Submitted – cmc.deusto.eus

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

L. Ignat, E. Zuazua (2025)Optimal convergence rates for the finite element approximation of the Sobolev constant Abstract. We establish optimal convergence rates for the P1 finite element approximation of the…

Boundary observation and control for fractional heat and wave equations

U. Biccari, E. Zuazua (2025) Boundary observation and control for fractional heat and wave equations, https://doi.org/10.48550/arXiv.2504.17413 Abstract. We establish boundary observability and control for the fractional heat equation over arbitrary…

Turnpike in optimal control and beyond: a survey

E. Trélat, E. Zuazua (2025)Turnpike in optimal control and beyond: a survey https://hal.science/hal-05002039 Abstract. The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide…

Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach

T. Crin-Barat, L. Liverani, Ly Shou, E. Zuazua (2025)Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach https://arxiv.org/abs/2503.00236 Abstract. We study the stability of one-dimensional linear hyperbolic systems…

Pointwise constraints for scalar conservation laws with positive wave velocity

N. De Nitti, D. Serre, E. Zuazua (2025) Pointwise constraints for scalar conservation laws with positive wave velocity Abstract. We consider a conservation law with strictly positive wave velocity and…

Exact Sequence Classification with Hardmax Transformers

A. Alcalde Zafra, G. Fantuzzi, E. Zuazua (2025)Exact Sequence Classification with Hardmax Transformers Abstract. We prove that hardmax attention transformers perfectly classify datasets of $N$ labeled sequences in , .…

Optimal L^p-approximation of convex sets by convex subsets

F. Zakaria, I. Ftouhi, E. Zuazua (2025)Optimal L^p-approximation of convex sets by convex subsets Abstract. Given a convex set Ω of R n, we consider the shape optimization problem of…

Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications

Z. Li, K. Liu, L. Liverani, E. Zuazua (2025)Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications Abstract. In this paper, we introduce semi-autonomous neural ordinary differential equations…

Pattern control via diffusion interaction

D. Ruiz-Balet, E. Zuazua (2025)Pattern control via diffusion interaction Abstract. We analyse a dynamic control problem for scalar reaction-diffusion equations, focusing on the emulation of pattern formation through the selection…

Local Stability and Convergence of Unconstrained Model Predictive Control

D. Veldman, E. Zuazua (2025) Local Stability and Convergence of Unconstrained Model Predictive Control Abstract. The local stability and convergence for Model Predictive Control (MPC) of unconstrained nonlinear dynamics based…

Deep Neural Networks: Multi-Classification and Universal Approximation

M. Hernández, E. Zuazua (2025)Approximate and Weighted Data Reconstruction Attack in Federated Learning arXiv:2409.06555 Abstract. We demonstrate that a ReLU deep neural network with a width of 2 and a…

Clustering in Pure-Attention Hardmax Transformers and its Role in Sentiment Analysis

A. Alcalde Zafra, G. Fantuzzi, E. Zuazua (2025)Clustering in Pure-Attention Hardmax Transformers and its Role in Sentiment Analysis Abstract. Transformers are extremely successful machine learning models whose mathematical properties remain…

Approximate and Weighted Data Reconstruction Attack in Federated Learning

E. Wang E, Y. Song, E. Zuazua (2025) Approximate and Weighted Data Reconstruction Attack in Federated Learning arXiv:2308.06822 Abstract. Federated Learning (FL) is a distributed learning paradigm that enables multiple…

Computational performance of the MMOC in the inverse design of the Doswell frontogenesis equation

A. Francisco, U. Biccari, E. Zuazua (2025)Computational performance of the MMOC in the inverse design of the Doswell frontogenesis equation Abstract. Inverse design of transport equations can be addressed by…

Tracking controllability for finite-dimensional linear systems

S. Zamorano, E. Zuazua (2024)Tracking controllability for finite-dimensional linear systems arXiv:2407.18641v1 Abstract. We introduce and analyze the concept of tracking controllability, where the objective is for the state of a…

Averaged observations and turnpike phenomenon for parameter-dependent systems

M. Hernandez, M. Lazar, S. Zamorano (2024) Averaged observations and turnpike phenomenon for parameter-dependent systems arXiv:2404.17455 Abstract. Our main contribution in this article is the achievement of the turnpike property…

Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications

Z. Li, K. Liu, L. Liverani, E. Zuazua (2024)Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications arXiv:2407.17092 Abstract. In this paper, we introduce semi-autonomous neural ordinary differential…

Random Batch Method (RBM) and Model Predictive Control (MPC) for Converter-Dominated Power Systems

C. Vázquez. Random Batch Method (RBM) and Model Predictive Control (MPC) for Converter-Dominated Power Systems (2024) Abstract. This paper investigates the application of Data-based Reduced Order Models, the Random Batch…

FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning

E. Zuazua. FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning (2024) https://doi.org/10.48550/arXiv.2402.13989 Abstract. Federated learning (FL) is a promising framework for learning from distributed data while maintaining privacy. The…

Large-time asymptotics in deep learning

deep learning, Neural ODEs, optimal control, Residual Neural Networks, Supervised Learning, turnpike property

C. Esteve, B. Geshkovski, D. Pighin, E. Zuazua (2025) Large-time asymptotics in deep learning, https://hal.archives-ouvertes.fr/hal-02912516 Abstract. We consider the neural ODE perspective of supervised learning and study the impact of the…

Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation

R. Bianchini, Crin-Barat T., M. Paicu. Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation (2022) Abstract. We prove the nonlinear asymptotic stability of stably stratified solutions…

Spectral inequalities for elliptic pseudo-differential operators on closed manifolds

D. Cardona. Spectral inequalities for elliptic pseudo-differential operators on closed manifolds (2022) Abstract. Let be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral…

Diffusive Relaxation Limit of the Multi-Dimensional Hyperbolic Jin-Xin System

Crin-Barat T., L. Shou. Diffusive Relaxation Limit of the Multi-Dimensional Hyperbolic Jin-Xin System (2022) Abstract. In this paper we study the diffusive relaxation limit of the Jin-Xin system toward viscous…

Controllability and Positivity Constraints in Population Dynamics with age, size Structuring and Diffusion

Yacouba Simporé, Umberto Biccari. Controllability and Positivity Constraints in Population Dynamics with age, size Structuring and Diffusion (2022) Abstract. In this article, we consider the infinite dimensional linear control system…

Null Controllability for a Degenerate Structured Population Model

Yacouba Simporé, Yassine El gantouh, Umberto Biccari. Null Controllability for a Degenerate Structured Population Model (2022) Abstract. In this paper, we consider the infinite dimensional linear control system describing population…

Sparse approximation in learning via neural ODEs

deep learning, Neural ODEs, Nonlinear systems, optimal control, Sparsity, Supervised Learning

Esteve C., Geshkovski B. Sparse approximation in learning via neural ODEs (2021) Abstract. We consider the continuous-time, neural ordinary differential equation (neural ODE) perspective of deep supervised learning, and study the…

Special issue in the honor of Enrique Zuazua’s 60th birthday

G. Buttazzo, E. Casas, L. de Teresa, R. Glowinski, G. Leugering, E. Trélat and X. Zhang (eds.). Special issue in the honor of Enrique Zuazua's 60th birthday (2021) ESAIM: Control, Optimisation…

Optimal control for neural ODE in a long time horizon and applications to the classification and simultaneous controllability problems

Jon Asier Bárcena-Petisco. Optimal control for neural ODE in a long time horizon and applications to the classification (2021) Abstract. We study the optimal control in a long time horizon…

Optimal control of linear non-local parabolic problems with an integral kernel

Umberto Biccari, Víctor Hernández-Santamaría, Loic Louison, Abdennebi Omrane. Optimal control of linear non-local parabolic problems with an integral kernel. (2021) Abstract. We consider a linear non-local heat equation in a bounded…

Blow-up results for a logarithmic pseudo-parabolic p(.)-Laplacian type equation

Abita Rahmoune, Umberto Biccari. Blow-up results for a logarithmic pseudo-parabolic -Laplacian equation. (2021) Abstract. In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic p(.)-Laplacian type equation…

Local null controllability of the penalized Boussinesq system with a reduced number of controls

