Nonlinear Control Sy.. - Free

134 CHAPTER 4. LYAPUNOV STABILITY II: NONAUTONOMOUS SYSTEMS
(iii) W3(x, t) = (xi + x2)e-t.
(iv) W4(x, t) = (xi + x2)(1 + e-t).
(V) W5(x, t) = (xi + x2) cos2 wt.
(vi) W6(x,t) = (xi + x2)(1 + cost wt).
(Vii) W7(x,t) =
(x2+x2)(l+e t).
xl{'1)
(4.5) It is known that a given dynamical system has an equilibrium point at the origin.
For this system, a function has been proposed, and its derivative has been
computed. Assuming that and are given below you are asked to classify the
origin, in each case, as (a) stable, (b) locally uniformly asymptotically stable, and/or
(c) globally uniformly asymptotically stable. Explain you answer in each case.
(i) W1 (X, t) = (xl + x2), W1(x, t) = -xi.
(ii) W2 (X, t) = (xi + x2), Wz(x, t) = -(xl + x2)e-t.
(iii) W3(x, t) = (xi + x2), W3(x, t) = -(xl + x2)et.
(iv) W4 (X, t) = (xi + x2)et, W4 (X, t) _ -(xl + x2)(1 + sine t).
(v) W5(x, t) = (xl + x2)e t W'5 (X, t) _ -(xi + x2)
(vi) W6 (X, t) = (xi + x2)(1 + e-t), W6 (x, t) _ -xle-t.
(Vii) W7(x, t) = (xi + x2) cos2 wt, W7(x, t) _ -(xi + x2).
(viii) W8 (X, t) _ (xl + x2)(1 + cos2 wt), 4Vs(x, t) _ -xi.
(ix) W9(x, t) = (xi +x2)(1 + cos2 wt), Ws(x, t) = -(x2 + x2)e-t.
(x) Wio(x,t) = (x2 + x2)(1 + cos2 wt), W10 (x,t) _ -(xi + x2)(1 + e-t).
(xi) W11(x, t) =
(x2+ )+1 , W11(x) t) _
(x1 + x2).
(4.6) Given the following system, study the stability of the equilibrium point at the origin:
(4.7) Prove Theorem 4.10.
(4.8) Prove Theorem 4.11.
21 = -x1 - x1x2 cos2 t
22 = -x2 - X1X2 sin2 t
(4.9) Given the following system, study the stability of the equilibrium point at the origin: