Nonlinear Control Sy.. - Free

90 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
Thus, V(x) > 0 and V(x) < 0 since V(x) = 0 for x = (xii0). Proceeding as in the previous
example, we assume that V = 0 and conclude that
V=0 x2=0, x2=0 Vt 22=0
X2=0 = -x2 - a X1 - (x1 + x2)2x2 = 0
and considering the fact that x2 = 0, the last equation implies that xl = 0. It follows that
V (x) does not vanish identically along any solution other than x = [0, 0]T . Moreover, since
V(.) is radially unbounded, we conclude that the origin is globally asymptotically stable.
Theorem 3.8 (LaSalle's theorem) Let V : D -* R be a continuously differentiable function
and assume that
(i) M C D is a compact set, invariant with respect to the solutions of (3.1).
(ii) V < 0 in M.
(iii) E : {x : x E M, and V = 0}; that is, E is the set of all points of M such that V = 0.
(iv) N: is the largest invariant set in E.
Then every solution starting in M approaches N as t -+ oo.
Proof: Consider a solution x(t) of (3.1) starting in M. Since V(x) < 0 E M, V(x) is a
decreasing function of t. Also, since V(.) is a continuous function, it is bounded from below
in the compact set M. It follows that V(x(t)) has a limit as t -+ oo. Let w be the limit
set of this trajectory. It follows that w C M since M is (an invariant) closed set. For any
p E w3 a sequence to with to -+ oo and x(tn) -+ p. By continuity of V(x), we have that
V(p) = lim V(x(tn)) = a (a constant).
n- oo
Hence, V (x) = a on w. Also, by Lemma 3.5 w is an invariant set, and moreover V (x) = 0
on w (since V(x) is constant on w). It follows that
wCNCEcM.
Since x(t) is bounded, Lemma 3.4 implies that x(t) approaches w (its positive limit set) as
t -+ oo. Hence x(t) approaches N as t -> oo.
Remarks: LaSalle's theorem goes beyond the Lyapunov stability theorems in two important
aspects. In the first place, V(-) is required to be continuously differentiable (and so