arXiv subjects

Enrique Zuazua

Publications and source records attributed to Enrique Zuazua.

At least 19 recordsLinked to original sources

arXiv2026

Two-term small-time spectral expansions for controllability Gramians

For a controllable single-input linear system in dimension $n$, the controllability Gramian eigenvalues, ordered decreasingly, are known to have the successive small-time orders $T, T^3, \ldots, T^{2n-1}$. We refine this leading-order hierarchy by explicitly computing the next-order coefficient of every eigenvalue and the first-order variation of its eigendirection and of the nested spectral subspaces. The resulting formulas have intrinsic expressions in the orthonormal Krylov basis: eigenvalue corrections describe the action of the dynamics along each Krylov direction, while variations of the spectral subspaces describe coupling to the next direction. We also establish local joint real-analytic dependence on $(T, A, b)$ near $T=0$ for the eigenvalues divided by their leading powers of $T$, consistently oriented eigenvectors, and spectral projectors. This provides convergent expansions with remainders uniform on compact families of controllable pairs. These results yield refined asymptotics for the worst-case minimum energy required to reach a unit target from the origin, the Gramian determinant and condition number, and the Ornstein--Uhlenbeck Gaussian profile in moving principal coordinates. The energy equals the reciprocal of the smallest Gramian eigenvalue, whose eigendirection identifies the most energy-demanding targets for sufficiently small times. We further derive an exact Gramian-weighted energy identity for the forward Fokker--Planck equation and sharp anisotropic short-time smoothing asymptotics using, respectively, the Lyapunov equation and an exact Fourier norm formula combined with the graded Gramian factorization. For symmetric dynamics, the spectral data reconstruct $A$ and determine $b$ up to its global sign.

math.NA↗

arXiv2026

Optimal Actuator Design across Ranks and Control Horizons

We study how actuator rank and control horizon jointly determine worst-case control performance for finite-dimensional linear systems under a fixed total actuator-gain budget. The design variable is the input covariance $X=BB^\top$. Scalar actuators give rank-one matrices $X=bb^\top$; dropping the rank constraint yields their convex hull and a semidefinite benchmark. We identify regimes in which low-rank designs necessarily fall short and others in which they attain the benchmark. At small time, the relaxed maximizer is unique and full rank. If $A$ is cyclic, then, for every $k

math.OC↗

arXiv2026

Transmutation based Quantum Simulation for Non-unitary Dynamics

We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form $A=L^\dagger L$, a structure that arises naturally in standard discretizations of elliptic operators. Our main tool is the Kannai transform, which represents the diffusion semigroup $e^{-AT}$, where $T$ is the final simulation time, as a Gaussian-weighted superposition of unitary wave propagators. For target accuracy $\varepsilon$, this representation leads to a linear-combination-of-unitaries implementation with a Gaussian tail and yields query complexity $\widetilde{O}(\sqrt{\|A\|\,T\,\log(1/\varepsilon)})$, up to the standard dependence on state-preparation and output-norm factors, improving the scaling in $\|A\|$, $T$, and $\varepsilon$ compared with generic Hamiltonian-simulation-based methods. We instantiate the method for the heat equation and biharmonic diffusion under non-periodic physical boundary conditions, and further use it as a subroutine for constant-coefficient linear parabolic surrogates arising in entropy-penalization schemes for the viscous Hamilton--Jacobi equations. In the long-time regime, under a spectral-gap assumption, the same framework gives a structured quantum linear solver by exploiting convergence to the steady state. For normalized positive definite systems $σ(A)\subset[1,κ]$, the solver outputs an $\varepsilon$-approximation to the state proportional to $\mathbf{x}=A^{-1}\mathbf{b}$ with query complexity $\widetilde{O}\left(\frac{\|\mathbf{b}\|}{\|\mathbf{x}\|}\sqrtκ\log^2\frac{\|\mathbf{b}\|}{\varepsilon\|\mathbf{x}\|}\right)$.

