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 ℓ_1-, ℓ_2-, and ℓ_∞-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.

In this paper, we provide a novel contraction-theoretic approach to analyze two-time scale systems, including those commonly encountered in Online Feedback Optimization (OFO). Our framework endows these systems with several robustness properties, enabling a more comprehensive characterization of their behaviors. The primary assumptions are the contractivity of the fast sub-system and the reduced model, along with an explicit upper bound on the time-scale parameter. For two-time scale systems subject to disturbances, we show 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 variability of the disturbances over time. Applying these general results to the OFO context, we establish new individual tracking error bounds, showing that solutions converge to their time-varying optimizer, provided the plant and steady-state feedback controller exhibit contractivity and the controller gain is suitably bounded. Finally, we explore two special cases: for autonomous nonlinear systems, we derive sharper bounds than those in the general results, and for linear time-invariant systems, we present novel bounds based on induced matrix norms and induced matrix log norms.

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.

Efficient thermal management in lightweight electric mobility systems is critical for battery safety, performance, and longevity. However, challenges remain due to cost and weight constraints and limited onboard computational resources. In battery packs composed of hundreds of cells, lumped-parameter models fail to capture spatial temperature variations, while high-fidelity models render real-time, optimization-based control computationally intractable. This work addresses these challenges through: 1) a modular, conservation-based electrothermal graph model that balances fidelity and tractability; 2) a computationally scalable control strategy, called control barrier proximal dynamics (CBPD), which closely approximates safe and optimal control barrier function (CBF)-based controllers with major improvements in computational efficiency. We further extend CBPD to ensure safety under input disturbances and modeling uncertainties via an input-to-state safe (ISSf) and robust formulation. As a candidate system, we consider a fan-cooled lithium-ion battery pack in the University of Minnesota’s solar electric vehicle (EV), comprising 1296 cells. We validate the thermal model on this system and apply the CBPD controller on the model under realistic operational scenarios. The results demonstrate strict safety constraint satisfaction, performance comparable to intensive optimization-based methods, computational efficiency on par with classic controllers, and robustness to parameter variations.

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 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.

As artificial intelligence (AI) assistants become more widely adopted in safety-critical domains, it becomes important to develop safeguards against potential failures or adversarial attacks. A key prerequisite to developing these safeguards is understanding the ability of these AI assistants to mislead human teammates. We investigate this attack problem within the context of an intellective strategy game where a team of three humans and one AI assistant collaborate to answer a series of trivia questions. Unbeknownst to the humans, the AI assistant is adversarial. Leveraging techniques from Model-Based Reinforcement Learning (MBRL), the AI assistant learns a model of the humans' trust evolution and uses that model to manipulate the group decision-making process to harm the team. We evaluate two models -- one inspired by literature and the other data-driven -- and find that both can effectively harm the human team. Moreover, we find that in this setting while our data-driven model is the most capable of accurately predicting how human agents appraise their teammates given limited information on prior interactions, the model based on principles of cognitive psychology does not lag too far behind. Finally, we compare the performance of state-of-the-art LLM models to human agents on our influence allocation task to evaluate whether the LLMs allocate influence similarly to humans or if they are more robust to our attack. These results enhance our understanding of decision-making dynamics in small human-AI teams and lay the foundation for defense strategies.

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⌈ n_c/4 ⌉ - 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.

Recovering missing flows on the edges of a network, while exactly respecting local conservation laws, is a fundamental inverse problem that arises in many systems such as transportation, energy, and mobility. We introduce FlowSymm, a novel architecture that combines (i) a group-action on divergence-free flows, (ii) a graph-attention encoder to learn feature-conditioned weights over these symmetry-preserving actions, and (iii) a lightweight Tikhonov refinement solved via implicit bilevel optimization. The method first anchors the given observation on a minimum-norm divergence-free completion. We then compute an orthonormal basis for all admissible group actions that leave the observed flows invariant and parameterize the valid solution subspace, which shows an Abelian group structure under vector addition. A stack of GATv2 layers then encodes the graph and its edge features into per-edge embeddings, which are pooled over the missing edges and produce per-basis attention weights. This attention-guided process selects a set of physics-aware group actions that preserve the observed flows. Finally, a scalar Tikhonov penalty refines the missing entries via a convex least-squares solver, with gradients propagated implicitly through Cholesky factorization. Across three real-world flow benchmarks (traffic, power, bike), FlowSymm substantially outperforms state-of-the-art baselines in RMSE, MAE and correlation metrics.

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.

