Background on Nonlinear Systems

Abstract

We review some basic concepts from the theory of ordinary differential equations and nonlinear control systems, as well as several notions of stability, including the input-to-state stability paradigm. An important feature is the distinction between uniform and non-uniform stability for timevarying systems. We also include an overview of the problem of stabilization of nonlinear systems, including the “virtual” obstacles to stabilization imposed by Brockett’s Necessary Condition. Brockett’s Criterion motivates our use of time-varying feedbacks to stabilize both autonomous and time-varying systems. We illustrate these notions in several examples. In later chapters, we revisit these notions using strict Lyapunov functions.