A phase diagram that will appear in Monday's lecture

The following will appear in an example in Eric's last optimal control lecture. (Just with a different letter for y.) That is, the maximum principle will lead to this system. So try to wrap your head a bit around the system itself first.

Let r be a constant, 0<r<1. Sketch a phase diagram for the system
$LaTeX: \dot x=(1-x)x+\min\{y-1,0\} \\ \dot y=2y\cdot\big(x-\frac{1-r}2\big)$ for the part of the plane where $LaTeX: 0\leq x\leq 1,\quad 0\leq y\leq 3/2.$

Note, the min{y-1,0} is zero above the line y=1 and y-1 below that line. You might want to consider the x-nullcline separately below this line and above this line. Or maybe even you want to first consider first an example with x'=(1-x)x+y-1, and patch up with the correct example afterwards.

An "enjoy a challenge" from Eric

The following is an adaptation of a "enjoy a challenge" from Eric last year. You are not asked to draw any phase diagram (but the solution to be posted, will show you one). System: $LaTeX: \dot x=(y+\tfrac15x)(1-x^2)$ and $LaTeX: \dot y=-x(1-y^2)$ .

State the Jacobian matrix.
Find the seven stationary states.
Which ones are saddle points?
(Hint: remember the "quickly stated" criterion for saddle points.)

About 8-06:

Look at 8-06, which you have solved as a variational problem.

Rewrite as an optimal control problem.
(Hint: call the adjoint/costate variable µ in place of p, since the "p" letter is already taken.)
Write out the necessary conditions from the maximum principle.

About 8-01:

Look at 8-01, which you have solved as a variational problem.

Rewrite as an optimal control problem and state the necessary conditions both in cases (a) x(1)=1 (as the problem says), (b) x(1) ≥ 1, and (c) x(1) free.
The conditions lead to a differential equation system. Deduce that system. (You are not asked to solve it, but it won't hurt you.)
[Vague question, but you can handle that for a seminar]
So you got three alternative problems, one for each of those conditions on x(1); what difference do they make on how to solve?

Compendium 9-01

(You can assume that the problem has a solution.)

Compendium 9-05

From Exam 2003:

We are at problem 4, the optimal control. But a bit of refreshing differential equations does not hurt, hence those from problem 2.

2(a)
2(b), but don't use the ordinary "Math 2" approach: first, solve the corresponding homogeneous equation, and then guess a particular solution (hint: it isn't complicated!) and then put things together.
Problem 4
Consider Problem 4 again:
- Rewrite it as a variational problem.
- State the Euler equation.
- Can you solve it using Math 3?

The following problem:

Consider the following problem (which resembles the "1e" video)
$LaTeX: \displaystyle \max_{u(t)\in [0,3]}\int_{2020}^{2021}e^{-\delta t}u(t)\: dt$ where $LaTeX: \dot x=-u,\quad x(2020)=5,\quad x(2021)\geq 4.$

State the conditions from the maximum principle.
Show that if these conditions hold, then
[semi-vague question] Was the optimal control determined by your usual first-order condition?
[super-vague question] Is the optimal control determined by a maybe less usual first-order condition?