Suivi de thèse Riccardo (1ère année)

[jbcaillau]

Point du 25/10/2024

@rdaluiso1

• regarding optimal control, have a look at the web page of the M2 course I'm giving right now, and the references therein: https://github.com/pns-mam/commande/
• more specifically, look at chapter 4 of the Evans notes below https://math.berkeley.edu/~evans/control.course.pdf
• also have a look at the more general book (in French) below, in particular chapter 5, 6, 7 and 9 https://github.com/jbcaillau/jbcaillau.github.io/blob/main/tmp/trelat-2015a.pdf
• Complex Days: submission deadline = 2/12/2024

[jbcaillau]

Point du 07/11/2024

@rdaluiso1

• for existence, look at Lee & Marcus chap. 4 and Agrachev & Sachkov chap. 10.3
• regarding measure theory (for Rademecher, e.g., look at Rudin chap. 8), please have a look at these references from the BSc course I am giving currently
• as for KKT (first order necessary conditions for optimality - actually for min or max, that is extremality), I like the following form (without constraint qualification):

Theorem. Let $\bar{x}$ be a local minimiser of $f$ under the constraint $h(x) = 0$ where $f : E \to \mathbf{R}$ and $h : E \to \mathbf{R}^m$ are smooth functions. Then there exists $(\lambda^0,\lambda) \neq (0,0)$ in $\mathbf{R} \times \mathbf{R}^m$ such that $\nabla_x L(\bar{x},\lambda^0,\lambda) = 0$ where $L$ is the Lagrangian

$$ L(x,\lambda^0,\lambda) := \lambda^0 f(x) + (\lambda | h(x)). $$

As a corollary, you get that $\lambda^0$ cannot be zero if $h$ is a submersion at $\bar{x}$.

• [x] @jbcaillau plan a groupe de travail (= working group) with colleagues in Sophia where you can tell us about Levi-Civita regularisation
• next meeting: is it OK for you to meet on Tuesday at 17:30 (instead of 14:00)?

[rdaluiso1]

Thank you for the references and the remarks. Well received the message, fixed the meeting on Tuesday at 17:30. I notice there is an interesting theorem on the Trélat notes about the existence of the optimal control (Theorem 6.2.1), maybe we can discuss about it on Tuesday. Points:

• [x] bounded functions in $L^1$ admit a subsequence convergent for the weak-topology in $L^\infty$
• [x] last inequality

[jbcaillau]

Hi @rdaluiso1; well seen for the book of Trélat, I did not remember there was an alternative proof for the general case. Regarding weak convergence: bounded sets in $L^\infty$ are weakly-$*$ compact (hence the existence of converging subsequences of bounded sequences in $L^\infty$ - a dual space, which is not the case of $L^1$).

NB. I've added the correct links to the references in the previous post, and added Lee & Markus.

[rdaluiso1]

Hi @jbcaillau, found another interesting reference of Rudin, Functional Analysis (1973). Chapter 3 is about convexity, Chapter 4 about duality (always in Banach spaces). There is also a proof of the Hahn-Banach theorem used in the proof of Trélat. I could read it well to conclude this part on the existence.

[jbcaillau]

@rdaluiso1 Yes, this is the "second Rudin book" I was telling you about. Actually nice, but not my favourite for functional analysis. L. Schwartz book is nicer, I think... but in French 🥲

Hi @jbcaillau, found another interesting reference of Rudin, Functional Analysis (1973). Chapter 3 is about convexity, Chapter 4 about duality (always in Banach spaces). There is also a proof of the Hahn-Banach theorem used in the proof of Trélat. I could read it well to conclude this part on the existence.

[jbcaillau]

@rdaluiso1 work group planned at Inria Sophia with JB Pomet and Lamberto dell'Elce next Tuesday, 19 November, at 14:30: please confirm it is OK for you! The idea is that you present in detail the Levi-Civita regularisation, revisited by / with explanations from Alain. There will a (white) board to write stuff.

[rdaluiso1]

@jbcaillau Fine, perfect for me! Thank you for the advice.

[rdaluiso1]

Hi @jbcaillau, as regards the poster, the deadline is on Monday, 2th December, should we fix a meeting in these days to organize it?

[rdaluiso1]

Hi @jbcaillau, one stupid remark about the correction of Albouy to the L-C regularisation (I should have thought about it before, write it below just as reminder for tomorrow). Given the equations of the 'regularised flow' without change of time we face to

 \dot \xi = \frac{\partial H}{\partial \varpi} = -\frac{\varpi}{2|\varpi|^4}\left(1-\frac{2}{|\xi|} \right) \qquad \dot \varpi =- \frac{\partial H}{\partial \xi} = - \frac{\xi}{|\varpi|^2 |\xi|^3} 

As pointed out last week, the flow is not defined as collision (i.e. when $|\varpi| = 0, |\xi| = 2$), since it lacks uniqueness. But by reparametrizing time and introducing the energy as a parameter we get

\xi' = -2H|\xi|\varpi \qquad \varpi' =- \frac{\xi}{|\xi|^2} 

So the remark at this point is simply that if the equations can be suitably modified in the context of the control problem, and we succeed with an auxiliary equation to control continuously the value of energy, we could obtain regularising equations. Also forgetting the other comment pointed out by Albouy, maybe this could be sufficient to obtain a regularisation which works for every energy level.

