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:

1. $M$ is positive definite.
2. All eigenvalues of $M$ are positive.
3. $M=N^{T}N$ 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

$$\begin{array}{rl}{\dot{x}}_1& =-x_1^3+e^{-x_2}{x}_2\\ {\dot{x}}_2& =x_1-x_1^2\end{array}$$

Problem 3

Consider the system $${\dot{x}}_1=x_2,{\dot{x}}_2=-x_2-\sin x_1$$ and the three candidate Lyapunov functions

$$\begin{array}{rl}{V}_1(x)& ={\displaystyle \frac{1}{2}}{x}_2^2-\cos x_1,\\ V_2(x)& =1+{\displaystyle \frac{1}{2}}{x}_2^2-\cos x_1,\\ V_3(x)& =1+{\displaystyle \frac{1}{4}}{x}_1^2+{\displaystyle \frac{1}{2}}{x}_1{x}_2+{\displaystyle \frac{1}{2}}{x}_2^2-\cos x_1\end{array}$$

1. For $i=1,2,3$, compute ${\dot{V}}_i(x)$.
2. In each case, explain what conclusions (if any) you can reach using Lyapunov’s second method.

Problem 4

Consider the system $$\begin{array}{rl}{\dot{x}}_1& =x_1-x_1^3+x_2\\ {\dot{x}}_2& =3x_1-x_2\end{array}$$

1. Find all equilibrium points of the system.
2. Comment on the stability of each equilibrium point (you may use linearization or Lyapunov functions)
3. 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.