### Optimization and Systems Theory Seminar

Friday, March 16, 2001, 11.00-12.00, Room 3721, Lindstedtsvägen 25

**Leonid Engelson **

Centre for Traffic Simulation Research

KTH

E-mail:
lee@infra.kth.se

####
Projected dynamical systems with delays: Definition, stability
and applications

The classical dynamical systems theory assumes that the right-hand
side of the ordinary differential equation is continuous and therefore
can not handle problems with constraints.
Seven years ago, Anna Nagurney introduced Projected Dynamical Systems
that recognise the constrained state space. She proved basic existence
and uniqueness theorems and developed methods for stability analysis
of such systems. The results were applied to investigate dynamic
properties of equilibrium problems in economics and transportation.

We extend some of Nagurney'ss results to dynamical systems with
delays. The right-hand side of the delay equation is not continuous
due to the projection operator that forces the system to stay within
the feasible set. Stationary points of the equation coincide with
solutions to the associated variational inequality problem. We show
that stability of a regular stationary point can be investigated via a
lower dimension ordinary dynamical equation with delay. Possible
applications of the stability criterion are illustrated by an example
of traffic network with a reactive route guidance system.

Calendar of seminars

*Last update: March 6, 2001 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*