Set Valued Convex And Nonsmooth Analysis In Dynamics And Control

SET-VALUED, CONVEX, AND NONSMOOTH ANALYSIS IN DYNAMICS AND CONTROL

Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control

Set-valued analysis, convex analysis, and nonsmooth analysis are relatively modern branches of mathematical analysis that have become increasingly relevant in current control theory and control engineering literature. This book serves as a broad introduction to analytical tools in these fields and to their applications in dynamical and control systems and is the first to cover these topics with this scope and at this level. Both continuous-time and discrete-time mutlivalued dynamics, modeled by differential and difference inclusions, are considered. Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction is aimed at graduate students in control engineering and applied mathematics and researchers in control engineering who have no prior exposure to set-valued, convex, and nonsmooth analysis. The book will also be of interest to advanced undergraduate mathematics students and mathematicians with no prior exposure to the topic. The expected mathematical background is a course on nonlinear differential equations / dynamical systems and a course on real analysis. Knowledge of some control theory is helpful, but not essential.

Nonsmooth Mechanics

Thank you for opening the second edition of this monograph, which is devoted to the study of a class of nonsmooth dynamical systems of the general form: ::i; = g(x,u) (0. 1) f(x, t) 2: 0 where x E JRn is the system's state vector, u E JRm is the vector of inputs, and the function f (-, . ) represents a unilateral constraint that is imposed on the state. More precisely, we shall restrict ourselves to a subclass of such systems, namely mechanical systems subject to unilateral constraints on the position, whose dynamical equations may be in a first instance written as: ii= g(q,q,u) (0. 2) f(q, t) 2: 0 where q E JRn is the vector of generalized coordinates of the system and u is an in put (or controller) that generally involves a state feedback loop, i. e. u= u(q, q, t, z), with z= Z(z, q, q, t) when the controller is a dynamic state feedback. Mechanical systems composed of rigid bodies interacting fall into this subclass. A general prop erty of systems as in (0. 1) and (0. 2) is that their solutions are nonsmooth (with respect to time): Nonsmoothness arises primarily from the occurence of impacts (or collisions, or percussions) in the dynamical behaviour, when the trajectories attain the surface f(x, t) = O. They are necessary to keep the trajectories within the subspace = {x : f(x, t) 2: O} of the system's state space.