[rdaluiso1]

Other two small side comments:

• I started studying more in depth the Knauf regularization for the Kepler problem. Is very recent (2024 in the updated version) and it does not require fixing of energy. However, is very abstract. It makes use of symplectic geometry and sees the phase space as the cotangent bundle. Don't know whether it can be used for applications in optimal control
• I could implement in Julia the Jacobi-Maupertuis surface of the Kepler problem. Not a big result, I followed the embedding suggested by Moeckel as a surface of revolution, but it is one of my first small programs in Julia, and maybe with some work we could update it in some way for the optimal control. Attach below the plot for E = -1:

kepler_manifold.pdf

[jbcaillau]

Hi @rdaluiso1 ; interesting, thanks. On top these points to be discussed today:

• averaging / Finsler metric, see Section 1 of the paper below (a preliminary work) http://caillau.perso.math.cnrs.fr/research/uca-complex-2018.pdf
• abstract Complex days
• Heidelberg Forum Laureate
• check your registration

[jbcaillau]

Point du 26/11/2024

@rdaluiso1

• Gauss equations (check in Zarrouati's book)

• you can also have a look at these two papers Geometric optimal control of elliptic Keplerian orbits Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 4, 929-956 (with Bonnard, B.; Trélat, E.)

  Non-integrability of the minimum-time Kepler problem J. Geom. Phys. 132 (2018), 452-459 (with Orieux, M.; Combot, T.; Féjoz, J.)

• also: suitable Mathieu transform for the extremal flow

[rdaluiso1]

Complex_days.pdf

[jbcaillau]

Thanks @rdaluiso1 ; can you please send the LaTeX source?

Complex_days.pdf

[rdaluiso1]

P.S. I can't upload .tex file here, it's ok this code? otherwise I send you the tex via mail [...]

[jbcaillau]

@rdaluiso1 as discussed yesterday:

• interesting to be able to treat the case of a coplanar transfer from a collision trajectory $q_0 \wedge \dot{q}_0 = 0$ towards a point $q_f$ s.t. $q_f \wedge q_0 = 0$, after $q = 0$
• if existence of minimisers, symmetry implies existence of cut points
• also interesting to compute first conjugate (e.g. for $\dot{q}_f = 0$) or focal (free $\dot{q}_f$) point on such a trajectory
• another global phenomenon: for $|qf|$ large enough and fixed $\varepsilon \geq ||u||\infty$, if existence, several revolutions might be needed
• on Lie derivatives (Lie / Poisson brackets) and their role for controllability (orbit theorem) and extremal analysis, you can have a look at Agrachev & Sachkov book; also to Jurdjevic and Bonnard-Chyba

[jbcaillau]

P.S. I can't upload .tex file here, it's ok this code? otherwise I send you the tex via mail [...]

Thanks, here it is: https://github.com/ljad-cnrs/lcr/blob/main/complex-days-2025/abstract.pdf

[jbcaillau]

@rdaluiso1 i've read them complex days abstract

• nice 👍🏽
• just updated a few minor things and added alain and me at the end (joint work)
• would be good to add a few references (\bibliography)
• please check that the number or words (apart from references) is still OK and submit it

[rdaluiso1]

Thank you Jean-Baptiste, I add the references then send the abstract. As regards ADUM, can you please ask Catherine Briet whether the application is now OK?

[rdaluiso1]

Just an additional reminder pointed out today during the phd welcome day: I need to find a 'thesis monitoring committee' with two professors not in my group of research for the superivision of the thesis during the three years. Maybe we can discuss about it next tuesday!

P.S. J'amerai aussi commencer à écrire en francais, mais c'est encore plus difficile que le parler/comprendre :)

[jbcaillau]

Just an additional reminder pointed out today during the phd welcome day: I need to find a 'thesis monitoring committee' with two professors not in my group of research for the superivision of the thesis during the three years. Maybe we can discuss about it next tuesday!

P.S. J'amerai aussi commencer à écrire en francais, mais c'est encore plus difficile que le parler/comprendre :)

Sure. the committee needs to be set for the end of first year (september / october)

[rdaluiso1]

Hi, Catherine told me the file I submitted for the application on ADUM is incomplete because it lacks a funding justification. Do you know which file should I attach?