Homework 5 - Due: 02/20
Problem 1
Let \(M\) be a symmetric real-valued \(n\times n\) matrix. Show that the following three statements are equivalent:
- \(M\) is positive definite.
- All eigenvalues of \(M\) are positive.
- \(M=N^TN\) for some nonsingular \(n\times n\) matrix \(N\).
Hint: Show \(a\Rightarrow b\Rightarrow c\Rightarrow a\).
Problem 2
Investigate asymptotic stability of the origin for the system
\[\begin{align*} \dot x_1&=-x_1^3+e^{-x_2}x_2\\ \dot x_2&=x_1-x_1^2 \end{align*}\]
Problem 3
Consider the system \[ \dot x_1=x_2,\qquad \dot x_2=-x_2-\sin x_1\] and the three candidate Lyapunov functions
\[\begin{align*} V_1(x)&=\dfrac{1}{2} x_2^2-\cos x_1,\\ V_2(x)&=1+\dfrac{1}{2} x_2^2-\cos x_1,\\ V_3(x)&=1+\dfrac{1}{4} x_1^2+ \dfrac{1}{2} x_1x_2+\dfrac{1}{2} x_2^2-\cos x_1 \end{align*}\]
- For \(i=1,2,3\), compute \(\dot V_i(x)\).
- In each case, explain what conclusions (if any) you can reach using Lyapunov’s second method.
Problem 4
Consider the system \[\begin{align*} \dot x_1 &= x_1 - x_1^3 +x_2 \\ \dot x_2 &= 3x_1 - x_2 \end{align*}\]
- Find all equilibrium points of the system.
- Comment on the stability of each equilibrium point (you may use linearization or Lyapunov functions)
- Using some CAS (Mathematica, Python, etc.) construct the phase portrait of the system and comment on the region of attractivity (if applicable) of each equilibrium.