Elements of Lyapunov Theory - Nonlinear Oligopolies: Stability and Bifurcations - page 275

Asymptotically stable

ε

1

x

x (0)

D

ε

Globally
asymptotically stable

Stable

Fig. A.1 Stability concepts

then the equilibrium is (locally) asymptotically stable. Global asymptotic stabil-
ity occurs if with any initial state x .0/ 2 D , the state trajectory converges to the

equilibrium as t !1 . Figure A.1 illustrates these concepts.
Assume now that

x is an equilibrium of the system (A.1) or (A.2) and let be a

subset of D such that x 2 .

Definition A.4. A real valued function V defined in is called a Lyapunov function

if is satisfies the following conditions:

(a) is continuous on ;

(b) The global minimum of

V

V on occurs at the equilibrium x ;

(c) For any state trajectory x .t/ contained in , V . x .t// is non-increasing in t.

Notice that the Lyapunov function concept is a straightforward generalization of
that of the energy function in mechanical systems. Condition (a) requires that V

has no discontinuities, (b) means that V has its smallest value at the equilibrium,