Full-Day Workshop, in conjunction with the 2024 Conference on Decision and Control in Milano, Italy

Organizer: Francesco Bullo, UC Santa Barbara

Date: Sunday, December 15, 2024

Time: 8:30-17:30, Central European Time

Location: to be announced

The workshop will feature numerous talks from worldwide leading scientists on multiple state-of-the-art topic on contraction theory.

Aminzare, Zahra, University of Iowa, USA (Confirmed)

Angeli, David, Imperial College, UK (Confirmed)

Astolfi, Daniele, Université de Lyon, France

Bullo, Francesco, UC Santa Barbara, USA (Confirmed)

Dall'Anese, Emiliano, Boston University, USA (Confirmed)

Giesl, Peter, University of Sussex, UK (Confirmed)

Kawano, Yu, Hiroshima University, Japan (Confirmed)

Manchester, Ian, University of Sydney, Australia

Margaliot, Michael, Tel Aviv University, Israel (Confirmed)

Proskurnikov, Anton, Politecnico di Torino, Italy (Confirmed)

Russo, Giovanni, Universita di Salerno, Italy (Confirmed, morning speaker 11-12:30)

Sepulchre, Rodolphe, Cambridge University, UK (Confirmed)

Slotine, Jean-Jacques, MIT, USA (Confirmed)

Sontag, Eduardo D., NorthEastern University, USA (Confirmed)

Recent research has increasingly focused on applying the Banach contraction principle in the broad area of systems and control. Similarly, these tools are essential for addressing key problems in machine learning and dynamical neuroscience. Contracting dynamical systems inherently offer numerous safety and stability guarantees. Additionally, the theory of monotone operators, which is being applied more frequently, serves as an important complement to these theoretical tools.

The workshop will feature an extensive list of presentations by leading scientists from around the world on:

The foundations of contraction theory

Theoretical developments for complex networks, including advances in synchronization and scalability

Computational advances in the design of contraction metrics and contracting dynamical systems for solving optimization problems

Applications to machine learning, planning, and robust control

Of particular interest to the CDC audience will be findings on robust stability analysis and control design for both deterministic and stochastic systems, as well as formal robustness and stability guarantees for various learning-based control problems.

This workshop will bring together experts from diverse backgrounds to discuss recent theoretical and computational advances, identify emerging challenges, and explore rapidly-developing application opportunities. It should appeal to both junior and senior researchers interested in systems, control, and learning. The control community's interest in these topics is evidenced by recent well-attended events, including a tutorial session at the 2021 IEEE CDC and a pre-conference workshop at the 2023 ACC.

Each talk is 30 minutes + 10 minutes for Q&A and open discussion. All times are listed in Central European Time.

08:30 to 10:30: Workshop (120 minutes)

10:30 to 11:00: Coffee break

11:00 to 12:30: Workshop (90 minutes)

12:30 to 13:30: Lunch break

13:30 to 15:30: Workshop (120 minutes)

15:30 to 16:00: Coffee break

16:00 to 17:30: Workshop (90 minutes)

There will be 14 presentations, each lasting a total of 30 minutes. This time includes 20 minutes for the talk, 5 minutes for questions, and 5 minutes for preparations and informal conversations. Altogether, the scientific part of the meeting will span 7 hours, divided into four segments: 120 minutes, 90 minutes, 120 minutes, and 90 minutes.

Tutorial session: “Contraction Theory for Machine Learning” (URL with PDFs and youtube videos) at the 2021 IEEE CDC conference, by Soon-Jo Chung (sjchung@caltech.edu) and Jean-Jacques Slotine (jjs@mit.edu) with help from Hiroyasu Tsukamoto (htsukamoto@caltech.edu)

Tutorial paper at CDC2021 “Contraction-Based Methods for Stable Identification and Robust Machine Learning: a Tutorial” by Ian Manchester and coauthors: Arxiv and IEEEXplore

Plenary presentation: “Contraction Theory in Systems and Control” (PDF) by Francesco Bullo at the 2022 IEEE CDC Conference

Youtube lectures: “Lectures on Nonlinear Systems” by Jean-Jacques Slotine, Fall 2013: Lectures 14-20 (approximately 1h20min each)

