Fabio Pasqualetti

Fabio Pasqualetti, Antonio Bicchi, Francesco Bullo

This work addresses the problem of ensuring trustworthy computation in a linear consensus network. A solution to this problem is relevant for several tasks in multi-agent systems including motion coordination, clock synchronization, and cooperative estimation. In a linear consensus network, we allow for the presence of misbehaving agents, whose behavior deviate from the nominal consensus evolution. We model misbehaviors as unknown and unmeasurable inputs affecting the network, and we cast the misbehavior detection and identification problem into an unknown-input system theoretic framework. We consider two extreme cases of misbehaving agents, namely faulty (non-colluding) and malicious (Byzantine) agents. First, we characterize the set of inputs that allow misbehaving agents to affect the consensus network while remaining undetected and/or unidentified from certain observing agents. Second, we provide worst-case bounds for the number of concurrent faulty or malicious agents that can be detected and identified. Precisely, the consensus network needs to be 2k+1 (resp. k+1) connected for k malicious (resp. faulty) agents to be generically detectable and identifiable by every well behaving agent. Third, we quantify the effect of undetectable inputs on the final consensus value. Fourth, we design three algorithms to detect and identify misbehaving agents. The first and the second algorithm apply fault detection techniques, and affords complete detection and identification if global knowledge of the network is available to each agent, at a high computational cost. The third algorithm is designed to exploit the presence in the network of weakly interconnected subparts, and provides local detection and identification of misbehaving agents whose behavior deviates more than a threshold, which is quantified in terms of the interconnection structure.

Fabio Pasqualetti, Florian Dörfler, Francesco Bullo

Future power networks will be characterized by safe and reliable functionality against physical malfunctions and cyber attacks. This paper proposes a unified framework and advanced monitoring procedures to detect and identify network components malfunction or measurements corruption caused by an omniscient adversary. We model a power system under cyber-physical attack as a linear time-invariant descriptor system with unknown inputs. Our attack model generalizes the prototypical stealth, (dynamic) false-data injection and replay attacks. We characterize the fundamental limitations of both static and dynamic procedures for attack detection and identification. Additionally, we design provably-correct (dynamic) detection and identification procedures based on tools from geometric control theory. Finally, we illustrate the effectiveness of our method through a comparison with existing (static) detection algorithms, and through a numerical study.

Fabio Pasqualetti, Antonio Franchi, Francesco Bullo

The subject of this work is the patrolling of an environment with the aid of a team of autonomous agents. We consider both the design of open-loop trajectories with optimal properties, and of distributed control laws converging to optimal trajectories. As performance criteria, the refresh time and the latency are considered, i.e., respectively, time gap between any two visits of the same region, and the time necessary to inform every agent about an event occurred in the environment. We associate a graph with the environment, and we study separately the case of a chain, tree, and cyclic graph. For the case of chain graph, we first describe a minimum refresh time and latency team trajectory, and we propose a polynomial time algorithm for its computation. Then, we describe a distributed procedure that steers the robots toward an optimal trajectory. For the case of tree graph, a polynomial time algorithm is developed for the minimum refresh time problem, under the technical assumption of a constant number of robots involved in the patrolling task. Finally, we show that the design of a minimum refresh time trajectory for a cyclic graph is NP-hard, and we develop a constant factor approximation algorithm.

Tommaso Menara, Giacomo Baggio, Danielle S. Bassett et al.

In this paper we study cluster synchronization in networks of oscillators with heterogenous Kuramoto dynamics, where multiple groups of oscillators with identical phases coexist in a connected network. Cluster synchronization is at the basis of several biological and technological processes; yet the underlying mechanisms to enable cluster synchronization of Kuramoto oscillators have remained elusive. In this paper we derive quantitative conditions on the network weights, cluster configuration, and oscillators' natural frequency that ensure asymptotic stability of the cluster synchronization manifold; that is, the ability to recover the desired cluster synchronization configuration following a perturbation of the oscillators' states. Qualitatively, our results show that cluster synchronization is stable when the intra-cluster coupling is sufficiently stronger than the inter-cluster coupling, the natural frequencies of the oscillators in distinct clusters are sufficiently different, or, in the case of two clusters, when the intra-cluster dynamics is homogeneous. We illustrate and validate the effectiveness of our theoretical results via numerical studies.

Eli J. Cornblath, Evelyn Tang, Graham L. Baum et al.

Executive function emerges late in development and displays different developmental trends in males and females. Sex differences in executive function in youth have been linked to vulnerability to psychopathology as well as to behaviors that impinge on health. Yet, the neurobiological basis of these differences is not well understood. Here we test the hypothesis that sex differences in executive function in youth stem from sex differences in the controllability of structural brain networks as they rewire over development. Combining methods from network neuroscience and network control theory, we characterize the network control properties of structural brain networks estimated from diffusion imaging data acquired in males and females in a sample of 882 youth aged 8-22 years. We summarize the control properties of these networks by estimating average and modal controllability, two statistics that probe the ease with which brain areas can drive the network towards easy- versus difficult-to-reach states. We find that females have higher modal controllability in frontal, parietal, and subcortical regions while males have higher average controllability in frontal and subcortical regions. Furthermore, average controllability values in the medial frontal cortex and subcortex, both higher in males, are negatively related to executive function. Finally, we find that average controllability predicts sex-dependent individual differences in activation during an n-back working memory task. Taken together, our findings support the notion that sex differences in the controllability of structural brain networks can partially explain sex differences in executive function. Controllability of structural brain networks also predicts features of task-relevant activation, suggesting the potential for controllability to represent context-specific constraints on network state more generally.

