#1 Consider the nonlinear scalar differential equation
x'(t) =ax(t)+f(t,x(t)),x(to =ro =
where a is real and constant and the function f is continuous with f(t,0) = 0.
Assume that there is a constant X such that
If (t, x) < 1/x%
Use the Lyapunov function
to show that
a) the zero solution of (1) is stable provided that
b) the zero solution of (1) is uniformly asymptotically stable (UAS) provided a + I<
for some positive constant a.
c) Use part b) to show that the zero solution of
#2 Find a Lyapunov function to show the zero solution of
= - -
y' = - 2y + y (x2 + y2)
#3 Find a Lyapunov function to show the zero solution of
x = y =
is stable. Hint: try V(x,y) = ax4 + by², and then chose a and b, accordingly.
#4 Show the zero solution of
y==r-yf(x,y) = -
where f is continuously differentiable. Show that
a) if f(x,y) > 0, then the origin is A.S,
b) whereas if f(x,y) < 0 the origin is unstable.
Hint: Use appropriate polynomial Lyapunov function.
#5 Consider the nonhomogenous system
I' (t) =
Were A(t) and B(t) are continuous on 0 prove that if all solutions are
bounded, then they are stable.
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