Synchronization of Stochastic Neural Oscillators Using Lyapunov Methods - Advanced Models of Neural Networks - page 175

Fig. 8.2 LaSalle's theorem:
C: invariant set, E C:
invariant set which satisfies
V.x/D 0, M E:invariant
set, which satisfies
V.x/D 0,andwhich
contains the limit points of
x.t/ 2 E, L C the set of limit
points of x.t/ 2 E

8.2.2.2 Convergence Analysis Using La Salle's Theorem

It has been shown that lim t!1 x.t/ D x and from Eq. ( 8.26 ) that each particle
will stay in a cycle C of center x and radius given by Eq. ( 8.27 ). The Lyapunov
function given by Eq. ( 8.23 ) is negative semi-definite, therefore asymptotic stability
cannot be guaranteed. It remains to make precise the area of convergence of each
particle in the cycle C of center x and radius . To this end, La Salle's theorem can

be employed [ 64 , 92 ].

La Salle's Theorem: Assume the autonomous system x D f.x/where f W D !
R n . Assume C D a compact set which is positively invariant with respect to
x D f.x/, i.e. if x.0/ 2 C ) x.t/ 2 C 8 t. Assume that V.x/ W D ! R is a
continuous and differentiable Lyapunov function such that V.x/ 0 for x 2 C, i.e.
V.x/is negative semi-definite in C. Denote by E the set of all points in C such that
V.x/D 0. Denote by M the largest invariant set in E and its boundary by L C , i.e.
for x.t/ 2 E W lim t!1 x.t/ D L C , or in other words L C is the positive limit set of
E(seeFig. 8.2 b). Then every solution x.t/ 2 C will converge to M as t !1.
La Salle's theorem is applicable to the case of the multi-particle system and helps
to describe more precisely the area round x to which the particle trajectories x i
will converge. A generalized Lyapunov function is introduced which is expected to
verify the stability analysis based on Eq. ( 8.26 ). It holds that