quant-ph↗

arXiv2026

Autonomous Flows: Exact Finite Interpolation and Uniform Approximation Obstructions

In dimension at least two, a class of locally Lipschitz vector fields realizes every finite correspondence between distinct inputs and distinct targets at any prescribed positive time, provided that it is linear, its members generate global flows, and it approximates every smooth vector field of bounded support uniformly on bounded sets. The input and target sets may overlap. The proof constructs a smooth autonomous reference flow and finitely many localized correction fields, then uses uniform stability and Brouwer degree to obtain exact endpoints within the approximating class. The argument uses only uniform approximation of the vector fields; it does not require control of their derivatives. Applied to shallow ReLU vector fields, the result gives exact finite interpolation without time-dependent coefficients or additional state variables, together with bounds on width and normalized coefficient strength. In contrast, no continuous autonomous semiflow can exchange and compress two disjoint balls. Its time maps therefore fail to approximate all continuous maps uniformly on compact sets. The results distinguish exact interpolation at finitely many points from uniform control of neighborhoods.

math.OC↗

arXiv2026

Crossover asymptotics and a sharp confinement rate for the viscous Burgers equation

We study the one-dimensional viscous Burgers equation on $(0,L)$ with the conservative boundary conditions $u_x=-u^2$. On the half-line, solutions approach a nonlinear self-similar profile, whereas on a bounded interval they converge to a nonconstant equilibrium. We describe explicitly how the dynamics passes between these states at the critical diffusive scale $t=cL^2$. For compactly supported initial data of mass $M$, the rescaled solution converges as $L\to\infty$ to an explicit crossover profile $Φ_c$. In similarity variables, $Φ_c$ converges to the half-line profile $f_M$ as $c\downarrow0$; after rescaling to domain variables, it converges to the interval equilibrium as $c\to\infty$. We also determine the sharp onset of confinement: \[ \lim_{c\downarrow0}-c\log|Φ_c(0)-f_M(0)|=1\qquad(M\ne0). \] Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in $C^k$ by at most $C_{k,\varepsilon}\exp(-(1-\varepsilon)L^2/t)$, uniformly in $L$, and the exponential constant is optimal. Thus we identify not only the transition scale $L^2$, but also the profile governing the crossover and the sharp rate at which the remote boundary becomes visible. We finally discuss the conclusions that persist for space-dependent diffusivity and illustrate the three asymptotic regimes numerically.

math.AP↗

arXiv2026

Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations

Physical and synthetic models may describe complementary aspects of the same PDE-governed system while receiving different, possibly fragmented, observations. We propose Bi-Objective HYCO (Bi-HYCO), a cooperative framework that retains both representations and their local observational objectives while coupling their predicted states at unlabeled interaction points. These points contain no measurements and do not augment the data; they provide a communication mechanism in the common state space. The two criteria form a vector-valued objective, and weighted scalarizations provide computational realizations. For the deterministic shared-observation algorithm with fixed interaction points, we prove sufficient decrease and finite length of the whole alternating sequence, which converges to a mixed critical point under the stated Kurdyka-Lojasiewicz-type assumptions. Elliptic transmission and two-dimensional Navier-Stokes experiments assess parameter and state reconstruction, noise and scalarization effects, and PINN/XPINN references. Ablations show that removing state interaction while retaining aggregation deteriorates parameter recovery in the tested configurations, particularly for Navier-Stokes.

cs.LG↗

arXiv2026

Boundary observation and control for fractional heat and wave equations

We establish boundary null controllability of the heat equation driven by the integral fractional Laplacian $(-Δ)^s$ on a bounded smooth domain for every $T>0$ and every fractional exponent $s\in(1/2,1)$. The control acts through the singular boundary trace naturally associated with the fractional Dirichlet problem. The main ingredient is a frequency-dependent boundary observability inequality for the associated fractional wave equation, obtained by combining multiplier arguments with the fractional Pohozaev identity. In contrast with the classical wave equation, the observability time deteriorates with the spectral cutoff, reflecting the slow propagation of high-frequency fractional waves. We transfer this estimate to the parabolic problem by transmutation and combine the resulting low-frequency controllability estimates with the high-frequency dissipation of the fractional heat semigroup through a Lebeau-Robbiano iteration. The balance between these two mechanisms yields null controllability precisely in the range $s>1/2$.

