arXiv subjects

Jorge Cortés

Publications and source records attributed to Jorge Cortés.

At least 19 recordsLinked to original sources

arXiv2026

Learning the Omega-Limit Partition from Approximate Koopman Invariants

Nonlinear dynamical systems with multiple omega-limit sets pose a fundamental challenge for global finite-dimensional Koopman modeling with continuous observables. A natural way to address this challenge is to decompose the state space into forward-invariant regions associated with distinct asymptotic behaviors and to learn a separate Koopman model for each region. This paper studies how to compute such partitions from data. We establish conditions under which Koopman-invariant observables, namely eigenfunctions associated with the unit eigenvalue, characterize the stationary partition induced by omega-limit sets. Building on this characterization, we develop OPAL (Omega-limit Partitions from Approximate invariants Learned from data), an algorithm that extracts partitions from approximate invariant observables. Numerical examples show that the recovered partitions capture the relevant asymptotic regions and support multiple regional linear Koopman models with significantly improved long-horizon prediction capabilities.

eess.SY↗

arXiv2026

On the Regularization of Primal-Dual Dynamics in Time-Dependent Nonconvex Optimization

We study continuous-time primal-dual dynamics for tracking solutions of time-dependent constrained optimization problems. First, for uniformly strongly convex-strongly concave saddle functions, we establish explicit finite-time tracking and additive-disturbance bounds. For convex problems, we then introduce a directional primal regularization, combined with dual Tikhonov regularization, and quantify the resulting optimization bias and constraint violation. For smooth nonconvex problems, we couple a proximal local convexification with a two-time-scale primal-dual flow and obtain a local tracking result under constraint qualification, second-order sufficiency, and temporal regularity. Numerical traffic-signal experiments at a single intersection evaluate the proposed method under vehicle-throughput and pedestrian-aware objectives, comparing it with standard signal-control schemes.

math.OC↗

arXiv2026

Koopman operator theory: fundamentals, control, and applications

The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.

eess.SY↗

arXiv2026

Learning infinite context windows in recurrent architectures via spatial neural computing

Recurrent neural networks (RNNs) offer linear-time scaling with sequence length while requiring only constant memory, yet they struggle to capture long-range dependencies due to vanishing gradients and limited receptive fields. To address these limitations, we introduce a second-order recurrent model in which the standard neuron-to-neuron communication is replaced by a spatially evolving field governed by (discretized) partial differential equations. Drawing inspiration from the role of cortical waves in brain computation, this mechanism allows structured spatiotemporal patterns to serve as an implicit, high-capacity memory. We show that the resulting model is equivalent to a structured infinite-order RNN in which the current state depends explicitly on its entire history of past states, yielding an effectively unbounded receptive field with a fixed number of parameters. We further derive constructive conditions to ensure marginal stability, constraining the gradient spectrum on the unit circle and thereby eliminating vanishing and exploding gradients. Empirically, the proposed architecture outperforms other recurrent models on long-horizon benchmarks while using substantially fewer parameters, demonstrating that spatial dynamics can effectively bridge the gap between efficient inference and long-term memory.

cs.LG↗

arXiv2026

Gramian-Informed Framework for Edge Flow Network Analysis Against Nodal Attacks

We study the first-order effects of nodal inputs on edge flows of network systems through controllability Gramians. We consider both discrete- and continuous-time dynamics subject to impulse and step input disturbances. To characterize their impact on network behavior, we introduce the notion of a vulnerability matrix (VM) and provide explicit Gramian-based expressions that reveal the critical role played by the interplay between network topology and dynamics. For the class of directed line networks, we particularize these closed-form expressions in terms of edge weights, input duration, and graph distance to the nodal input. Numerical simulations on directed line and random Erdős-Rényi networks show a tight correspondence between influential nodes identified with the VM and conventional time-domain performance metrics, such as $\mathscr{H}_2$ and $\mathscr{H}_{\infty}$.