Fabio Pasqualetti, Ruggero Carli, Francesco Bullo

This work presents a distributed method for control centers to monitor the operating condition of a power network, i.e., to estimate the network state, and to ultimately determine the occurrence of threatening situations. State estimation has been recognized to be a fundamental task for network control centers to ensure correct and safe functionalities of power grids. We consider (static) state estimation problems, in which the state vector consists of the voltage magnitude and angle at all network buses. We consider the state to be linearly related to network measurements, which include power flows, current injections, and voltages phasors at some buses. We admit the presence of several cooperating control centers, and we design two distributed methods for them to compute the minimum variance estimate of the state given the network measurements. The two distributed methods rely on different modes of cooperation among control centers: in the first method an incremental mode of cooperation is used, whereas, in the second method, a diffusive interaction is implemented. Our procedures, which require each control center to know only the measurements and structure of a subpart of the whole network, are computationally efficient and scalable with respect to the network dimension, provided that the number of control centers also increases with the network cardinality. Additionally, a finite-memory approximation of our diffusive algorithm is proposed, and its accuracy is characterized. Finally, our estimation methods are exploited to develop a distributed algorithm to detect corrupted data among the network measurements.

Lorenzo Tiberi, Chiara Favaretto, Mario Innocenti et al.

Synchronization is crucial for the correct functionality of many natural and man-made complex systems. In this work we characterize the formation of synchronization patterns in networks of Kuramoto oscillators. Specifically, we reveal conditions on the network weights and structure and on the oscillators' natural frequencies that allow the phases of a group of oscillators to evolve cohesively, yet independently from the phases of oscillators in different clusters. Our conditions are applicable to general directed and weighted networks of heterogeneous oscillators. Surprisingly, although the oscillators exhibit nonlinear dynamics, our approach relies entirely on tools from linear algebra and graph theory. Further, we develop a control mechanism to determine the smallest (as measured by the Frobenius norm) network perturbation to ensure the formation of a desired synchronization pattern. Our procedure allows us to constrain the set of edges that can be modified, thus enforcing the sparsity structure of the network perturbation. The results are validated through a set of numerical examples.

Fabio Pasqualetti, Domenica Borra, Francesco Bullo

This work studies consensus strategies for networks of agents with limited memory, computation, and communication capabilities. We assume that agents can process only values from a finite alphabet, and we adopt the framework of finite fields, where the alphabet consists of the integers {0,...,p-1}, for some prime number p, and operations are performed modulo p. Thus, we define a new class of consensus dynamics, which can be exploited in certain applications such as pose estimation in capacity and memory constrained sensor networks. For consensus networks over finite fields, we provide necessary and sufficient conditions on the network topology and weights to ensure convergence. We show that consensus networks over finite fields converge in finite time, a feature that can be hardly achieved over the field of real numbers. For the design of finite-field consensus networks, we propose a general design method, with high computational complexity, and a network composition rule to generate large consensus networks from smaller components. Finally, we discuss the application of finite-field consensus networks to distributed averaging and pose estimation in sensor networks.

Tommaso Menara, Giacomo Baggio, Danielle S. Bassett et al.

In this paper, we propose a framework to control brain-wide functional connectivity by selectively acting on the brain's structure and parameters. Functional connectivity, which measures the degree of correlation between neural activities in different brain regions, can be used to distinguish between healthy and certain diseased brain dynamics and, possibly, as a control parameter to restore healthy functions. In this work, we use a collection of interconnected Kuramoto oscillators to model oscillatory neural activity, and show that functional connectivity is essentially regulated by the degree of synchronization between different clusters of oscillators. Then, we propose a minimally invasive method to correct the oscillators' interconnections and frequencies to enforce arbitrary and stable synchronization patterns among the oscillators and, consequently, a desired pattern of functional connectivity. Additionally, we show that our synchronization-based framework is robust to parameter mismatches and numerical inaccuracies, and validate it using a realistic neurovascular model to simulate neural activity and functional connectivity in the human brain.

Cheng-Zong Bai, Fabio Pasqualetti, Vijay Gupta

Consider a stochastic process being controlled across a communication channel. The control signal that is transmitted across the control channel can be replaced by a malicious attacker. The controller is allowed to implement any arbitrary detection algorithm to detect if an attacker is present. This work characterizes some fundamental limitations of when such an attack can be detected, and quantifies the performance degradation that an attacker that seeks to be undetected or stealthy can introduce.