math.AP↗

arXiv2026

HYCO: Hybrid-Cooperative Learning for Data-Driven PDE Modeling

We introduce Hybrid-Cooperative Learning (HYCO), a framework for data-driven PDE modeling in which a physics-based solver and a flexible synthetic model are trained as two independent but cooperating components. Rather than imposing the governing equation as a residual on a single network, HYCO alternates between updating the physical parameters and the synthetic model, coupling them through agreement of their predictions.Each component therefore fits the information available to it while acting as a regularizer for the other. This modular formulation accommodates sparse, heterogeneous, or disjoint datasets, avoids differentiating continuous PDE residuals, and combines standard solvers with general data-driven architectures. We further show that HYCO defines an exact potential game and establish equilibrium existence for a convex measure-relaxed model, providing a first structural interpretation of the alternating procedure. On inverse problems for reaction-diffusion systems, a heterogeneous Helmholtz equation, and a shock-forming traffic-flow model, HYCO reconstructs solutions and identifies parameters from sparse or localized observations, improving parameter recovery and extrapolation over uncoupled models, PINN-type methods, and classical solver-based inversion.

math.OC↗

arXiv2026

Interior interpretability with attention rollout: contraction and propagation profiles in Transformers

Feature-attribution methods assign scores relating input variables to a model's output, but do not by themselves characterize how explicitly defined interaction operators compose across its intermediate layers. We introduce \emph{interior interpretability}, a propagation-based perspective on internal model organization, and instantiate it for tabular Transformers using attention rollout. We interpret rollout as a row-stochastic operator encoding attention-mediated propagation between feature tokens. By applying classical Doeblin--Dobrushin contraction theory, we show that a rollout operator with a small Dobrushin coefficient is quantitatively close to a rank-one stochastic matrix whose common row is determined by its normalized column sums. This result gives a structural interpretation to the corresponding rollout propagation profile. In Transformers trained for metabolomic age prediction, the measured rollout contraction strengthens with depth. Trained and randomly initialized models also exhibit different propagation profiles, although the present experiments do not establish the predictive relevance of individual rollout-ranked variables. Exploratory comparisons with PCA and GradientExplainer approximations to SHAP reveal localized agreement among highly ranked variables but weak agreement across complete rankings. Attention rollout is therefore used here as a diagnostic of attention-mediated propagation, not as a causal explanation or faithful attribution of the complete Transformer.

cs.LG↗

arXiv2026

Tracking controllability for finite-dimensional linear systems

This paper develops a functional-analytic characterization of output tracking controllability for finite-dimensional linear systems. By formulating tracking as the surjectivity of the control-to-output map on suitable trajectory spaces, we show that exact tracking is equivalent to a trajectory-space observability inequality associated with the dual input-output structure. This characterization enables a Hilbert Uniqueness Method (HUM) type variational construction of minimum-norm tracking controls and makes explicit the intrinsic regularity requirements on reference trajectories induced by the system dynamics and the output operator. The same framework also yields a natural notion of approximate tracking when exact tracking fails. We provide explicit formulas in the scalar case and report illustrative numerical examples for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and nonsmooth targets.

math.OC↗

arXiv2026

Federated Learning for Object Detection: Enabling Collaborative Drone Learning Without Centralizing Data

