Nonlinear Control Sy.. - Free

106 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
(3.15) Given the following system, discuss the stability of the equilibrium point at the origin:
X2
xix2 + 2x1x2 + xi
_ -X3 + x2
(3.16) (Lagrange stability) Consider the following notion of stability:
Definition 3.14 The equilibrium point x = 0 of the system (3.1) is said to be bounded
or Lagrange stable if there exist a bound A such that
Prove the following theorem
lx(t)II0.
Theorem 3.13 [49] (Lagrange stability theorem) Let fl be a bounded neighborhood of
the origin and let 1 be its complement. Assume that V(x) : R^ -* R be continuously
differentiable in S2' and satisfying:
(i) V(x) > 0 VxE52C.
(i) V(x) < 0 Vx E Q`.
(i) V is radially unbounded.
Then the equilibrium point at the origin is Lagrange stable.
Notes and References
Good sources for the material of this chapter are References [48], [27], [41] [88] [68]
and [95] among others. The proof of theorem 3.1 is based on Reference [32]. Section 3.7
as well as lemmas 3.4 and 3.5 follow closely the presentation in Reference [95]. Section
3.8 is based on LaSalle, [49], and Khalil, [41]. The beautiful Example 3.20 was taken from
Reference [68].