Fabio Pasqualetti, Florian Dörfler, Francesco Bullo

Cyber-physical systems integrate computation, communication, and physical capabilities to interact with the physical world and humans. Besides failures of components, cyber-physical systems are prone to malignant attacks, and specific analysis tools as well as monitoring mechanisms need to be developed to enforce system security and reliability. This paper proposes a unified framework to analyze the resilience of cyber-physical systems against attacks cast by an omniscient adversary. We model cyber-physical systems as linear descriptor systems, and attacks as exogenous unknown inputs. Despite its simplicity, our model captures various real-world cyber-physical systems, and it includes and generalizes many prototypical attacks, including stealth, (dynamic) false-data injection and replay attacks. First, we characterize fundamental limitations of static, dynamic, and active monitors for attack detection and identification. Second, we provide constructive algebraic conditions to cast undetectable and unidentifiable attacks. Third, by using the system interconnection structure, we describe graph-theoretic conditions for the existence of undetectable and unidentifiable attacks. Finally, we validate our findings through some illustrative examples with different cyber-physical systems, such as a municipal water supply network and two electrical power grids.

Fabio Pasqualetti, Florian Dörfler, Francesco Bullo

Cyber-physical systems integrate computation, communication, and physical capabilities to interact with the physical world and humans. Besides failures of components, cyber-physical systems are prone to malicious attacks so that specific analysis tools and monitoring mechanisms need to be developed to enforce system security and reliability. This paper builds upon the results presented in our companion paper [1] and proposes centralized and distributed monitors for attack detection and identification. First, we design optimal centralized attack detection and identification monitors. Optimality refers to the ability of detecting (respectively identifying) every detectable (respectively identifiable) attack. Second, we design an optimal distributed attack detection filter based upon a waveform relaxation technique. Third, we show that the attack identification problem is computationally hard, and we design a sub-optimal distributed attack identification procedure with performance guarantees. Finally, we illustrate the robustness of our monitors to system noise and unmodeled dynamics through a simulation study.

Tommaso Menara, Shi Gu, Danielle S. Bassett et al.

The question of controllability of natural and man-made network systems has recently received considerable attention. In the context of the human brain, the study of controllability may not only shed light into the organization and function of different neural circuits, but also inform the design and implementation of minimally invasive yet effective intervention protocols to treat neurological disorders. While the characterization of brain controllability is still in its infancy, some results have recently appeared and given rise to scientific debate. Among these, [1] has numerically shown that a class of brain networks constructed from DSI/DTI imaging data are controllable from one brain region. That is, a single brain region is theoretically capable of moving the whole brain network towards any desired target state. In this note we provide evidence supporting controllability of brain networks from a single region as discussed in [1], thus contradicting the main conclusion and methods developed in [2].

Gianluca Bianchin, Paolo Frasca, Andrea Gasparri et al.

This paper studies the observability radius of network systems, which measures the robustness of a network to perturbations of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix, and dedicated sensors are positioned at a subset of nodes. We allow for perturbations of certain edge weights, with the objective of preventing observability of some modes of the network dynamics. To comply with the network setting, our work considers perturbations with a desired sparsity structure, thus extending the classic literature on the observability radius of linear systems. The paper proposes two sets of results. First, we propose an optimization framework to determine a perturbation with smallest Frobenius norm that renders a desired mode unobservable from the existing sensor nodes. Second, we study the expected observability radius of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more robust than star networks.

Erfan Nozari, Fabio Pasqualetti, Jorge Cortes

Despite extensive research and remarkable advancements in the control of complex dynamical networks, most studies and practical control methods limit their focus to time-invariant control schedules (TICS). This is both due to their simplicity and the fact that the benefits of time-varying control schedules (TVCS) have remained largely uncharacterized. In this paper we study networks with linear and discrete-time dynamics and analyze the role of network structure in TVCS. First, we show that TVCS can significantly enhance network controllability over TICS, especially when applied to large networks. Through the analysis of a scale-dependent notion of nodal centrality, we then show that optimal TVCS involves the actuation of the most central nodes at appropriate spatial scales at all times. Consequently, it is the scale-heterogeneity of the central-nodes in a network that determine whether, and to what extent, TVCS outperforms conventional policies based on TICS. Here, scale-heterogeneity of a network refers to how diverse the central nodes of the network are at different spatial (local vs. global) scales. Several analytical results and case studies support and illustrate this relationship.

Gianluca Bianchin, Fabio Pasqualetti

This paper proposes a simplified version of classical models for urban transportation networks, and studies the problem of controlling intersections with the goal of optimizing network-wide congestion. Differently from traditional approaches to control traffic signaling, a simplified framework allows for a more tractable analysis of the network overall dynamics, and enables the design of critical parameters while considering network-wide measures of efficiency. Motivated by the increasing availability of real-time high-resolution traffic data, we cast an optimization problem that formalizes the goal of minimizing the overall network congestion by optimally controlling the durations of green lights at intersections. Our formulation allows us to relate congestion objectives with the problem of optimizing a metric of controllability of an associated dynamical network. We then provide a technique to efficiently solve the optimization by parallelizing the computation among a group of distributed agents. Lastly, we assess the benefits of the proposed modeling and optimization framework through microscopic simulations on typical traffic commute scenarios for the area of Manhattan. The optimization framework proposed in this study is made available online on a Sumo microscopic simulator based interface [1].

