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.