Hope you are well and safe
The following question is to find the shortest distance between 2 points avoiding a circle between them using the Maximum principle (see photo 1 & photo 2).
the solutions are divided into 3 cases, before the obstacle (the circle), touching the circle and finally leaving the circle.
Our Hamiltonian Equation is $H(x,p,a) = p(a) -1 = p^1a^1 + p^2a^2 -1$ and that because we are is $R^2$ , all and our dynamic is $x` = a$. all of these detail are shown in photo 2.
in photo 2: the showed the case 1 (before the obstacle) and they started with case 2.
My problem is in case 2: I was able to get all the 5 equations he mentioned, but at end of the page he says: "we introduce the angle theta as illustrated (in photo 4), and note that $\frac{d}{d \theta} = r \frac{d}{dt}$. A calculation then confirms that the solutions are...". The obtained solutions at the end of photo 3 are wrong, and I tried to get them right but I couldn`t.
MY Goal: I need the calculations to get the right solutions for the 5 unknows (the red box in photo 3). it is an algebra problem more than a concept of optimal control theory. if you have any intuition or suggestion please tell me.
photo 1: https://i.stack.imgur.com/BkGwn.jpg
photo 2:https://i.stack.imgur.com/jeeiq.jpg
If the line connecting the two points passes through the circle, then the shortest path avoiding the circle will look like the blue path in this image:
Can you see why the shortest path can only contain straight lines and arcs of the circle? Hint: if the path contains a curved line which is not bordering the circle, you can tweak it into a shorter path.
To see why the shortest path connects the points to the circle using tangent lines, look at the image below. The path along the blue line is shorter than the path along the red line and green arc.