Vaibhav Katewa, Fabio Pasqualetti

In this paper, we study robust stability of sparse LTI systems using the stability radius (SR) as a robustness measure. We consider real perturbations with an arbitrary and pre-specified sparsity pattern of the system matrix and measure their size using the Frobenius norm. We formulate the SR problem as an equality-constrained minimization problem. Using the Lagrangian method for optimization, we characterize the optimality conditions of the SR problem, thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system. Further, we use the Sylvester equation based parametrization to develop a penalty based gradient/Newton descent algorithm which converges to the local minima of the optimization problem. Finally, we illustrate how our framework provides structural insights into the robust stability of sparse networks.

Sarah Feldt Muldoon, Fabio Pasqualetti, Shi Gu et al.

The ability to modulate brain states using targeted stimulation is increasingly being employed to treat neurological disorders and to enhance human performance. Despite the growing interest in brain stimulation as a form of neuromodulation, much remains unknown about the network-level impact of these focal perturbations. To study the system wide impact of regional stimulation, we employ a data-driven computational model of nonlinear brain dynamics to systematically explore the effects of targeted stimulation. Validating predictions from network control theory, we uncover the relationship between regional controllability and the focal versus global impact of stimulation, and we relate these findings to differences in the underlying network architecture. Finally, by mapping brain regions to cognitive systems, we observe that the default mode system imparts large global change despite being highly constrained by structural connectivity. This work forms an important step towards the development of personalized stimulation protocols for medical treatment or performance enhancement.

Yuzhen Qin, Yingcong Li, Fabio Pasqualetti et al.

The growing interest in complex decision-making and language modeling problems highlights the importance of sample-efficient learning over very long horizons. This work takes a step in this direction by investigating contextual linear bandits where the current reward depends on at most $s$ prior actions and contexts (not necessarily consecutive), up to a time horizon of $h$. In order to avoid polynomial dependence on $h$, we propose new algorithms that leverage sparsity to discover the dependence pattern and arm parameters jointly. We consider both the data-poor ($T<h$) and data-rich ($T\ge h$) regimes, and derive respective regret upper bounds $\tilde O(d\sqrt{sT} +\min\{ q, T\})$ and $\tilde O(\sqrt{sdT})$, with sparsity $s$, feature dimension $d$, total time horizon $T$, and $q$ that is adaptive to the reward dependence pattern. Complementing upper bounds, we also show that learning over a single trajectory brings inherent challenges: While the dependence pattern and arm parameters form a rank-1 matrix, circulant matrices are not isometric over rank-1 manifolds and sample complexity indeed benefits from the sparse reward dependence structure. Our results necessitate a new analysis to address long-range temporal dependencies across data and avoid polynomial dependence on the reward horizon $h$. Specifically, we utilize connections to the restricted isometry property of circulant matrices formed by dependent sub-Gaussian vectors and establish new guarantees that are also of independent interest.

Yuzhen Qin, Tommaso Menara, Samet Oymak et al.

Humans are capable of adjusting to changing environments flexibly and quickly. Empirical evidence has revealed that representation learning plays a crucial role in endowing humans with such a capability. Inspired by this observation, we study representation learning in the sequential decision-making scenario with contextual changes. We propose an online algorithm that is able to learn and transfer context-dependent representations and show that it significantly outperforms the existing ones that do not learn representations adaptively. As a case study, we apply our algorithm to the Wisconsin Card Sorting Task, a well-established test for the mental flexibility of humans in sequential decision-making. By comparing our algorithm with the standard Q-learning and Deep-Q learning algorithms, we demonstrate the benefits of adaptive representation learning.

Lulu Gong, Fabio Pasqualetti, Thomas Papouin et al.

Astrocytes are a ubiquitous and enigmatic type of non-neuronal cell and are found in the brain of all vertebrates. While traditionally viewed as being supportive of neurons, it is increasingly recognized that astrocytes may play a more direct and active role in brain function and neural computation. On account of their sensitivity to a host of physiological covariates and ability to modulate neuronal activity and connectivity on slower time scales, astrocytes may be particularly well poised to modulate the dynamics of neural circuits in functionally salient ways. In the current paper, we seek to capture these features via actionable abstractions within computational models of neuron-astrocyte interaction. Specifically, we engage how nested feedback loops of neuron-astrocyte interaction, acting over separated time-scales may endow astrocytes with the capability to enable learning in context-dependent settings, where fluctuations in task parameters may occur much more slowly than within-task requirements. We pose a general model of neuron-synapse-astrocyte interaction and use formal analysis to characterize how astrocytic modulation may constitute a form of meta-plasticity, altering the ways in which synapses and neurons adapt as a function of time. We then embed this model in a bandit-based reinforcement learning task environment, and show how the presence of time-scale separated astrocytic modulation enables learning over multiple fluctuating contexts. Indeed, these networks learn far more reliably versus dynamically homogeneous networks and conventional non-network-based bandit algorithms. Our results indicate how the presence of neuron-astrocyte interaction in the brain may benefit learning over different time-scales and the conveyance of task-relevant contextual information onto circuit dynamics.