math.OC↗

arXiv2026

A Unified Perspective on Conformal Prediction and Wasserstein Distributionally Robust Optimization for Uncertainty Quantification

Uncertainty quantification from finite data is central to machine learning, optimization, and automation systems, where decisions must remain reliable under limited samples and test-time distribution shift. Conformal prediction (CP) and distributionally robust optimization (DRO) offer two complementary approaches: CP constructs data-dependent prediction sets with distribution-free finite-sample validity under exchangeability, while DRO optimizes worst-case performance over an ambiguity set around an empirical distribution. We develop a unified probabilistic perspective on CP and DRO by viewing both as ways to turn finite calibration data into a data-dependent quantile estimator that a test score falls below with high probability. From this perspective, CP and DRO correct the empirical quantile along two coordinates of the same family of estimators: CP inflates the quantile level, whereas DRO shifts the quantile value through an ambiguity radius. Both methods provide the same calibration-conditional guarantee for the true distribution, requiring the target coverage to hold with high probability over the calibration sample. Their constructions differ, however: CP uses a closed-form, distribution-free level correction, while DRO uses a value-space correction whose certified radius depends on properties of the unknown distribution and additionally guarantees coverage uniformly over the ambiguity set. This distinction emerges in the tails of the score distribution. Because CP relies on sparse upper-tail order statistics of the calibration samples, its level inflation barely moves the estimator when those samples are dense near the target quantile but overshoots when they are sparse, whereas a well-chosen DRO radius corrects in value space and may avoid this overshoot.

math.OC↗

arXiv2026

Subspace Pruning via Principal Vectors for Accurate Koopman-Based Approximations

The accuracy of Koopman operator approximations over finite-dimensional spaces relies critically on their invariance properties. These can be rigorously quantified via the principal angles between a candidate subspace and its image under the Koopman operator. This paper proposes a unified algebraic framework for subspace pruning designed to systematically refine the invariance error. We establish the geometric equivalence between consistency-based methods and principal-vector pruning, and build on this insight to introduce a hybrid strategy that balances between multiple and single principal vector pruning for improved numerical stability and scalability. We derive error bounds for the retention of approximate and external eigenfunctions, demonstrating that the multi-vector approach mitigates the numerical drift inherent to sequential pruning. To ensure scalability, we develop an efficient numerical update scheme based on rank-one modifications that reduces the computational complexity of tracking principal angles by an order of magnitude. Finally, we exploit the subspace obtained from the pruning algorithms to build a lifted linear model for state prediction that accounts for the trade-offs between improving invariance and minimizing state reconstruction error. Simulations demonstrate the effectiveness of our approach.

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

Constrained Variational Inference via Safe Particle Flow

We propose a control barrier function (CBF) formulation for enforcing equality and inequality constraints in variational inference. The key idea is to define a barrier functional on the space of probability density functions that encode the desired constraints imposed on the variational density. By leveraging the Liouville equation, we establish a connection between the time derivative of the variational density and the particle drift, which enables the systematic construction of corresponding CBFs associated to the particle drift. Enforcing these CBFs gives rise to the safe particle flow and ensures that the variational density satisfies the original constraints imposed by the barrier functional. This formulation provides a principled and computationally tractable solution to constrained variational inference, with theoretical guarantees of constraint satisfaction. The effectiveness of the method is demonstrated through numerical simulations.

math.OC↗

arXiv2026

Reachability-Based Design Optimization for Aircraft Maneuverability

This paper presents a method for incorporating control analysis into design optimization for highly-maneuverable aircraft. By studying reachable sets for aircraft dynamics, we ensure that the optimizer will take the aircraft's controlled capabilities into account. We compute reachable sets of linear dynamics for computational efficiency, and account for aircraft trim points to factor in asymmetric magnitude bounds on the input signals. We demonstrate the proposed method in design optimization of a blended-wing-body aircraft. Considering its wing half-span and center half-span as design variables, we optimize the aircraft based on its longitudinal dynamics' reachable sets to yield improvements in its controlled performance. When designing a reference tracking controller, we find up to 30\% less tracking error for angle of attack of the optimized model's nonlinear dynamics.

