arXiv subjects

Francesco Bullo

Publications and source records attributed to Francesco Bullo.

At least 19 recordsLinked to original sources

arXiv2026

Mean-Field Oscillator Ising Machines: Gradient Flows and Classification of Limit Solutions

Oscillator Ising Machines (OIMs) have emerged as promising computational architectures for approximating solutions to combinatorial optimization problems. We derive and analyze the mean-field limit of an OIM model and show that it inherits the gradient flow structure of the finite-dimensional dynamics. We identify conditions under which this mean-field evolution admits an Eulerian formulation as a gradient flow on the Wasserstein space of probability measures and contrast this with a Lagrangian formulation which is always available. This gradient flow structure strongly constrains the long-time dynamics and enables a complete classification of limit solutions and their stability in the totally symmetric case. In particular, all limit solutions are fixed points whose phases cluster into at most four groups, and for almost all parameter values, only binarized fixed points -- those with clusters at $0$ and/or $π$ -- can be stable. Since binarized states are exactly those for which a feasible solution to the original problem can be read out, this shows that feasible solutions can almost always be recovered. We provide tight bounds on the parameter thresholds for which fixed points in this binarized family are stable, thereby identifying the threshold for binarization in this model, and present numerical evidence that the mean-field model correctly predicts behavioral regimes in certain large random networks, including Erdős-Rényi networks. Moreover, this problem provides a case study in the application of mean-field models, gradient flows, and optimal transport to the study of large-scale systems, and we highlight a number of features of this problem which are informative in this respect.

math.OC↗

arXiv2026

Deceptive Stochastic Patrolling via Markov Chain Lifting

In this paper we propose lifted Markov Chains (MCs) as a new paradigm for deriving patrol strategies for mobile agents on an environment represented as a graph. Lifted MCs operate on a state space that can be larger than the set of nodes of the graph, and a projection operation maps the MC state to the corresponding node of the graph. We prove bounds on the performance improvement of lifted MCs over non-lifted MCs, where the performance is measured by the Kemeny constant, and show a corresponding monotonicity for the Stackelberg game capture probability and return-time entropy. These bounds yield a result of independent interest that relates the Kemeny constant and the MC conductance for general MCs. We also provide a tractable method for optimizing lifted MCs on general graphs with edge weights encoding travel times. Simulation results on randomly generated and real-world graphs verify that lifted MCs achieve substantial improvements, particularly on sparse graphs.

eess.SY↗

arXiv2026

An Energy-Based Mechanism for Compositional Behavior

Flexible intelligence relies on the ability to reuse previously acquired behaviors and combine them differently as circumstances change. In biological and artificial systems, this ability is often attributed to gating mechanisms that determine how much each available behavior should contribute at a given time. Yet these gating rules, the dynamics that compute them, and the neural circuits that may implement them are usually introduced separately, leaving unclear whether they reflect a common underlying principle. Here, we show that they can all be derived from a single variational principle for behavioral composition. The resulting mechanism naturally gives rise to softmax gating, evolves as an energy-based dynamical system with guaranteed convergence, and admits a recurrent neural network instantiation featuring context-dependent and local interactions. Across collective behavior, human decision-making, and layered control, the same mechanism reproduces characteristic behavioral patterns, provides interpretable accounts of how different behaviors are combined, and matches or outperforms established approaches. These results provide a unified account of how behavioral composition can emerge from a common principle, with implications for understanding flexible behavior in natural systems and for designing artificial agents that can adapt by recombining existing capabilities.

math.OC↗

arXiv2026

Structured LLM Reasoning for Zero-Shot Human--Robot Coordination Under Hidden Goals

