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

2005 ◽  
Vol 3 (3) ◽  
pp. 251-262 ◽  
Author(s):  
Monica-Gabriela Cojocaru
Keyword(s):  
Dynamical Systems ◽  
Variational Inequalities ◽  
Dynamic Problem ◽  
Obstacle Problem ◽  
Direct Application ◽  
Projected Dynamical Systems ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
Elastic String

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.

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

scholarly journals Nonpivot and Implicit Projected Dynamical Systems on Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Monica Gabriela Cojocaru ◽  
Stephane Pia
Keyword(s):  
Dynamical Systems ◽  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Dual Space ◽  
Variational Problems ◽  
Base Space ◽  
Projected Dynamical Systems

This paper presents a generalization of the concept and uses of projected dynamical systems to the case of nonpivot Hilbert spaces. These are Hilbert spaces in which the topological dual space is not identified with the base space. The generalization consists of showing the existence of such systems and their relation to variational problems, such as variational inequalities. In the case of usual Hilbert spaces these systems have been extensively studied, and, as in previous works, this new generalization has been motivated by applications, as shown below.

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