Search Results

This chapter discusses the notion of action-minimizing measures, recalling the needed measure–theoretical material. In particular, this allows the definition of a first family of invariant sets, the so-called Mather sets. It discusses their main dynamical and symplectic properties, and introduces the minimal average actions, sometimes called Mather's α‎- and β‎-functions. A thorough discussion of their properties (differentiability, strict convexity or lack thereof) is provided and related to the dynamical and structural properties of the Mather sets. The chapter also describes these concepts in a concrete physical example: the simple pendulum.

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

This chapter describes another interesting approach to the study of invariant sets provided by the so-called weak KAM theory, developed by Albert Fathi. This approach can be considered as the functional analytic counterpart of the variational methods discussed in the previous chapters. The starting point is the relation between KAM tori (or more generally, invariant Lagrangian graphs) and classical solutions and subsolutions of the Hamilton–Jacobi equation. It introduces the notion of weak (non-classical) solutions of the Hamilton–Jacobi equation and a special class of subsolutions (critical subsolutions). In particular, it highlights their relation to Aubry–Mather theory.

Existence of stationary measures and ergodic behavior of stochastic systems form an active and important research area with several applications in engineering, physical and biological sciences. A mean-square ergodic theorem is proved for one-dimensional diffusions. Existence and uniqueness of invariant measures, and ergodicity for Markov processes are established under fairly general hypotheses.

This chapter describes a stability and control design framework for time-varying and time-invariant sets of nonlinear dynamical systems. The framework is applied to the problem of coordination control for multiagent interconnected systems predicated on vector Lyapunov functions. In multiagent systems, several Lyapunov functions arise naturally where each agent can be associated with a generalized energy function corresponding to a component of a vector Lyapunov function. The chapter characterizes a moving formation of vehicles as a time-varying set in the state space to develop a distributed control design framework for multivehicle coordinated motion control by designing stabilizing controllers for time-varying sets of nonlinear dynamical systems. The proposed cooperative control algorithms are shown to globally exponentially stabilize both moving and static formations.