Two point boundary problem - Shooting method

[RamosPinto]

I'm currently trying to solve the following two-point boundary problem by means of the shooting method:

To clarify, I'm investigating the optimal route of aerial/marine vehicles from one point to another point, considering a flow field.
* The starting and ending locations are set as x₀, y₀ and x_f, y_f respectively.
* v is the constant speed of the vehicle relative to the field.
* A vector v=[v_cx,v_cy]^T is used to describe the drift velocity of the field with respect to some coordinate system fixed to the ground.
* ψ is the vehicles navigation angle.
* The optimal change rate of the navigation angle has been found and is as follows:

There are two unknowns in this two-point boundary value problem, which are the initial navigation angle ψ(0) and t_f, which is the final time.

The problem that I would like some help with is thus the two-point boundary value problem that I want to solve by means of the shooting method. I haven't found useful and applicable sources that show me how to use the shooting method for this problem. I would very much appreciate a help in the right direction, either by some explanation or my directing me to useful sources.

Kind regards,

Ramos Pinto

 

[Chestermiller]

Why is it necessary to specify the final time?

 

[RamosPinto]

Why is it necessary to specify the final time?

First of all, thanks for your reply, Chestermiller!

The final time is unknown and can be anything. However, the initial goal is to minimize the final time, such that the most optimal routing is obtained.
This is the performance measure I want to minimize:

By means of Pontryagin's minimum principle I have obtained the rate of change of the optimal navigation angle, which I stated in my initial post. The next step is now to get the initial navigation angle ψ(0) by solving the two point boundary problem.

I hope this further clarifies my problem.

 

[Chestermiller]

Then what is being held constant, the total distance?

 

[RamosPinto]

Then what is being held constant, the total distance?

x₀, y₀, x_f, y_f are constant. These are the starting and ending location in a Cartesian plane.
The thrust speed of the vehicle, v, is also constant. I'm not exactly sure how to say this in English, but what I mean is that when the vehicle has a thrust speed of unit 2, and there is a flow field flowing in the opposite direction with unit 1, then the vehicle moves with a velocity of unit 1.

This is an example where you can see the starting and ending location, as well as the flow field that is present.

 

[Chestermiller]

Sorry, I still don't understand. Maybe someone else can help.

 

[RamosPinto]

Sorry, I still don't understand. Maybe someone else can help.

No problem, thanks anyway.
Hopefully someone else can help me

 

[SlowThinker]

The optimal change rate of the navigation angle has been found and is as follows

I can't help you with the math, but I don't think this is the optimal solution.
Imagine the starting point on an island. There is a stream of finite width, going around this island, say between 2 circles or radii $r_1$ and $r_2$, centered at the island. The destination is behind this stream.
I'm pretty sure that the optimal path starts at some angle "upstream" from the destination, continues without steering through the stream, and emerges just at the point nearest to the destination, where the ship continues, again without steering.
Your solution will clearly produce steering when entering and leaving the stream. Thus I have doubts about its optimality.

I'm not sure if the optimal path is always a "no steering" path, but I can't think of a counterexample. It is, however, possible that there are several "no steering" paths leading to the same destination, some longer than others.

 

[RamosPinto]

I can't help you with the math, but I don't think this is the optimal solution.
Imagine the starting point on an island. There is a stream of finite width, going around this island, say between 2 circles or radii $r_1$ and $r_2$, centered at the island. The destination is behind this stream.
I'm pretty sure that the optimal path starts at some angle "upstream" from the destination, continues without steering through the stream, and emerges just at the point nearest to the destination, where the ship continues, again without steering.
Your solution will clearly produce steering when entering and leaving the stream. Thus I have doubts about its optimality.

I'm not sure if the optimal path is always a "no steering" path, but I can't think of a counterexample. It is, however, possible that there are several "no steering" paths leading to the same destination, some longer than others.

Thanks for your answer!
The optimal path will have steering, also in the stream. I could go and show you the preliminary math, but then we'd go into part of the research that isn't a problem at this point. I'm struggling with solving the two point boundary problem and would very much appreciate any help regarding this issue.