Infinite-dimensional projected dynamics and the 1-dimensional obstacle problem

Monica-Gabriela Cojocaru

doi:10.1155/2005/379248

Abstract

In this paper we present a direct application of the theory of infinite-dimensional projected dynamical systems (PDS) related to the well-knownobstacle problem, i.e., the problem of determining the shape of an elastic string stretched over a body (obstacle). While the obstacle problem is static in nature and is solved via variational inequalities theory, we show here that the dynamic problem of describing the vibration movement of the string around the obstacle is solved via the infinite-dimensional theory of projected dynamical systems.

Highlights

The rigorous theory of infinite-dimensional projected dynamical systems (PDS) started in 2002, with the works of Isac and Cojocaru, and continued in [5], [15]
The finite-dimensional theory has its roots in equilibrium problems and their relation to variational inequality problems (VI)
The intimate relation between a PDS and an associated VI, in both finite and infinite-dimensional settings, is the catalyst for viewing a PDS as the natural dynamics which “tells the story” of a system around steady states. This relation consists of the fact that any critical point of a PDS coincides with a solution to the associated VI, which solution, in turn, is known to describe a steady state of a physical system

Summary

Introduction

The rigorous theory of infinite-dimensional PDS started in 2002, with the works of Isac and Cojocaru (see [14], [4]), and continued in [5], [15]. Infinite-dimensional projected dynamics infinite-dimensional Hilbert spaces of the theory of PDS in finite dimensions which started in the early 90’s with the papers of Dupuis and Nagurney [8] and [9]. The intimate relation between a PDS and an associated VI, in both finite and infinite-dimensional settings, is the catalyst for viewing a PDS as the natural dynamics which “tells the story” of a system around steady states. This relation consists of the fact that any critical point of a PDS coincides with a solution to the associated VI (and viceversa), which solution, in turn, is known to describe a steady state of a physical system.

Projected dynamical systems and variational inequalities

The 1-dimensional obstacle problem as a variational inequality

The 1-dimensional obstacle problem and PDS

Concluding remarks and Acknowledgments