Youtube lectures: “Minicourse on Contraction Theory” by Francesco Bullo, Fall 2022–present: introductory slides (PDF) and youtube lectures (12h in 6 lectures, with chapters)

Textbook: Contraction Theory for Dynamical Systems, Francesco Bullo, rev 1.2, Aug 2024. (Book and slides freely available)

Speaker: **Zahra Aminzare**, University of Iowa, USA

Title: **Transverse Contraction in Oscillatory Autonomous Systems: Extending to Lp Norms**

Abstract: Dynamic systems with periodic solutions play crucial roles
across disciplines like ecology, neuroscience, and engineering. Despite the
effectiveness of classical contraction theory in dynamical system analysis,
its application to oscillatory autonomous systems has been
limited. Contraction to limit cycles, initially explored by Borg in 1960
and further investigated by Manchester and Slotine in 2014 (Transverse
Contraction) as well as Forni and Sepulchre in 2014 (Horizontal
Contraction), emerges as a powerful approach for assessing the existence,
stability, and robustness of limit cycles. This talk provides a
comprehensive overview of existing findings regarding contraction to
periodic solutions, primarily employing L2 norms, Riemannian metric, and
Finsler metric. Additionally, novel extensions to Lp norms will be
introduced. This is a joint project with Saber Jafarpour.

Speaker: **Angeli, David**, Imperial College, UK

Title: **On 2-Contraction and Non-Oscillatory Systems: Some Theory and Applications**

Abstract: The notion of 2-contraction arises when considering pairs of
infinitesimal perturbations in initial conditions with respect to a nominal
trajectory, and following the propagation in time of the surface area
spanned by these in all projected two-dimensional quadrants. If these
propagations contracts in some norm, then the system is 2-contractive and
this prevents existence of oscillatory behaviours in the original nonlinear
dynamics. We will review some of the definitions and main results,
including non-existence of periodic or almost periodic solutions, a priori
bound on Lyapunov exponents (no chaos), and applications of such results to
convergence analysis.

Speaker: **Astolfi, Daniele**, Université de Lyon, France

Title: **Contraction Theory of Output Regulation**

Abstract: The problem of output regulation consists in tracking desired references while rejecting disturbances affecting the system dynamics. A possible manner to solve such a problem relies on the internal model principle, i.e. by incorporating in the controller a copy of the model generating such references*disturbances. For instance, in the context of constant references*perturbations, an integral action is needed. Once the integral action is added, then one needs to stabilize the extended cascade system in an appropriate manner so that closed-loop trajectories possess an equilibrium in the presence of external constant signals. One of the main difficulties in the nonlinear context is precisely such a step. In this talk we will show how contraction theory plays a fundamental role and we propose an extension of the forwarding method used to stabilize cascade forms in such an incremental context. The proposed paradigm is applied in the context of integral action and harmonic regulation for the rejection of periodic signals.

Speaker: **Francesco Bullo**, UC Santa Barbara, USA

Coworkers: **Alexander Davydov, Veronica Centorrino, Giovanni Russo, Anand Gokhale**

Title: **Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach**

Abstract: This talk presents a novel and broadly-applicable contraction-theoretic approach to continuous-time time-varying convex optimization. For any parameter-dependent contracting dynamics, we show that the
tracking error is asymptotically proportional to the rate of change of the parameter with proportionality constant upper bounded by Lipschitz constant in which the parameter appears divided by the contraction rate of the dynamics squared.
We additionally establish that any parameter-dependent contracting dynamics can be augmented with a feedforward prediction term to ensure that the tracking error converges to zero exponentially quickly.
To apply these results to time-varying convex optimization problems, we establish the strong infinitesimal contractivity of dynamics solving three canonical problems, namely monotone inclusions, linear equality-constrained problems, and composite minimization problems. For each of these problems, we prove the sharpest-known rates of contraction and provide explicit tracking error bounds between solution trajectories and minimizing trajectories. We validate our theoretical results on numerical examples including an application to control-barrier function based controller design.

