### Optimization and Systems Theory Seminar

Friday, October 6, 2000, 11.00-12.00, Room 3721, Lindstedtsv. 25

**Professor Anders
Rantzer**

Department of Automatic Control

Lund Institute of Technology

Lund, Sweden

E-mail: rantzer@control.lth.se

####
On stability and convergence of nonlinear systems

Lyapunov's second theorem is a standard tool for stability analysis of
nonlinear ordinary differential equations. In this lecture, we discuss
a theorem which can be viewed as a dual to Lyapunov's
result. Assuming existence of a scalar function satisfying certain
inequalities it is possible to prove that "almost
all trajectories" of the system tend to zero.
The scalar function has a physical interpretation as the stationary
density of a substance that is generated in all points of the state
space and flows along the system trajectories. If the stationary
density is bounded everywhere except at a singularity in the origin,
then almost all trajectories tend towards the origin.

The weaker notion of stability allows for applications also in
situations where Lyapunov's theorem can not be used. Moreover, the use
of density functions has advantages in control theory, for the
synthesis of stabilizing controllers. These issues will be addressed
in the seminar and the relation between density functions and Lyapunov
functions will be further explained.

Calendar of seminars

*Last update: October 2, 2000 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*