We present a structured large-language-model (LLM) architecture for zero-shot human--robot coordination in a cooperative construction task with private goal views. Guided by a Dec-POMDP formulation, the architecture decomposes decision-making into (i) action-conditioned Theory-of-Mind (ToM) inference, (ii) hierarchical planning, (iii) conversation interpretation, (iv) action verification, and (v) feedback-based replanning. We compare the proposed method with an ablation without ToM inference and a multi-agent reinforcement-learning policy trained offline over many goal pairs. In human-participant experiments, the proposed method required fewer interaction steps and yielded higher post-interaction trust ratings than both baselines. These results suggest that systematically decomposing the team decision problem, using LLMs as tractable surrogates for otherwise intractable inference and planning computations, and retaining conventional verification for physical feasibility can improve both task coordination and the human experience.

cs.RO↗

arXiv2026

Mind the Gap: Where Analog Ising Machines Cease to Minimize the Ising Hamiltonian

The design of nonlinear dynamical systems whose gradient flows minimize the Ising Hamiltonian has emerged as a compelling paradigm for realizing Ising machines, forming the foundation of architectures including coherent Ising machines, simulated bifurcation machines, oscillator-based Ising machines, and dynamical Ising machines. Here, we identify a fundamental structural feature shared by these systems a functional parameter gap defined by the separation between the destabilization of the trivial state and the stabilization of Ising-encoded states. We demonstrate that this separation creates a finite parameter interval in which convergence to an Ising-encoded solution is no longer functionally guaranteed, and the resulting evolution is dictated by the spectral structure of the Jacobian at bifurcation. Subsequently, by introducing a hybrid dynamical framework that reshapes the bifurcation topology, we establish a principled pathway for modulating this parameter gap. The parameter gap thus emerges as a unifying structural principle for the analysis, design and optimization of analog Ising machines.

physics.comp-ph↗

arXiv2026

Input-to-State Stability Implications in Contraction Theory

For nonlinear control systems on normed vector spaces, we characterize an incremental input-to-state stability (ISS) type property in which the overshoot constant multiplies both the initial-condition and the input terms. Working through the associated variational system, we show that two properties are equivalent: an ISS-type bound on the variational system, and the incremental ISS-type bound on the original system. We further establish the equivalence between an infinitesimal contraction condition, expressed through a Lyapunov-type function, and an incremental Lyapunov condition. Each of these equivalent conditions yields a necessary condition and a sufficient condition for the ISS-type bounds, differing only in the input Lipschitz constant of the vector field. When the overshoot constant equals one, the infinitesimal contraction condition reduces to the standard norm-based contraction conditions. We establish these implications under mere continuous differentiability of the vector field, and we illustrate the results through sensitivity matrices and Lyapunov characteristic exponents. Moreover, we develop similar implications for discrete-time systems.

eess.SY↗

arXiv2026

Sampled-data Systems: Stability, Contractivity and Single-iteration Suboptimal MPC

This paper analyzes the stability of interconnected continuous-time (CT) and discrete-time (DT) systems coupled through sampling and zero-order hold mechanisms. The DT system updates its output at regular intervals $T>0$ by applying an $n$-fold composition of a given map. This setup is motivated by online and sampled-data implementations of optimization-based controllers - particularly model predictive control (MPC) - where the DT system models $n$ iterations of an algorithm approximating the solution of an optimization problem. We introduce the concept of a reduced model, defined as the limiting behavior of the sampled-data system as $T \to 0^+$ and $n \to +\infty$. Our main theoretical contribution establishes that when the reduced model is contractive, there exists a threshold duration $T(n)$ for each iteration count $n$ such that the CT-DT interconnection achieves exponential stability for all sampling periods $T < T(n)$. Finally, under the stronger condition that both the CT and DT systems are contractive, we show exponential stability of their interconnection using a small-gain argument. Our theoretical results provide new insights into suboptimal MPC stability, showing that convergence guarantees hold even when using a single iteration of the optimization algorithm - a practically significant finding for real-time control applications.

eess.SY↗

arXiv2026

Regularized Model Predictive Control via Contractivity and Implicit Lur'e Analysis