math.OC↗

arXiv2026

Universal Formulas for Safe Control and Their Neural Network Approximations

We study the problem of designing a controller that satisfies an arbitrary number of affine inequalities at every point in the state space. This is motivated by the fact that a variety of key control objectives, such as stability, safety, and input saturation, are guaranteed by closed-loop systems whose controllers satisfy such inequalities. Many works in the literature design such controllers as the solution to a state-dependent quadratic program (QP) whose constraints are precisely the inequalities. When the input dimension and number of constraints are high, computing a solution of this QP in real time can become computationally burdensome. Additionally, the solution of such optimization problems is not smooth in general, which can degrade the performance of the system. This paper provides a novel method to design a smooth controller that satisfies an arbitrary number of affine constraints. The controller is given at every state as the minimizer of a strictly convex function. To avoid computing the minimizer of such function in real time, we introduce a method based on neural networks (NN) to approximate the controller. Remarkably, this NN can be used to solve the controller design problem for any task with less than a fixed input dimension and number of affine constraints, and is completely independent of the state dimension. This is why we refer to such NN approximation as a NN-based universal formula for control. Additionally, we show that the NN-based controller only needs to be trained with datapoints from a bounded set in the state space, which significantly simplifies the training process.

math.OC↗

arXiv2026

Control Forward-Backward Consistency: Quantifying the Accuracy of Koopman Control Family Models

This paper extends the forward-backward consistency index, originally introduced in Koopman modeling of systems without input, to the setting of control systems, providing a closed-form computable measure of accuracy for data-driven models associated with the Koopman Control Family (KCF). Building on a forward-backward regression perspective, we introduce the control forward-backward consistency matrix and demonstrate that it possesses several favorable properties. Our main result establishes that the relative root-mean-square error of KCF function predictors is strictly bounded by the square root of the control consistency index, defined as the maximum eigenvalue of the consistency matrix. This provides a sharp, closed-form computable error bound for finite-dimensional KCF models. We further specialize this bound to the widely used lifted linear and bilinear models. We also discuss how the control consistency index can be incorporated into optimization-based modeling and illustrate the methodology via simulations.

math.OC↗

arXiv2026

Safe Policy Optimization via Control Barrier Function-based Safety Filters

Control barrier function (CBF)-based safety filters provide a systematic way to enforce state constraints, but they can significantly alter the closed-loop dynamics induced by a nominal, stabilizing controller. In particular, the resulting safety-filtered system may exhibit undesirable behaviors including limit cycles, unbounded trajectories, and undesired equilibria. This paper develops a policy optimization framework to maximally enhance the stability properties of safety-filtered controllers. Focusing on linear systems with linear nominal controllers, we jointly parameterize the nominal feedback gain and safety-filter components, and optimize them using trajectory-based objectives computed from closed-loop rollouts. To ensure that the nominal controller remains stabilizing throughout training, we encode Lyapunov-based stability conditions as smooth scalar constraints and enforce them using robust safe gradient flow. This guarantees feasibility of the stability constraints along the optimization iterates and therefore avoids instability during training. Numerical experiments on obstacle-avoidance problems show that the proposed approach can remove asymptotically stable undesired equilibria and improve convergence behavior while maintaining forward invariance of the safe set.

eess.SY↗

arXiv2026

Reachability Analysis for Design Optimization

We present an approach to approximate reachable sets for linear systems with bounded L-infinity controls in finite time. Our first approach investigates the boundaries of these sets and reveals an exact characterization for single-input, planar systems with real, distinct eigenvalues. The second approach leverages convergence of the Lp-norms to L-infinity and uses Lp-norm reachable sets as an approximation of the L-infinity-norm reachable sets. Our optimal control results yield insights that make computational approximations of the Lp-norm reachable sets more tractable, and yield exact characterizations for L-infinity with the previous assumptions on the system. As an example, we incorporate our reachability analysis into the design optimization of a highly-maneuverable aircraft. Introducing constraints based on reachability allow us to factor physical limitations to desired flight maneuvers into the design process.

