Add tutorial on regularization

[YouriRenoud]

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[YouriRenoud]

@ocots @jbcaillau I tried various ways to initialize the orbital transfer problem, but I couldn't find one where the logarithm doesn't become negative. I also tried to start the indirect resolution by following the Kepler example, but I'm not sure if that's the right approach.

[ocots]

@YouriRenoud if you have some difficulties you can switch to this issue: #28

See here: https://control-toolbox.org/Tutorials.jl/stable/tutorial-iss.html#Resolution-of-the-shooting-equation

[ocots]

@YouriRenoud pour aider à la convergence ne pas hésiter à jouer sur la taille de la discrétisation. Mettre 100, 500, 1000, 2000 points pour voir.

[jbcaillau]

@YouriRenoud pour initialiser :

• résoudre le pb en temps min par méthode directe pour une poussée assez forte, par ex. T = 60 N, comme au début de ce tuto
• cette solution est admissible pour un problème conso. min. avec barrière log et T > 60 N ; essayer avec T = c * 60, c = 1.1 (pas de pb pour le log, du coup, puisqu'on est à l'intérieur de la contrainte)
• pour la conso. min, le temps final doit être fixé (pas de solution sinon) et strictement supérieur au temps minimum : tu peux donc fixer ce temps final au temps min trouvé pour T = 60 (puisque pour T > 60 le temps min correspondant est strictement plus petit)

[YouriRenoud]

@ocots @jbcaillau I finally could solve the minimal conso problem what are you thinking of this solution ?

[ocots]

Can you plot the norm of the control? See the plot documentation.

You can plot also in 3D, see the end of the Kepler application.

[YouriRenoud]

Here the norm of the control :

https://github-production-user-asset-6210df.s3.amazonaws.com/168093425/463251937-d77c9d6d-3402-4955-b5c3-f655b8177fc4.PNG?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAVCODYLSA53PQK4ZA%2F20250707%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20250707T134126Z&X-Amz-Expires=300&X-Amz-Signature=b80d74a832bc15ca94c7bf8d8cf8bb25544dea16b738bec708f7667b4e3c9256&X-Amz-SignedHeaders=host

[YouriRenoud]

For the 3D plot I only get one revolution before it stops ?

[jbcaillau]

Can you plot the norm of the control? See the plot documentation.

You can plot also in 3D, see the end of the Kepler application.

hi @YouriRenoud cannot see the plot of the norm. for a strong enough log penalty and a large enough final time (20% larger than min time, e.g.), you should see a control subarc close to a zero arc (= zero norm)

[YouriRenoud]

Here the norm

[ocots]

Todo:

• Continuation sur $\varepsilon$ : direct et indirect.
• Méthode indirecte sur le problème à consommation minimale.

[YouriRenoud]

@ocots @jbcaillau Here the norm of the control with the continuation of epsilon at the last iteration :

[jbcaillau]

@ocots @jbcaillau Here the norm of the control with the continuation of epsilon at the last iteration :

nice @YouriRenoud can you please add (on the same plot) control norms for previous values of $\varepsilon$? compare Fig. 1 from this ref.

[ocots]

@YouriRenoud Can you please make a point on what you still have to do?

[YouriRenoud]

@ocots The direct part is all finished and working but I am stuck in the indirect resolution and I don't know how to solve my problems. For example if I start two times the indirect resolution without changing anything the two solution won't give the same trajectory and it is not even close.

Maybe I should start the other part of regularization with the Malisani's code ?

[ocots]

@ocots The direct part is all finished and working but I am stuck in the indirect resolution and I don't know how to solve my problems. For example if I start two times the indirect resolution without changing anything the two solution won't give the same trajectory and it is not even close.

Maybe I should start the other part of regularization with the Malisani's code ?

@YouriRenoud Actually, writing the shooting function for the minimum consumption problem is not easy. Since, I won't be there before you finish, I think you should switch to Malisani's regularization. You can create another file for the tutorial.

[YouriRenoud]

@jbcaillau I could plot this figure is that what you were expecting ?

[jbcaillau]

@YouriRenoud exactly: very nice! 👍🏽