Darshan Gadginmath, Ahmed Allibhoy, Fabio Pasqualetti

We propose a constricting Control Barrier Function (CBF) framework for prescribed-time control of control-affine systems with input constraints. Given a system starting outside a target safe set, we construct a time-varying safety tube that shrinks from a relaxed set containing the initial condition to the target set at a user-specified deadline. Any controller rendering this tube forward invariant guarantees prescribed-time recovery by construction. The constriction schedule is bounded and tunable by design, in contrast to prescribed-time methods where control effort diverges near the deadline. Feasibility under input constraints reduces to a single verifiable condition on the constriction rate, yielding a closed-form minimum recovery time as a function of control authority and initial violation. The framework imposes a single affine constraint per timestep regardless of state dimension, scaling to settings where grid-based reachability methods are intractable. We validate on a 16-dimensional multi-agent system and a unicycle reach-avoid problem, demonstrating prescribed-time recovery with bounded control effort.

Darshan Gadginmath, Ahmed Allibhoy, Fabio Pasqualetti

Flow-based generative models, such as diffusion models and flow matching models, have achieved remarkable success in learning complex data distributions. However, a critical gap remains for their deployment in safety-critical domains: the lack of formal guarantees that generated samples will satisfy hard constraints. We address this by proposing a safety filtering framework that acts as an online shield for any pre-trained generative model. Our key insight is to cooperate with the generative process rather than override it. We define a constricting safety tube that is relaxed at the initial noise distribution and progressively tightens to the target safe set at the final data distribution, mirroring the coarse-to-fine structure of the generative process itself. By characterizing this tube via Control Barrier Functions (CBFs), we synthesize a feedback control input through a convex Quadratic Program (QP) at each sampling step. As the tube is loosest when noise is high and intervention is cheapest in terms of control energy, most constraint enforcement occurs when it least disrupts the model's learned structure. We prove that this mechanism guarantees safe sampling while minimizing the distributional shift from the original model at each sampling step, as quantified by the KL divergence. Our framework applies to any pre-trained flow-based generative scheme requiring no retraining or architectural modifications. We validate the approach across constrained image generation, physically-consistent trajectory sampling, and safe robotic manipulation policies, achieving 100% constraint satisfaction while preserving semantic fidelity.

Arie Ogranovich, Taosha Guo, Arvind R. Venkatakrishnan et al.

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.

Karthik Elamvazhuthi, Darshan Gadginmath, Fabio Pasqualetti

We propose a novel approach based on Denoising Diffusion Probabilistic Models (DDPMs) to control nonlinear dynamical systems. DDPMs are the state-of-art of generative models that have achieved success in a wide variety of sampling tasks. In our framework, we pose the feedback control problem as a generative task of drawing samples from a target set under control system constraints. The forward process of DDPMs constructs trajectories originating from a target set by adding noise. We learn to control a dynamical system in reverse such that the terminal state belongs to the target set. For control-affine systems without drift, we prove that the control system can exactly track the trajectory of the forward process in reverse, whenever the the Lie bracket based condition for controllability holds. We numerically study our approach on various nonlinear systems and verify our theoretical results. We also conduct numerical experiments for cases beyond our theoretical results on a physics-engine.

Karthik Elamvazhuthi, Darshan Gadginmath, Fabio Pasqualetti

We propose a novel control-theoretic framework that leverages principles from generative modeling -- specifically, Denoising Diffusion Probabilistic Models (DDPMs) -- to stabilize control-affine systems with nonholonomic constraints. Unlike traditional stochastic approaches, which rely on noise-driven dynamics in both forward and reverse processes, our method crucially eliminates the need for noise in the reverse phase, making it particularly relevant for control applications. We introduce two formulations: one where noise perturbs all state dimensions during the forward phase while the control system enforces time reversal deterministically, and another where noise is restricted to the control channels, embedding system constraints directly into the forward process. For controllable nonlinear drift-free systems, we prove that deterministic feedback laws can exactly reverse the forward process, ensuring that the system's probability density evolves correctly without requiring artificial diffusion in the reverse phase. Furthermore, for linear time-invariant systems, we establish a time-reversal result under the second formulation. By eliminating noise in the backward process, our approach provides a more practical alternative to machine learning-based denoising methods, which are unsuitable for control applications due to the presence of stochasticity. We validate our results through numerical simulations on benchmark systems, including a unicycle model in a domain with obstacles, a driftless five-dimensional system, and a four-dimensional linear system, demonstrating the potential for applying diffusion-inspired techniques in linear, nonlinear, and settings with state space constraints.