math.OC↗

arXiv2026

Variational Formulation of Particle Flow

This paper provides a formulation of the log-homotopy particle flow from the perspective of variational inference. We show that the transient density used to derive the particle flow follows a time-scaled trajectory of the Fisher-Rao gradient flow in the space of probability densities. The Fisher-Rao gradient flow is obtained as a continuous-time algorithm for variational inference, minimizing the Kullback-Leibler divergence between a variational density and the true posterior density. When considering a parametric family of variational densities, the function space Fisher-Rao gradient flow simplifies to the natural gradient flow of the variational density parameters. By adopting a Gaussian variational density, we derive a Gaussian approximated Fisher-Rao particle flow and show that, under linear Gaussian assumptions, it reduces to the Exact Daum and Huang particle flow. Additionally, we introduce a Gaussian mixture approximated Fisher-Rao particle flow to enhance the expressive power of our model through a multi-modal variational density. Simulations on low- and high-dimensional estimation problems illustrate our results.

stat.ML↗

arXiv2026

Enhancing network resilience through topological switching

This work studies how to preemptively increase the resilience of a network by means of time-varying topological actuation. To do this, we focus on linear dynamical systems that are compatible with a given network, and consider policies that switch periodically between the given one and an alternative, topologically-compatible dynamics. In particular, we seek to solve design problems aimed at finding a) the optimal switching schedule between two preselected topologies, and b) an optimal topology and optimal switching schedule. By imposing periodicity, we first provide a metric of resilience in terms of the spectral abscissa of the averaged linear time-invariant dynamics. By restricting our policies to commutative networks, we then show how the optimal scheduling problem reduces to a convex optimization, providing a bound on the net resilience that can be achieved. After this, we find that the optimal, sparse commutative network to switch with is fully disconnected and allocates the spectral sum among the nodes of the network equally. We then impose additional restrictions on topology edge selection, which leads to a biconvex optimization for which certain matrix rank conditions guide the choice of weighting parameters to obtain desirable solutions. Finally, we provide two methods to solve this problem efficiently (based on a McCormick relaxation, and alternating minimization), and illustrate the results in simulations.

math.OC↗

arXiv2026

On the Existence of Koopman Linear Embeddings for Controlled Nonlinear Systems

Koopman linear representations have become a popular tool for control design of nonlinear systems, yet it remains unclear when such representations are exact. In this paper, we establish sufficient and necessary conditions under which a controlled nonlinear system admits an exact finite-dimensional Koopman linear representation, which we term Koopman linear embedding. We show that such a system must be transformable into a special control-affine preserved (CAP) structure, which enforces affine dependence of the state on the control input and isolates all nonlinearities into an autonomous subsystem. We further prove that this autonomous subsystem must itself admit a finite-dimensional Koopman linear model with a sufficiently-rich Koopman invariant subspace. Finally, we introduce a symbolic procedure to determine whether a given controlled nonlinear system admits the CAP structure, thereby elucidating whether Koopman approximation errors arise from intrinsic system dynamics or from the choice of lifting functions.

math.OC↗

arXiv2026

Safe Feedback Optimization through Control Barrier Functions

Feedback optimization refers to a class of methods that steer a control system to a steady state that solves an optimization problem. Despite tremendous progress on the topic, an important problem remains open: enforcing state constraints at all times. The difficulty in addressing it lies on mediating between the safety enforcement and the closed-loop stability, and ensuring the equivalence between closed-loop equilibria and the optimization problem's critical points. In this work, we present a feedback-optimization method that enforces state constraints at all times employing high-order control-barrier functions. We provide several results on the proposed controller dynamics, including well-posedness, safety guarantees, equivalence between equilibria and critical points, and local and global (in certain convex cases) asymptotic stability of optima. Various simulations illustrate our results.

math.OC↗