Object detection is a fundamental capability for AI-driven perception in safety-critical drone and edge-vision systems, including disaster response, operational security environments, infrastructure monitoring and defense applications. Robust model performance in such environments depends on large, continuously updated datasets. However, training high-performing detectors typically requires centralizing aerial imagery, which raises privacy, regulatory, storage, and bandwidth challenges. This is especially problematic in distributed drone deployments, where visual data is generated onboard and is often impractical or undesirable to transfer to a centralized infrastructure. In this work, we apply Federated Learning (FL) for object detection, enabling drones to improve a shared model while keeping image data local and private. We implement a federated object detection pipeline using the Sherpa.ai FL platform on the KIIT-MiTA dataset, and compare it with Single-drone and Centralized baselines using mean Average Precision (mAP) at IoU thresholds of 0.50 and 0.50-0.95. In our experiments, the proposed FL approach remains close to Centralized training while dramatically improving over Single-drone training, with the best lightweight model (YOLO26 nano), suitable for deployment even on very limited edge infrastructure, achieving relative gains of 52.89% and 67.80% in mAP@0.50 and mAP@0.50:0.95, respectively. These results show that FL enables scalable, high-performing, and privacy-preserving object detection across distributed drone fleets without data centralization.

cs.LG↗

arXiv2026

Turnpike in optimal control and beyond: a survey

The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide class of long-horizon optimal control problems, the optimal trajectory spends most of its time near a steady-state solution (the ''turnpike'') rather than being influenced by the initial or final conditions. In this article, we provide a survey on the turnpike property in optimal control, adding several recent and novel considerations. After some historical insights, we present an elementary proof of the exponential turnpike property for linear-quadratic optimal control problems in finite dimension. Next, we show an extension to nonlinear optimal control problems, with a local exponential turnpike property. On simple but meaningful examples, we illustrate the local and global aspects of the turnpike theory, clarifying the global picture and raising new questions. We discuss key generalizations, in infinite dimension and other various settings, and review several applications of the turnpike theory across different fields.

math.OC↗

arXiv2026

The Coercivity Gap in Neural PDE Solvers: Parameter Escape and Functional Convergence

We study neural approximation of elliptic PDE solutions from a variational perspective. The central point is the distinction between the geometry of neural parameters and the convergence of the corresponding physical states. Even when the original elliptic energy is coercive and strictly convex in the natural energy space, its restriction to a nonlinear neural ansatz may fail to be coercive in parameter space. This failure is caused by non-closedness of neural approximation manifolds and by condensation of neurons, which may generate limiting profiles outside the fixed ansatz class. Nevertheless, the associated state functions may remain bounded and converge strongly to the exact PDE solution. We prove this mechanism for Gaussian wave-packet approximations of a prototypical elliptic model in the whole space, derive convergence rates, and explain how the same state-level stability principle applies to residual minimization methods of PINN type, and HYCO-type hybrid methods. We also discuss relaxation and Tikhonov regularization.

math.NA↗

arXiv2026

Constructive interpolation and generalization rates for neural ODEs: a control perspective

We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NODEs (SA-NODEs). We constructively prove that SA-NODEs are capable of \emph{exact} interpolation of admissible finite datasets, and even satisfy a stronger property that we call \emph{simultaneous cell controllability} (SCC): their flows can map prescribed disjoint cells into arbitrarily small target balls. This property is the mechanism that upgrades interpolation into quantitative generalization, by allowing SA-NODEs to emulate piecewise-constant nonparametric estimators. Consequently, our risk bounds recover the rates of histogram and nearest-neighbor estimators, provided the network width satisfies a conservative scaling with the sample size. Numerical experiments show that trained SA-NODEs achieve competitive -- often lower -- test errors than these baselines. Finally, we show that the explicit time dependence is essential. Although two-layer autonomous NODEs can interpolate geometrically nondegenerate datasets, structural obstructions prevent them from achieving SCC. These limitations, further confirmed numerically, support the view that SA-NODEs provide a minimal effective architecture for learning.

math.OC↗

arXiv2026

Nonlinear Equilibrium Transitions in a Potential Game Model for Federated Learning