This paper develops a contraction-based stability analysis for regularized model predictive control (MPC), whose feedback law is defined implicitly by a finite-horizon optimal control problem with an additional regularizing cost. The proposed approach interprets regularized MPC as an implicit Lur'e system, in which the regularizing cost perturbs the optimality conditions. We develop a multiplier-based contraction framework for implicit Lur'e systems and derive linear matrix inequality conditions for regularized MPC with three broad classes of regularizers: convex smooth stage costs, convex closed proper stage costs, and differentiable regularizers with Lipschitz gradients. Numerical studies on input and state soft penalties, hard input constraints, and sparsity-promoting penalties illustrate that regularization shapes closed-loop performance while retaining formal contraction-based stability guarantees.

math.OC↗

arXiv2026

Proximal Gradient Dynamics and Feedback Control for Equality-Constrained Composite Optimization

This paper studies equality-constrained composite minimization problems. This class of problems, capturing regularization terms and inequality constraints, naturally arises in a wide range of engineering and machine learning applications. To tackle these optimization problems, inspired by recent results, we introduce the \emph{proportional--integral proximal gradient dynamics} (PI--PGD): a closed-loop system where the Lagrange multipliers are control inputs and states are the problem decision variables. First, we establish the equivalence between the stationary points of the minimization problem and the equilibria of the PI--PGD. Then for the case of affine constraints, by leveraging tools from contraction theory we give a comprehensive convergence analysis for the dynamics, showing convergence to a stationary point. Moreover, under suitable assumptions, we show linear--exponential convergence towards the equilibrium. That is, the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. Our findings are illustrated numerically on a set of representative examples, which include an exploratory application to nonlinear equality constraints.

math.OC↗

arXiv2026

Predictive Coding with Bayesian Priors via Proximal Gradients

We recast predictive coding as continuous-time proximal gradient descent applied to a regularized maximum-a-posteriori (MAP) objective. We study first a single-level problem and then a multi-level hierarchy. For the single-level problem, we show that proximal gradient descent is precisely a leaky firing-rate network: the membrane leak, the effective recurrent matrix, the local synaptic drive, and the static nonlinearity all follow from one optimization principle, and the resulting circuit is the one proposed by Rao and Ballard. The prior selects the nonlinearity through its proximal operator, and the likelihood precision sets the gain on the observation. For the hierarchy, we show that a classical variable-splitting relaxation of the deep MAP problem yields hierarchical predictive coding as the interconnection of local and distributed solvers. In probabilistic modeling terms, this relaxation replaces the directed generative chain by an undirected Markov random field whose node potentials are the level-wise priors. Each level then applies its own activation function, namely the proximal operator of its prior.

eess.SY↗

arXiv2026

Competition, stability, and functionality in excitatory-inhibitory neural circuits

Energy-based models have become a central paradigm for understanding computation and stability in both theoretical neuroscience and machine learning. However, the energetic framework typically relies on symmetry in synaptic or weight matrices - a constraint that excludes biologically realistic systems such as excitatory-inhibitory (E-I) networks. When symmetry is relaxed, the classical notion of a global energy landscape fails, leaving the dynamics of asymmetric neural systems conceptually unanchored. In this work, we extend the energetic framework to asymmetric firing rate networks, revealing an underlying game-theoretic structure for the neural dynamics in which each neuron is an agent that seeks to minimize its own energy. In addition, we exploit rigorous stability principles from network theory to study regulation and balancing of neural activity in E-I networks. We combine the novel game-energetic interpretation and the stability results to revisit standard frameworks in theoretical neuroscience, such as the Wilson-Cowan and lateral inhibition models. These insights allow us to study cortical columns of lateral inhibition microcircuits as contrast enhancer - with the ability to selectively sharpen subtle differences in the environment through hierarchical excitation-inhibition interplay. Our results bridge energetic and game-theoretic views of neural computation, offering a pathway toward the systematic engineering of biologically grounded, dynamically stable neural architectures.