Darshan Gadginmath, Shivanshu Tripathi, Fabio Pasqualetti

This study addresses the challenge of online learning in contexts where agents accumulate disparate data, face resource constraints, and use different local algorithms. This paper introduces the Switched Online Learning Algorithm (SOLA), designed to solve the heterogeneous online learning problem by amalgamating updates from diverse agents through a dynamic switching mechanism contingent upon their respective performance and available resources. We theoretically analyze the design of the selecting mechanism to ensure that the regret of SOLA is bounded. Our findings show that the number of changes in selection needs to be bounded by a parameter dependent on the performance of the different local algorithms. Additionally, two test cases are presented to emphasize the effectiveness of SOLA, first on an online linear regression problem and then on an online classification problem with the MNIST dataset.

Umberto Casti, Sandro Zampieri, Fabio Pasqualetti

Recent advancements in language modeling tasks have been driven by architectures such as Transformers and, more recently, by Selective State Space Models (SSMs). In this paper, we introduce an alternative selection mechanism inspired by control theory methodologies. Specifically, we propose a novel residual generator for selection, drawing an analogy to fault detection strategies in Linear Time-Invariant (LTI) systems. Unlike Mamba, which utilizes Linear Time-Varying (LTV) systems, our approach combines multiple LTI systems, preserving their beneficial properties during training while achieving comparable selectivity. To evaluate the effectiveness of the proposed architecture, we test its performance on synthetic tasks. While these tasks are not inherently critical, they serve as benchmarks to test the selectivity properties of different cores architecture. This work highlights the potential of integrating theoretical insights with experimental advancements, offering a complementary perspective to deep learning innovations at the intersection of control theory and machine learning.

Shiqi Zhang, Darshan Gadginmath, Fabio Pasqualetti

Predicting the behavior of AI-driven agents is particularly challenging without a preexisting model. In our paper, we address this by treating AI agents as nonlinear dynamical systems and adopting a probabilistic perspective to predict their statistical behavior using the Perron-Frobenius (PF) operator. We formulate the approximation of the PF operator as an entropy minimization problem, which can be solved by leveraging the Markovian property of the operator and decomposing its spectrum. Our data-driven methodology simultaneously approximates the PF operator to perform prediction of the evolution of the agents and also predicts the terminal probability density of AI agents, such as robotic systems and generative models. We demonstrate the effectiveness of our prediction model through extensive experiments on practical systems driven by AI algorithms.

Karthik Elamvazhuthi, Samet Oymak, Fabio Pasqualetti

In Score based Generative Modeling (SGMs), the state-of-the-art in generative modeling, stochastic reverse processes are known to perform better than their deterministic counterparts. This paper delves into the heart of this phenomenon, comparing neural ordinary differential equations (ODEs) and neural stochastic differential equations (SDEs) as reverse processes. We use a control theoretic perspective by posing the approximation of the reverse process as a trajectory tracking problem. We analyze the ability of neural SDEs to approximate trajectories of the Fokker-Planck equation, revealing the advantages of stochasticity. First, neural SDEs exhibit a powerful regularizing effect, enabling $L^2$ norm trajectory approximation surpassing the Wasserstein metric approximation achieved by neural ODEs under similar conditions, even when the reference vector field or score function is not Lipschitz. Applying this result, we establish the class of distributions that can be sampled using score matching in SGMs, relaxing the Lipschitz requirement on the gradient of the data distribution in existing literature. Second, we show that this approximation property is preserved when network width is limited to the input dimension of the network. In this limited width case, the weights act as control inputs, framing our analysis as a controllability problem for neural SDEs in probability density space. This sheds light on how noise helps to steer the system towards the desired solution and illuminates the empirical success of stochasticity in generative modeling.

Karthik Elamvazhuthi, Xuechen Zhang, Samet Oymak et al.

In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural ordinary differential equations (ODEs), however, typically fail to satisfy these manifold constraints and perform poorly for these applications. To address this shortcoming, in this paper we study a class of neural ordinary differential equations that, by design, leave a given manifold invariant, and characterize their properties by leveraging the controllability properties of control affine systems. In particular, using a result due to Agrachev and Caponigro on approximating diffeomorphisms with flows of feedback control systems, we show that any map that can be represented as the flow of a manifold-constrained dynamical system can also be approximated using the flow of manifold-constrained neural ODE, whenever a certain controllability condition is satisfied. Additionally, we show that this universal approximation property holds when the neural ODE has limited width in each layer, thus leveraging the depth of network instead for approximation. We verify our theoretical findings using numerical experiments on PyTorch for the manifolds S2 and the 3-dimensional orthogonal group SO(3), which are model manifolds for mechanical systems such as spacecrafts and satellites. We also compare the performance of the manifold invariant neural ODE with classical neural ODEs that ignore the manifold invariant properties and show the superiority of our approach in terms of accuracy and sample complexity.

Yuzhen Qin, Tommaso Menara, Samet Oymak et al.