In federated learning (FL), a central server typically allocates training efforts to clients. However, from a market-oriented perspective, clients may independently choose their training efforts based on rational self-interest. To study this setting, we propose a potential game framework in which each client's payoff is determined by its individual effort and the rewards provided by the server. The rewards are influenced by the collective efforts of all clients and can be modulated by a reward factor. We first establish the existence of Nash equilibria (NEs) and then investigate their uniqueness in a stationary setting. We show that the NEs depend nonlinearly on the reward factor and exhibit a nonsmooth transition at a critical value, where the stationary potential loses strict curvature, leading to nonunique NEs and a jump between low-effort and high-effort branches. Furthermore, we prove the convergence of the best-response algorithm for computing NEs in our FL game. Finally, we apply the clients' rational efforts derived from the NEs to FL training with various datasets and models, thereby validating the effectiveness of the identified critical reward factor.

cs.LG↗

arXiv2026

Reachability and asymptotics of Gaussian Transformer dynamics

We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability measures. For the mean-field Transformer model with self-attention and affine feed-forward layers, we prove that Gaussian distributions remain exactly Gaussian along the induced flow. This invariance reduces the infinite-dimensional measure dynamics to a finite-dimensional bilinear control system governing the evolution of the mean and covariance, reformulates the expressive capacity of Transformers as a reachability problem for prescribed Gaussian moments, and reveals a novel connection with Riccati-type equations from classical filtering and control. For time-varying controls, we prove exact finite-time reachability of any target Gaussian distribution whose covariance matrix has the same rank as the initial one, this rank constraint being an intrinsic invariant of the dynamics. For time-invariant parameters, we derive explicit spectral conditions leading either to asymptotic stability toward positive-definite equilibria or to finite-time blow-up of the covariance. Numerical experiments complement the theory by showing that practical Transformers with Gaussian inputs remain close to moment-matched Gaussian distributions through early and intermediate layers, while Transformers with prescribed attention matrices reproduce the predicted covariance regimes: bounded evolution in stabilizing configurations and blow-up in destabilizing ones.

cs.LG↗

arXiv2026

Optimal convergence rates for the finite element approximation of the Sobolev constant

We establish optimal convergence rates for the continuous piecewise affine finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p-Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.

math.NA↗

arXiv2026

Sherpa.ai Privacy-Preserving Multi-Party Entity Alignment without Intersection Disclosure for Noisy Identifiers

Federated Learning (FL) enables collaborative model training among multiple parties without centralizing raw data. There are two main paradigms in FL: Horizontal FL (HFL), where all participants share the same feature space but hold different samples, and Vertical FL (VFL), where parties possess complementary features for the same set of samples. A prerequisite for VFL training is privacy-preserving entity alignment (PPEA), which establishes a common index of samples across parties (alignment) without revealing which samples are shared between them. Conventional private set intersection (PSI) achieves alignment but leaks intersection membership, exposing sensitive relationships between datasets. The standard private set union (PSU) mitigates this risk by aligning on the union of identifiers rather than the intersection. However, existing approaches are often limited to two parties or lack support for typo-tolerant matching. In this paper, we introduce the Sherpa.ai multi-party PSU protocol for VFL, a PPEA method that hides intersection membership and enables both exact and noisy matching. The protocol generalizes two-party approaches to multiple parties with low communication overhead and offers two variants: an order-preserving version for exact alignment and an unordered version tolerant to typographical and formatting discrepancies. We prove correctness and privacy, analyze communication and computational (exponentiation) complexity, and formalize a universal index mapping from local records to a shared index space. This multi-party PSU offers a scalable, mathematically grounded protocol for PPEA in real-world VFL deployments, such as multi-institutional healthcare disease detection, collaborative risk modeling between banks and insurers, and cross-domain fraud detection between telecommunications and financial institutions, while preserving intersection privacy.

cs.CR↗