Karthik Elamvazhuthi

The generative modeling for control lectures are proceeding in nonlinear order. Here is one on a control generalization of gradient descent that I refer to as nonholonomic gradient descent. I explain why it's doomed to fail due to Brockett, and how noise might save us.

hackmd.io/pTtk6ohHSBSf...

Encoding geometry in neural network architectures has wide applications, from robots' configuration spaces to safety. We present natural ways to enforce constraints by design.

Link to preprint: www.researchgate.net/publication/...

#geometricdeeplearning

Designing the optimal slide for a lion, of course.

Also curious if you are going to be rightfully referring to the maximum principle as the Pontryagin–Boltyanskii–Gamkrelidze–Mishchenko Maximum Principle it time you mention it. 😆

A bit of a propaganda talk I gave online, recently at the University of Washington, Control-X seminar. Thanks to Amirhossein Taghvaei for the invitation.

Link to recording: washington.zoom.us/rec/share/ig...

What is a good source for this? And is there a non-finance analogue of arbitrage opportunities that motivates this?

All hail the Lie bracket 🙌

I’m writing notes on how generative modeling connects with control theory.
Starting with intuition: simulations, nonlinear control, and how stochastic dynamics can shape long-term behavior.

LN1 (Fokker–Planck): hackmd.io/@clopenloop/...
LN2 (Nonholonomic Fokker-Planck): hackmd.io/@clopenloop/...

The road to the maximum principle turns out to be more non-smooth than one might expect.
Boltyanski, Martini & Soltan’s book ( Geometric Methods and Optimization Problems) has far more drama than you’d guess from the Sussmann–Willems survey.

Job season in academia can take a toll on confidence, identity, and energy. You’re not alone.

Sometimes it helps to zoom out.

Hunter Wapman’s thesis highlights some interesting hiring patterns:
www.hne.golf/static/pdfs/...

Non-US trends may differ, perhaps shaping distinct knowledge bubbles.

I suspect a case of chaos or non-uniqueness. 🧐

Multi-agent control often feels like control theorists doing dynamical systems instead of control theory. What if we posed it like classic control problems.

Collective decision-making as feedback stabilizability. Turns out, it leads to a strange kind of control problem.

hackmd.io/@clopenloop/...

Wowza!
arxiv.org/html/2510.07...

It’s hard enough following Stroock when he’s explaining his own perspective.😵‍💫

A basin of attraction

(PDF) Flow Matching for Measure Transport and Feedback Stabilization of Control-Affine Systems PDF | We develop a flow-matching framework for transporting probability measures under control-affine dynamics and for stabilizing systems to points or... | Find, read and cite all the research you ne...

We adapt flow matching to measure transport in control systems.

We also introduce a control-oriented version of denoising. No classical white noise needed.
Uncertainty flows through the (sub-Riemannian) exponential map or via regular (but random) controls.

www.researchgate.net/publication/...

An interesting result on how LQ problems in optimal control can fail to have minimizers: arxiv.org/pdf/1311.200... (in an almost merciless way).

Naive random sampling of initial conditions and control.

Optimal Transport (OT) is emerging as a versatile tool for control questions. We show an OT-based way to sample from the reachable set (RS), especially useful when systems have strong attractors.

www.researchgate.net/publication/...

RS of Van der Pol vs. naive random sampling (below).

YouTube video by Automatic Control Engineering Network (ACE) ACE AI and Data Talk Series-Dr. Karthik Elamvazhuthi

Finally, my YouTube debut! A recording of my talk at the AI and data seminar series organized by the Automatic Control Engineering Network.
m.youtube.com/watch?v=o-4B...

Here I pontificate on using time reversals of diffusions for stabilization and planning problems.

I am starting to compile some notes on Generative Modeling for Control Theory. The goal is to include topics such as degenerate Fokker-Planck equations (FPEs) and Optimal transport theory. Here is a first post on the classical (nicer) FPE and its long-term behavior.

hackmd.io/@clopenloop/...

That’s an amazing stat 😵

Fun tangential fact: The idea of velocity-averaging in the flow matching can be found in Lacker's notes on mean-field control. He refers to it as "mimicking".

Probably a good chapter if it starts with Nietzsche.

- Tanja Eisner, Bálint Farkas, Markus Haase and
Rainer Nagel (Operator Theoretic Aspects
of Ergodic Theory)

I like how some inequalities were derived for fun. 😄

Universal Approximation of Mean-Field Models via Transformers This paper investigates the use of transformers to approximate the mean-field dynamics of interacting particle systems exhibiting collective behavior. Such systems are fundamental in modeling phenomen...

Permutation equivariance is a common property of multi-agent systems. In a new paper with Shiba Biswal and
@rishisonthalia.bsky.social
‬, accepted to ICML, we leverage this property shared by transformers, for mean field models of collective behavior.

#machinelearning

arxiv.org/abs/2410.16295

From Combinatorics to Partial Differential Equations The optimal matching of point clouds in $\mathbb{R}^d$ is a combinatorial problem; applications in statistics motivate to consider random point clouds, like the Poisson point process. There is a cruci...

Nice set of notes connecting the finite particle matching problem to the continuum optimal transport solution.

arxiv.org/abs/2505.10175

Analysis of Heat Equations on Domains. (LMS-31)

A reference for Markov semigroups that I have found extremely useful.
press.princeton.edu/books/hardco...

People are talking about controls in Bluesky?! Where?

Nice timeline! Adjoint equations arose in optimal control of finite dimensional ODE systems initially due to Pontryagin and coworkers extending methods of calculus of variations. Lions popularized them for infinite dimensional systems like PDEs.

It’s known that the Laplacian operator, describing how heat flows through a domain, carries rich information about the domain’s underlying geometry. Similarly, one can understand the geometry of a control system using its sub-Laplacian.

See

comptes-rendus.academie-sciences.fr/mathematique...