Speaker: **Dall'Anese, Emiliano**, Boston University, USA

Title: **A Contraction-Theoretic Approach for the Analysis of Two-time Scale Systems: Theory and Application to Online Feedback Optimization**

Coworkers: **Lily Cothren and Francesco Bullo**
Abstract: The talk presents a novel approach for the analysis of two-time
scale systems using contraction theory. Mirroring the setup used in
classical Lyapunov-based approaches, key assumptions on the two-time-scale
system pertain to the contractivity of the fast sub-system and of the
so-called reduced model. For two-time scale systems subject to disturbances
or exogenous inputs, the first result shows that the distance between
solutions of the nominal system and solutions of its reduced model is
uniformly upper bounded by a function of contraction rates, Lipschitz
constants, the time-scale parameter, and the time variability of the
disturbances. Secondly, results in terms of local contractivity of the
two-time scale system and sufficient conditions for global contractivity
are presented. Finally, the contraction-theoretic analysis is applied to
online feedback optimization methods; in this case, the
contraction-theoretic approach allows one to obtain new tracking error
bounds showing that solutions converge to their (time-varying) optimizer,
with explicit computation of the overall contraction rates.

Speaker: **Giesl, Peter**, University of Sussex, UK

Title: **Computation of Contraction Metrics with Meshfree Collocation**

Abstract: In this talk, we consider a general dynamical system given by
solutions of an autonomous ordinary differential equation. The long-term
behaviour can be characterised by attractors and their corresponding basins
of attraction. Examples of attractors are equilibria or periodic orbits,
and their respective basins of attraction consist of all initial points
such that the corresponding solutions converge towards them.

A contraction metric is a metric such that the distance between adjacent solutions decreases with respect to the metric, and thus they share the same long-term behaviour. Hence, a contraction metric is a tool to determine the basin of attraction of an attractor without the need to know its location. A contraction metric can be described by a matrix-valued function, which is a symmetric and positive definite matrix at every point.

We will discuss the numerical construction of contraction metrics for the case of an equilibrium and a periodic orbit using meshfree collocation with Radial Basis Functions. First, we will establish existence results for contraction metrics which satisfy a certain partial differential equation (PDE), and then we will use meshfree collocation to compute an approximate solution of the PDE. Error estimates ensure that the approximation, if sufficiently close, is itself a contraction metric.

Much of this is joint work with Sigurdur Hafstein and Holger Wendland.

Speaker: **Kawano, Yu**, Hiroshima University, Japan

Title: **Youla-Kucera Parametrization in the Contraction Framework**

Coworkers: **Arjan J. van der Schaft (University of Groningen), Jacquelien M.A. Scherpen (University of Groningen)**
Abstract: In this talk, we study incrementally exponentially stable (IES)
image and kernel representations for nonlinear systems with the aim of
generalizing the Youla-Kucera parametrization in the contraction
framework. We first construct them and their stable inverses in the
contraction framework and then provide a parametrization of stabilizing
controllers by additionally assuming incremental input-to-state stability
for the image representation. Focusing on constant metrics results in a
parametrization of all stabilizing controllers rendering the closed-loop
systems IES with respect to constant metrics.

Speaker: **Ian R. Manchester**, University of Sydney, Australia

Title: **A Robust Learning Framework built on Contraction and Monotonicity**

Abstract: This talk will discuss an emerging set of tools allowing precise control of behavioural properties of neural network models. Neural networks have enormous representational capacity, but in many applications it is important to ensure properties such as robustness, dynamical stability, energy dissipation, invertibility of models, solvability of optimization models, and others. We will describe a new framework that builds on notions of contracting systems and monotone operators, and present novel links to the concepts of Koopman operators for dynamical systems theory, KKL observers, Polyak-Lojasiewicz functions in optimization theory, and Hamiltonian mechanics.

Speaker: **Margaliot, Michael**, Tel Aviv University, Israel

Title: **Compound Matrices and Dynamical Systems**

Abstract: Compound matrices have found applications in combinatorics,
graph theory, multilinear algebra, and ordinary differential
equations. More recently, they were used to generalize important classes of
dynamical systems including contractive systems and cooperative systems. We
describe these applications as well as several promising research
directions.

