Nonlinear Control Sy.. - Free

3.12. EXERCISES
(3.8) Prove the following properties of class IC and K, functions:
(i) If a : [0, a) -> IRE K, then a-1 : [0, a(a)) --> R E K.
(ii) If al, a2 E K, then al o a2 E K.
(iii) If a E Kim, then a-1 E K.
(iv) If al, a2 E Kim, then a1 o a2 E K.
(3.9) Consider the system defined by the following equations:
xl
2 = -x1
(X3
x2+Q3 1 -xl
Study the stability of the equilibrium point xe = (0, 0), in the following cases:
(i) /3 > 0.
(ii) /3 = 0.
(iii) Q < 0.
(3.10) Provide a detailed proof of Theorem 3.7.
(3.11) Provide a detailed proof of Corollary 3.1.
(3.12) Provide a detailed proof of Corollary 3.2.
(3.13) Consider the system defined by the following equations:
Il = x2
12 = -x2 - axl - (2x2 + 3x1)2x2
Study the stability of the equilibrium point xe = (0, 0).
(3.14) Consider the system defined by the following equations:
(i)
2
11 = 3x2
±2 = -xl + x2(1 - 3x1 - 2x2)
Show that the points defined by (a) x = (0, 0) and (b) 1 - (3x1 + 2x2) = 0 are
invariant sets.
(ii) Study the stability of the origin x = (0, 0).
(iii) Study the stability of the invariant set 1 - (3x1 + 2x2) = 0.
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