In this paper, we study representation learning for multi-task decision-making in non-stationary environments. We consider the framework of sequential linear bandits, where the agent performs a series of tasks drawn from distinct sets associated with different environments. The embeddings of tasks in each set share a low-dimensional feature extractor called representation, and representations are different across sets. We propose an online algorithm that facilitates efficient decision-making by learning and transferring non-stationary representations in an adaptive fashion. We prove that our algorithm significantly outperforms the existing ones that treat tasks independently. We also conduct experiments using both synthetic and real data to validate our theoretical insights and demonstrate the efficacy of our algorithm.

Abed AlRahman Al Makdah, Vaibhav Katewa, Fabio Pasqualetti

In this work, we consider a binary classification problem and cast it into a binary hypothesis testing framework, where the observations can be perturbed by an adversary. To improve the adversarial robustness of a classifier, we include an abstain option, where the classifier abstains from making a decision when it has low confidence about the prediction. We propose metrics to quantify the nominal performance of a classifier with an abstain option and its robustness against adversarial perturbations. We show that there exist a tradeoff between the two metrics regardless of what method is used to choose the abstain region. Our results imply that the robustness of a classifier with an abstain option can only be improved at the expense of its nominal performance. Further, we provide necessary conditions to design the abstain region for a 1- dimensional binary classification problem. We validate our theoretical results on the MNIST dataset, where we numerically show that the tradeoff between performance and robustness also exist for the general multi-class classification problems.

Abed AlRahman Al Makdah, Vishaal Krishnan, Fabio Pasqualetti

In this work, we propose a framework to learn feedback control policies with guarantees on closed-loop generalization and adversarial robustness. These policies are learned directly from expert demonstrations, contained in a dataset of state-control input pairs, without any prior knowledge of the task and system model. We use a Lipschitz-constrained loss minimization scheme to learn feedback policies with certified closed-loop robustness, wherein the Lipschitz constraint serves as a mechanism to tune the generalization performance and robustness to adversarial disturbances. Our analysis exploits the Lipschitz property to obtain closed-loop guarantees on generalization and robustness of the learned policies. In particular, we derive a finite sample bound on the policy learning error and establish robust closed-loop stability under the learned control policy. We also derive bounds on the closed-loop regret with respect to the expert policy and the deterioration of closed-loop performance under bounded (adversarial) disturbances to the state measurements. Numerical results validate our analysis and demonstrate the effectiveness of our robust feedback policy learning framework. Finally, our results suggest the existence of a potential tradeoff between nominal closed-loop performance and adversarial robustness, and that improvements in nominal closed-loop performance can only be made at the expense of robustness to adversarial perturbations.

Giacomo Baggio, Fabio Pasqualetti

In this paper we study the problem of learning minimum-energy controls for linear systems from heterogeneous data. Specifically, we consider datasets comprising input, initial and final state measurements collected using experiments with different time horizons and arbitrary initial conditions. In this setting, we first establish a general representation of input and sampled state trajectories of the system based on the available data. Then, we leverage this data-based representation to derive closed-form data-driven expressions of minimum-energy controls for a wide range of control horizons. Further, we characterize the minimum number of data required to reconstruct the minimum-energy inputs, and discuss the numerical properties of our expressions. Finally, we investigate the effect of noise on our data-driven formulas, and, in the case of noise with known second-order statistics, we provide corrected expressions that converge asymptotically to the true optimal control inputs.

Vishaal Krishnan, Abed AlRahman Al Makdah, Fabio Pasqualetti

In this work we propose a graph-based learning framework to train models with provable robustness to adversarial perturbations. In contrast to regularization-based approaches, we formulate the adversarially robust learning problem as one of loss minimization with a Lipschitz constraint, and show that the saddle point of the associated Lagrangian is characterized by a Poisson equation with weighted Laplace operator. Further, the weighting for the Laplace operator is given by the Lagrange multiplier for the Lipschitz constraint, which modulates the sensitivity of the minimizer to perturbations. We then design a provably robust training scheme using graph-based discretization of the input space and a primal-dual algorithm to converge to the Lagrangian's saddle point. Our analysis establishes a novel connection between elliptic operators with constraint-enforced weighting and adversarial learning. We also study the complementary problem of improving the robustness of minimizers with a margin on their loss, formulated as a loss-constrained minimization problem of the Lipschitz constant. We propose a technique to obtain robustified minimizers, and evaluate fundamental Lipschitz lower bounds by approaching Lipschitz constant minimization via a sequence of gradient $p$-norm minimization problems. Ultimately, our results show that, for a desired nominal performance, there exists a fundamental lower bound on the sensitivity to adversarial perturbations that depends only on the loss function and the data distribution, and that improvements in robustness beyond this bound can only be made at the expense of nominal performance. Our training schemes provably achieve these bounds both under constraints on performance and~robustness.

