Abstract:  Feedback is ubiquitous and is a core concept in control systems, where the main objective of using feedback is to deal with the influences of various uncertainties on the performance of the dynamical systems to be controlled. Although much progress has been made in control theory over the past 50 years, especially in such areas as adaptive control and robust control, the following fundamental problem remains less explored: How much uncertainty can be dealt with by the feedback mechanism? To answer this, we need not only to investigate what the feedback mechanism can do, but also need to understand, the more challenging and difficult issue, what the feedback mechanism cannot do. The feedback mechanism is defined as the class of all possible feedback laws (which are not restricted to a certain particular subclass), and the maximum capability of feedback is measured as the maximum size of uncertainties that can be dealt with by the feedback mechanism. In this lecture, we will mainly consider discrete-time (or sampled-data) nonlinear dynamical control systems with both structural and environmental uncertainties. We will present a series of “Critical Values” and “Impossibility Theorems” concerning the maximum capability of the feedback mechanism for several basic classes of uncertain nonlinear control systems, and will expand on their theoretical implications as well as their practical significances. 

Abstract. To understand how locally interacting agents (or particles) lead to collective behaviors (or structures) is a fundamental issue in complex systems. Such problems arise from diverse fields ranging from material and life sciences to social and engineering systems, and have attracted much research attention in recent years. In this lecture, we will consider the synchronization problem of some basic classes of flocks with large population, without resorting to any connectivity assumptions imposed on the dynamical trajectories of the flocks. By working in a stochastic framework with large population and by establishing some kind of dynamical connectivity, we are able to provide a rigorous theory for synchronization of the flocks with both geometric and topological interaction distances. In particular, we will show the smallest possible interaction radius needed for synchronization of flocks can be as small as that for the connectivity of the initial static graphs. Furthermore, we will show how the collective behaviors of the flocks may be intervened in a “leader-follower” mode or in the general “soft control” framework, without changing the existing interaction rules of the agents. The main theorems are established based on analyses of the nonlinear dynamical equations involved and of the asymptotical properties of the spectrum of the corresponding random geometric graphs.