Homework 2

Due at 11:59pm on Jan 30th. Submit to https://canvas.uw.edu/courses/1862386/assignments/11123228.

Problem 1

Consider the system \(\dot{x}=A x\). Show that the nonnegative orthant (\(\mathbb{R}_+^n = \{x:x_i \geq 0 \; \forall i\}\)) is invariant if and only if \(A_{ij} \geq 0\) for all \(i \neq j\). That is, show that nonegative trajectories stay nonengative if and only if the off diagonal entries of \(A\) are nonnegative.

Problem 2

The system \(\dot{x}=Ax\) is called constant norm if for every trajectory, \(\Vert x(t) \Vert\) is constant, that is, it doesn’t depend on \(t\). The system is called constant speed if for every trajectory, \(\Vert \dot{x}(t)\Vert\) is constant.

a) Find the conditions on \(A\) under which the system is constant norm.

b) Find the conditions on \(A\) under which the system is constant speed.

c) Is every constant norm system a constant speed system?

d) Is every constant speed system a constant norm system?

Problem 3

Consider the nonlinear system

\(\ddot{x} = -g(x)\),

where \(g\) is some increasing function with \(g(0)=0\). Find a conserved quantity for this system. Hint: think back to the case of \(g(x)=x\), and see how the conserved quantity was constructed (think about integrals).

Problem 4

Compute the graident of the following functions:

a) \(f(x_1,x_2)= x_1 x_2 + x_1 \cos(x_2)\)

b) \(f(x)=\frac{1}{\Vert x \Vert} a^Tx\), where \(x\) is a vector in \(\mathbb{R}^n\) and \(a\) is a constant vector. To be tehnical, we can define \(f(0)=0\), although it doesn’t really matter in the gradident calculation.

Problem 5

Consider the single bus swing equation:

\[\begin{align} \dot{\theta} &=\omega \\ M \dot{\omega} &= -D \omega - B \theta \end{align}\]

Take \(M\), \(D\) and \(B\) to all be \(1\). We know the system is stable. But let’s say we have a more strigent requirement: the frequency should stay within \(\pm 0.1\). Find all initial points such that all the frequency is within this bound for all \(t\geq 0\).