Vishaal Krishnan, Fabio Pasqualetti

This paper studies the attack detection problem in a data-driven and model-free setting, for deterministic systems with linear and time-invariant dynamics. Differently from existing studies that leverage knowledge of the system dynamics to derive security bounds and monitoring schemes, we focus on the case where the system dynamics, as well as the attack strategy and attack location, are unknown. We derive fundamental security limitations as a function of only the observed data and without estimating the system dynamics (in fact, no assumption is made on the identifiability of the system). In particular, (i) we derive detection limitations as a function of the informativity and length of the observed data, (ii) provide a data-driven characterization of undetectable attacks, and (iii) construct a data-driven detection monitor. Surprisingly, and in accordance with recent studies on data-driven control, our results show that model-based and data-driven security techniques share the same fundamental limitations, provided that the collected data remains sufficiently informative.

Rajasekhar Anguluri, Abed AlRahman Al Makdah, Vaibhav Katewa et al.

This paper proposes a new framework and several results to quantify the performance of data-driven state-feedback controllers for linear systems against targeted perturbations of the training data. We focus on the case where subsets of the training data are randomly corrupted by an adversary, and derive lower and upper bounds for the stability of the closed-loop system with compromised controller as a function of the perturbation statistics, size of the training data, sensitivity of the data-driven algorithm to perturbation of the training data, and properties of the nominal closed-loop system. Our stability and convergence bounds are probabilistic in nature, and rely on a first-order approximation of the data-driven procedure that designs the state-feedback controller, which can be computed directly using the training data. We illustrate our findings via multiple numerical studies.

Abed AlRahman Al Makdah, Vaibhav Katewa, Fabio Pasqualetti

Despite the widespread use of machine learning algorithms to solve problems of technological, economic, and social relevance, provable guarantees on the performance of these data-driven algorithms are critically lacking, especially when the data originates from unreliable sources and is transmitted over unprotected and easily accessible channels. In this paper we take an important step to bridge this gap and formally show that, in a quest to optimize their accuracy, binary classification algorithms -- including those based on machine-learning techniques -- inevitably become more sensitive to adversarial manipulation of the data. Further, for a given class of algorithms with the same complexity (i.e., number of classification boundaries), the fundamental tradeoff curve between accuracy and sensitivity depends solely on the statistics of the data, and cannot be improved by tuning the algorithm.

Giacomo Baggio, Vaibhav Katewa, Fabio Pasqualetti

In this paper we study the problem of computing minimum-energy controls for linear systems from experimental data. The design of open-loop minimum-energy control inputs to steer a linear system between two different states in finite time is a classic problem in control theory, whose solution can be computed in closed form using the system matrices and its controllability Gramian. Yet, the computation of these inputs is known to be ill-conditioned, especially when the system is large, the control horizon long, and the system model uncertain. Due to these limitations, open-loop minimum-energy controls and the associated state trajectories have remained primarily of theoretical value. Surprisingly, in this paper we show that open-loop minimum-energy controls can be learned exactly from experimental data, with a finite number of control experiments over the same time horizon, without knowledge or estimation of the system model, and with an algorithm that is significantly more reliable than the direct model-based computation. These findings promote a new philosophy of controlling large, uncertain, linear systems where data is abundantly available.

Gianluca Bianchin, Fabio Pasqualetti, Soumya Kundu

This paper investigates the use of Infrastructure-To-Vehicle (I2V) communication to generate routing suggestions for drivers in transportation systems, with the goal of optimizing a measure of overall network congestion. We define link-wise levels of trust to tolerate the non-cooperative behavior of part of the driver population, and we propose a real-time optimization mechanism that adapts to the instantaneous network conditions and to sudden changes in the levels of trust. Our framework allows us to quantify the improvement in travel time in relation to the degree at which drivers follow the routing suggestions. We then study the resilience of the system, measured as the smallest change in routing choices that results in roads reaching their maximum capacity. Interestingly, our findings suggest that fluctuations in the extent to which drivers follow the provided routing suggestions can cause failures of certain links. These results imply that the benefits of using Infrastructure-To-Vehicle communication come at the cost of new fragilities, that should be appropriately addressed in order to guarantee the reliable operation of the infrastructure.

Vaibhav Srivastava, Fabio Pasqualetti, Francesco Bullo

We design persistent surveillance strategies for the quickest detection of anomalies taking place in an environment of interest. From a set of predefined regions in the environment, a team of autonomous vehicles collects noisy observations, which a control center processes. The overall objective is to minimize detection delay while maintaining the false alarm rate below a desired threshold. We present joint (i) anomaly detection algorithms for the control center and (ii) vehicle routing policies. For the control center, we propose parallel cumulative sum (CUSUM) algorithms (one for each region) to detect anomalies from noisy observations. For the vehicles, we propose a stochastic routing policy, in which the regions to be visited are chosen according to a probability vector. We study stationary routing policy (the probability vector is constant) as well as adaptive routing policies (the probability vector varies in time as a function of the likelihood of regional anomalies). In the context of stationary policies, we design a performance metric and minimize it to design an efficient stationary routing policy. Our adaptive policy improves upon the stationary counterpart by adaptively increasing the selection probability of regions with high likelihood of anomaly. Finally, we show the effectiveness of the proposed algorithms through numerical simulations and a persistent surveillance experiment.