Speaker: **Anton Proskurnikov**, Politecnico di Torino, Italy

Coworkers: **Alexander Davydov and Francesco Bullo**

Title: **Towards Non-quadratic Absolute Stability Theory**

Abstract: Classical absolute stability and contraction theory, originating from Lur'e and Postnikov, relies on quadratic Lyapunov functions. Their existence is typically ensured by Yakubovich's S-lemma and LMI feasibility, traditionally validated via semidefinite programming or the Kalman-Yakubovich-Popov lemma. A quadratic Lyapunov function is essentially a squared Euclidean norm (associated with an inner product). Thus, the classical absolute stability/contractivity problem requires finding a robust estimate for some Euclidean norm of a solution (or the deviation between two solutions) in the presence of uncertain nonlinearities.

While the Euclidean norm is standard in control theory, it may be inconvenient for higher-dimensional systems like recurrent neural networks. In some machine learning problems, estimating the Manhattan (L1) or L-inf norm is necessary. Although formally equivalent, different norms differ by multiplicative constants that grow with dimensionality. Consequently, estimates based solely on Euclidean norms and quadratic Lyapunov functions can be overly conservative.

New mathematical tools introduced by Davydov et al. within the framework of contraction analysis open up the possibility for a “non-quadratic” absolute stability theory, providing less conservative estimates for non-Euclidean norms of the solutions. This talk gives a brief introduction to this new direction of research and presents some open problems.

Speaker: **Giovanni Russo**, Universita’ di Salerno, Italy

Title: **Towards Contracting Biologically Plausible Neural Networks**

Coworkers: **Veronica Centorrino, Francesco Bullo, Alexander Davydov, Anand Gokhale**

Abstract: In this talk, we will present a normative framework for the
design of continuous-time biologically plausible neural networks to tackle
sparse reconstruction problems. These problems, which naturally arise in
the context of e.g., signal processing and neuroscience, can be formalized
as regularized least squares optimization and involve approximating (e.g.,
sensory) inputs with just a few active neurons given an available
dictionary. First, we link the optimal solution of these problems to the
equilibria of a suitably defined proximal gradient dynamics. Then, using
contraction theory, we show that this dynamics exhibits a
linear-exponential convergence property under a standard assumption on the
dictionary. We conclude by discussing the integration of Hebbian
learning. Throughout the talk, the results are illustrated via numerical
examples.

Speaker: **Rodolphe Sepulchre**, Cambridge University, UK

Title: **Regulation Without Calibration**

Abstract: The internal model principle suggests that perfect regulation
requires an exact internal copy of the exosystem. How to reconcile this
calibration principle with systems made of uncertain and variable
components? The talk will suggest that machines that synchronize events
rather than trajectories considerably release the calibration requirements
of regulation. I will highlight the role of excitability and synaptic
coupling to synchronize the events of uncertain trajectories.

Speaker: **Jean-Jacques Slotine**, MIT, USA

Title: **A Quarter Century of Contraction Analysis**

Abstract: It has been more than a quarter century since the paper in
Automatica by Lohmiller and Slotine introduced contraction analysis to the
nonlinear control community, outlining the role of differential analysis
using state-dependent Riemannian metrics and its many potential
applications. Research in this domain is now very active, and we will
review basics as well as some recent work in our group on applications to
machine learning and to non-autonomous partial differential equations.

Speaker: **Eduardo Sontag**, Northeastern University, USA

Title: **Contractions for Interaction Networks**

Abstract: This talk will focus on the dynamics of “interaction networks” which follow the formalism of chemical reaction networks (CRN’s). We will present results on the construction of polyhedral norms that provide contraction, as well as study relations to monotone systems. Some of these norms provide only non-expansivity, as opposed to strict contraction, so we also study omega-limit sets of non-expansive systems, showing that they tend to be well-behaved. Much of this work has been jointly performed with M. Ali Al-Radhawi, David Angeli, and Alon Duvall.

After covering costs and taxes, all remaining proceeds from the workshop will be donated to the Graduate Student Fellowship Fund of the Mechanical Engineering Department at UC Santa Barbara.