Spectral Representations of Graphons in Very Large Network Systems Control

Graphon-based control has recently been proposed and developed to solve control problems for dynamical systems on networks which are very large or growing without bound (see Gao and Caines, CDC 2017, CDC 2018). In this paper, spectral representations, eigenfunctions and approximations of graphons, and their applications to graphon-based control are studied. First, spectral properties of graphons are presented and then approximations based on Fourier approximated eigenfunctions are analyzed. Within this framework, two classes of graphons with simple spectral representations are given. Applications to graphon-based control analysis are next presented; in particular, the controllability of systems distributed over very large networks is expressed in terms of the properties of the corresponding graphon dynamical systems. Moreover, spectral analysis based upon real-world network data is presented, which demonstrates that low-dimensional spectral approximations of networks are possible. Finally, an initial, exploratory investigation of the utility of the spectral analysis methodology in graphon systems control to study the control of epidemic spread is presented.Comment: 8 pages, 58th IEEE Conference on Decision and Control (CDC 2019

Analytic Controllability of Time-Dependent Quantum Control Systems

The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrodinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space -- and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrodinger control problem. The applicability of the resulting theorem is illustrated with selected examples.Comment: 13 page

Novel Results on the Factorization and Estimation of Spectral Densities

This dissertation is divided into two main parts. The first part is concerned with one of the most classical and central problems in Systems and Control Theory, namely the factorization of rational matrix-valued spectral densities, commonly known as the spectral factorization problem. Spectral factorization is a fundamental tool for the solution of a variety of problems involving second-order statistics and quadratic cost functions in control, estimation, signal processing and communications. It can be thought of as the frequency-domain counterpart of the ubiquitous Algebraic Riccati Equation and it is intimately connected with the celebrated Kálmán-Yakubovich-Popov Lemma and, therefore, to passivity theory. Here, we provide a rather in-depth and comprehensive analysis of this problem in the discrete-time setting, a scenario which is becoming increasingly pervasive in control applications. The starting point in our analysis is a general spectral factorization result in the same vein of Dante C. Youla. Building on this fundamental result, we then investigate some key issues related to minimality and parametrization of minimal spectral factors of a given spectral density. To conclude, we show how to extend some of the ideas and results to the more general indefinite or J-spectral factorization problem, a technique of paramount importance in robust control and estimation theory. In the second part of the dissertation, we consider the problem of estimating a spectral density from a finite set of measurements. Following the Byrnes-Georgiou-Lindquist THREE (Tunable High REsolution Estimation) paradigm, we look at spectral estimation as an optimization problem subjected to a generalized moment constraint. In this framework, we examine the global convergence of an efficient algorithm for the estimation of scalar spectral densities that hinges on the Kullback-Leibler criterion. We then move to the multivariate setting by addressing the delicate issue of existence of solutions to a parametric spectral estimation problem. Eventually, we study the geometry of the space of spectral densities by revisiting two natural distances defined in cones for the case of rational spectra. These new distances are used to formulate a "robust" version of THREE-like spectral estimation