and a bachelor’s degree in computer systems engineering and mathematics from the University of Massachusetts, Amherst in 2017.