J.A. Bárcena-Petisco, Kevin Le Balc'H. Local null controllability of the penalized Boussinesq system with a reduced number of controls (2021) Abstract. In this paper we consider the Boussinesq system with homogeneous…

Multiplicity of solutions for fractional q(.)-Laplacian equations

Abita Rahmoune, Umberto Biccari. Multiplicity of solutions for fractional -Laplacian equations. (2021) Abstract. In this paper, we deal with the following elliptic type problem where is a measurable function and is…

Control and Deep Learning: Some connections

B. Geshkovski, E. Zuazua. Control and Deep Learning: Some connections (2021) This note is an extended abstract for a talk given by the second author during the workshop ”Challenges in…

Reachable subspaces, control regions and heat equations with memory

control regions, Controlled heat equations with memory, reachable subspaces

G. Wang, Y. Zhang, E. Zuazua. Reachable subspaces, control regions and heat equations with memory. (2021) Abstract. We study the controlled heat equations with analytic memory from two perspectives: reachable…

Analysis and numerics solvability of backward-forward conservation laws

Backward-forward method, Conservation Laws, Identification problems, Non-smooth optimization problem, Wave-front tracking algorithm, Weak-entropy solutions

Thibault Liard, Enrique Zuazua. Analysis and numerics solvability of backward-forward conservation laws. (2020) Abstract. In this paper, we study the problem of initial data identification for the one-dimensional Burgers equation.…

The turnpike with lack of observability

Pighin D., Sakamoto N. The turnpike with lack of observability (2020) Abstract: We obtain turnpike results for optimal control problems with lack of stabilizability in the state equation and/or detectability in…

Stochastic optimization methods for the simultaneous control of parameter-dependent systems

Umberto Biccari, Ana Navarro-Quiles, Enrique Zuazua. Stochastic optimization methods for the simultaneous control of parameter-dependent systems (2020) Abstract. We address the application of stochastic optimization methods for the simultaneous control of…

Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term

heat equation, singular limits, spectral decomposition, transport equation, uniform controllability

Bárcena-Petisco J.A., Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term (2020). HAL Id: hal-02455632 Abstract. In this paper we consider the heat equation with…

A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diﬀusive models

I. Mazari, D. Ruiz-Balet. A fragmentation phenomenon for a non-energetic optimal control problem: optimisation of the total population size in logistic diﬀusive models (2020) Abstract. Following the recent works [9, 17,…

Asymptotic behavior of scalar convection-diffusion equations

Enrique Zuazua. Asymptotic behavior of scalar convection-diffusion equations (2020) Abstract. In these lecture notes, we address the problem of large-time asymptotic behaviour of the solutions to scalar convection-diffusion equations set in…

A PDE-ODE model for traffic control with autonomous vehicles

Thibault Liard, Raphael Stern, Maria Laura Delle Monache. A PDE-ODE model for traffic control with autonomous vehicles. (2020) Abstract. We introduce a coupled PDE-ODEs model to describe mixed traffic with…

A controlled multiscale model for traffic regulation via autonomous vehicles

T. Liard, . A controlled multiscale model for traffic regulation via autonomous vehicles. (2019) Abstract. Autonomous vehicles ($AV$s) allow new ways of regulating the traffic flow on road networks. Most…

On entropic solutions to conservation laws coupled with moving bottlenecks

T. Liard, Benedetto Piccoli. On entropic solutions to conservation laws coupled with moving bottlenecks. (2019) Abstract. Moving bottlenecks in road traffic represent an interesting mathematical problem, which can be modeled…

Inverse design for the one-dimensional Burgers equation

T. Liard, E. Zuazua. Inverse design for the one-dimensional Burgers equation. (2019) Abstract. In this paper, we study the problem of inverse design for the one-dimensional Burgers equation. This problem…

Model reduction of converter-dominated power systems by Singular Perturbation Theory

U. Biccari, N. Sakamoto, E. Unamuno, D. Madariaga, E. Zuazua, J.A. Barrena. Model reduction of converter-dominated power systems by Singular Perturbation Theory Abstract: The increasing integration of power electronic devices is driving the development of more advanced tools and methods for…

Mathematical model to asses impact of vih/aids infection in the spread of tuberculosis

Nkamba L. Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis (2019) Abstract: Tuberculosis is a serious problem of public health in several countries.…