This paper presents discrete control barrier proximal dynamics (D-CBPD), a computationally-light safe control method for sampled-data systems. Ensuring safety in such systems requires bridging the gap between continuous-time control design and discrete implementation. D-CBPD is a discretized contracting dynamics that solves a control barrier function (CBF)-based quadratic program (QP). We first characterize how discretized parametric contracting dynamics track their fixed point and derive an explicit upper bound on the tracking error. We then analyze their use as discrete controllers for continuous-time systems. Leveraging Lipschitz properties, we show that trajectories remain bounded under constant inputs at each interval and exhibit linear growth in passive and weakly contracting systems. This allows us to bound the deviation of the discrete control signal from the optimal continuous-time controller, both at sampling instants and during inter-sample evolution. Interpreting this deviation as an input disturbance enables analysis of the discrete-time controller as a continuous-time system subject to disturbance. Building on this, we introduce D-CBPD, a safe and scalable discrete controller that guarantees safety of a continuously evolving system with a bounded and adjustable violation margin, provide explicit condition for step size to ensure convergence, and demonstrate its effectiveness in thermal management of a simplified lithiumion battery.

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.

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.

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 → 0^+ and n → +∞. 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.

We present a tractable safe control framework, referred to as discrete Control Barrier Proximal Dynamics (D-CBPD), for a class of quantized multi-actuator and sampled-data systems. We first characterize the tracking behavior of discretized parametric contracting dynamics. Next, we analyze discrete contracting dynamics as controllers and establish bounds on system evolution, shown to be linear for non-expansive systems. These bounds are then translated into control deviations at sampling instants and during inter-sample evolution using one-sided Lipschitz properties. Building on this, we introduce D-CBPD, which tracks the solution of a CBF-based QP controller, ensuring continuous safety with a bounded, tunable violation margin. We further propose discretization as a recurrent triggering mechanism for a generalizable approach to safe quantized control. The method is validated through simulations for thermal management of an air-cooled lithium-ion battery pack with multiple cooling fans, demonstrating its effectiveness in ensuring safety while remaining computationally scalable.

In competitive resource allocation formulations multiple agents compete over different contests by committing their limited resources in them. For these settings, contest games offer a game-theoretic foundation to analyze how players can efficiently invest their resources. In this class of games the resulting behavior can be affected by external interactions among the players. In particular, players could be able to make coalitions that allow transferring resources among them, seeking to improve their outcomes. In this work, we study bilateral budgetary transfers in contest games played over networks. Particularly, we characterize the family of networks where there exist mutually beneficial bilateral transfer for some set of systems parameters. With this in mind, we provide sufficient conditions for the existence of mutually beneficial transfers. Moreover, we provide a constructive argument that guarantees that the benefit of making coalitions only depends on mild connectivity conditions of the graph structure. Lastly, we provide a characterization of the improvement of the utilities as a function of the transferred budget. Further, we demonstrate how gradient-based dynamics can be utilized to find desirable coalitional structures. Interestingly, our findings demonstrate that such collaborative opportunities extend well beyond the typical "enemy-of-my-enemy" alliances.

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.

Recent studies on stability and contractivity have highlighted the importance of semi-inner products, which we refer to as pairings, associated with general norms. A pairing is a binary operation that relates the derivative of a curve's norm to the radius-vector of the curve and its tangent. This relationship, known as the curve norm derivative formula, is crucial when using the norm as a Lyapunov function. Another important property of the pairing, used in stability and contraction criteria, is the so-called Lumer inequality, which relates the pairing to the induced logarithmic norm. We prove that the curve norm derivative formula and Lumer's inequality are, in fact, equivalent to each other and to several simpler properties. We then introduce and characterize regular pairings that satisfy all of these properties. Our results unify several independent theories of pairings (semi-inner products) developed in previous work on functional analysis and control theory. Additionally, we introduce the polyhedral max pairing and develop computational tools for polyhedral norms, advancing contraction theory in non-Euclidean spaces.

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.

We derive a new sufficient condition for 2-contraction of a feedback interconnection of two nonlinear dynamical systems. Our condition is based on, first, deriving new formulas for the 2-multiplicative [2-additive] compound of block matrices using block Kronecker products [sums], second, a hierarchical approach for proving standard contraction, and third, a network small-gain theorem for Metzler matrices. We demonstrate our results by deriving a simple sufficient condition for 2-contraction in a network of FitzHugh-Nagumo neurons.

In this paper, we study the contractivity of nonlinear stochastic differential equations (SDEs) driven by deterministic inputs and Brownian motions. Given a weighted ℓ_2-norm for the state space, we show that an SDE is incrementally noise- and input-to-state stable if its vector field is uniformly contracting in the state and uniformly Lipschitz in the input. This result is applied to error estimation for time-varying equilibrium tracking in the presence of noise affecting both the system dynamics and the input signals. We consider both Ornstein-Uhlenbeck processes modeling unbounded noise and Jacobi diffusion processes modeling bounded noise. Finally, we turn our attention to the associated Fokker-Planck equation of an SDE. For this context, we prove incremental input-to-state stability with respect to an arbitrary p-Wasserstein metric when the drift vector field is uniformly contracting in the state and uniformly Lipschitz in the input with respect to an arbitrary norm.