q-bio.NC↗

arXiv2026

A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning

This paper establishes a nonlinear separation principle based on contraction theory and derives sharp stability conditions for recurrent neural networks (RNNs). First, we introduce a nonlinear separation principle that guarantees global exponential stability for the interconnection of a contracting state-feedback controller and a contracting observer, alongside parametric extensions for robustness and equilibrium tracking. Second, we derive sharp linear matrix inequality (LMI) conditions that guarantee the contractivity of both firing rate and Hopfield neural network architectures. We establish structural relationships among these certificates-demonstrating that continuous-time models with monotone non-decreasing activations maximize the admissible weight space-and extend these stability guarantees to interconnected systems and Graph RNNs. Third, we combine our separation principle and LMI framework to solve the output reference tracking problem for RNN-modeled plants. We provide LMI synthesis methods for feedback controllers and observers, and rigorously design a low-gain integral controller to eliminate steady-state error. Finally, we derive an exact, unconstrained algebraic parameterization of our contraction LMIs to design highly expressive implicit neural networks, achieving competitive accuracy and parameter efficiency on standard image classification benchmarks.

eess.SY↗

arXiv2026

Timescale Limits of Linear-Threshold Networks

Linear-threshold networks (LTNs) capture the mesoscale behavior of interacting populations of neurons and are of particular interest to control theorists due to their dynamical richness and relative ease of analysis. The aim of this paper is to advance the study of global asymptotic stability in LTNs with asymmetric neural interactions and heterogeneous dissipation under the structural Lyapunov diagonal stability (LDS) condition. To this end, we introduce a one-parameter family of LTNs that preserves the LDS condition and has a parameter-independent equilibrium set. In the fast limit, this family converges to a projected dynamical system (PDS), while in the slow limit, it converges to a discontinuous hard-selector system (HSS). Under LDS, we prove that the fast PDS limit is globally exponentially stable and that the HSS limit is globally asymptotically stable. This alignment suggests that the limiting systems capture essential mechanisms governing stability across the entire LTN family. Together with numerical evidence, these findings indicate that resolving stability at the fast and slow endpoints provides a promising and structurally grounded path toward establishing global stability for LTNs with biologically plausible recurrence and diagonal dissipation.

eess.SY↗

arXiv2026

Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization

Recent advances at the intersection of control theory, neuroscience, and machine learning have revealed novel mechanisms by which dynamical systems perform computation. These advances encompass a wide range of conceptual, mathematical, and computational ideas, with applications for model learning and training, memory retrieval, data-driven control, and optimization. This tutorial focuses on neuro-inspired approaches to computation that aim to improve scalability, robustness, and energy efficiency across such tasks, bridging the gap between artificial and biological systems. Particular emphasis is placed on energy-based dynamical models that encode information through gradient flows and energy landscapes. We begin by reviewing classical formulations, such as continuous-time Hopfield networks and Boltzmann machines, and then extend the framework to modern developments. These include dense associative memory models for high-capacity storage, oscillator-based networks for large-scale optimization, and proximal-descent dynamics for composite and constrained reconstruction. The tutorial demonstrates how control-theoretic principles can guide the design of next-generation neurocomputing systems, steering the discussion beyond conventional feedforward and backpropagation-based approaches to artificial intelligence.

cs.LG↗

arXiv2026

Oscillator-Based Associative Memory with Exponential Capacity: Theory, Algorithms, and Hardware Implementation

Associative memory systems enable content-addressable storage and retrieval of patterns, a capability central to biological neural computation and artificial intelligence. Classical implementations such as Hopfield networks face fundamental limitations in memory capacity, scaling at most linearly with network size. We present an associative memory architecture based on Kuramoto oscillator networks with honeycomb topology in which memories are encoded as stable phase-locked configurations. The honeycomb network consists of multiple cycles that share nodes in a chain-like arrangement, creating a one-dimensional lattice of chained+loops. We prove that this architecture achieves exponential memory capacity: a network of $N$ oscillators can store $(2\lceil n_c/4 \rceil - 1)^m$ distinct patterns, where $m$ honeycomb cycles each contain $n_c$ oscillators. Moreover, we fully characterize all stable configurations and prove that each memory's basin of attraction maintains a guaranteed minimum size independent of network scale. Simulations using charge-density-wave (CDW) oscillators validate predicted phase-locking behavior, demonstrating practical realizability in neuromorphic hardware.

cs.NE↗

arXiv2026

Contracting Neural Networks: Sharp LMI Conditions with Applications to Integral Control and Deep Learning

This paper studies contractivity of firing-rate and Hopfield recurrent neural networks. We derive sharp LMI conditions on the synaptic matrices that characterize contractivity of both architectures, for activation functions that are either non-expansive or monotone non-expansive, in both continuous and discrete time. We establish structural relationships among these conditions, including connections to Schur diagonal stability and the recovery of optimal contraction rates for symmetric synaptic matrices. We demonstrate the utility of these results through two applications. First, we develop an LMI-based design procedure for low-gain integral controllers enabling reference tracking in contracting firing rate networks. Second, we provide an exact parameterization of weight matrices that guarantee contraction and use it to improve the expressivity of Implicit Neural Networks, achieving competitive performance on image classification benchmarks with fewer parameters.

eess.SY↗

arXiv2026

Detection of adversarial intent in Human-AI teams using LLMs

Large language models (LLMs) are increasingly deployed in human-AI teams as support agents for complex tasks such as information retrieval, programming, and decision-making assistance. While these agents' autonomy and contextual knowledge enables them to be useful, it also exposes them to a broad range of attacks, including data poisoning, prompt injection, and even prompt engineering. Through these attack vectors, malicious actors can manipulate an LLM agent to provide harmful information, potentially manipulating human agents to make harmful decisions. While prior work has focused on LLMs as attack targets or adversarial actors, this paper studies their potential role as defensive supervisors within mixed human-AI teams. Using a dataset consisting of multi-party conversations and decisions for a real human-AI team over a 25 round horizon, we formulate the problem of malicious behavior detection from interaction traces. We find that LLMs are capable of identifying malicious behavior in real-time, and without task-specific information, indicating the potential for task-agnostic defense. Moreover, we find that the malicious behavior of interest is not easily identified using simple heuristics, further suggesting the introduction of LLM defenders could render human teams more robust to certain classes of attack.

cs.LG↗

arXiv2026

Contractivity of Multi-Stage Runge-Kutta Dynamics

Many control, optimization, and learning algorithms rely on discretizations of continuous-time contracting systems, where preservation of contractivity under numerical integration is key for stability, robustness, and reliable fixed-point computation. In this paper, we establish conditions under which multi-stage Runge-Kutta methods preserve strong contractivity when discretizing infinitesimally contractive continuous-time systems. For explicit Runge-Kutta methods, preservation conditions are derived by bounding Lipschitz constants of the associated composite stage mappings, leading to coefficient-dependent criteria. For implicit methods, the algebraic structure of the stage equations enables explicit conditions on the Runge-Kutta coefficients that guarantee preservation of strong contractivity. In the implicit case, these results extend classical guarantees, typically limited to weak contractivity in the Euclidean metric, to strong contractivity with respect to the $\ell_1$-, $\ell_2$-, and $\ell_\infty$-norms. In addition, we study well-definedness of implicit methods through an auxiliary continuous-time system associated with the stage equations. We show that strong infinitesimal contractivity of this auxiliary system is sufficient to guarantee unique solvability of the stage equations. This analysis generalizes standard well-definedness conditions and provides a dynamic implementation approach that avoids direct solution of the implicit algebraic equations.

eess.SY↗