Umberto Biccari
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My research interests include the analysis, simulation, and control of dynamical systems and, more recently, the mathematical foundations of Machine Learning and the integration of Artificial Intelligence into control theory and dynamical modeling.
Associated Researcher
University of Deusto
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Associated Researcher at DeustoTech, University of Deusto, Basque Country, Spain. Former postdoctoral researcher at DeustoTech (2017-2022) and AI researcher at Sherpa.ai (2022–2024). I got my PhD summa cum laude from the University of the Basque Country (EHU) in 2016, within the ERC project NUMERIWAVES.
• Ph.D summa cum laude in Mathematics (Sep 2013 – Dec 2016), University of the Basque Country and BCAM – Basque Center for Applied Mathematics, Bilbao, Spain.
• Internship (Mar 2013 – Aug 2013), BCAM – Basque Center for Applied Mathematics, Bilbao, Spain, ERC Advanced Grant FP7-246775 NUMERIWAVES.
• Master degree in applied mathematics (2010 – 2012), University of Florence, Italy.
• Bachelor’s Degree in Mathematics (2007 – 2010), University of Florence, Italy.
PhD’s Thesis. On the controllability of Partial Differential Equations involving non-local terms and singular potentials
Advisor: Prof. Enrique Zuazua – FAU Erlangen-Nürnberg (Germany), Deusto Foundation-University of Deusto (Bilbao, Basque Country, Spain) and UAM (Madrid, Spain)
Description: In this thesis we study the controllability and observability of certain types of Partial Differential Equations, that describes several phenomena arising in many fields of the applied sciences, such as elasticity theory, ecology, anomalous transport and diffusion, material science, filtration in porus media and quantum mechanics. In particular, we focus on the analysis of PDEs with non-local and singular terms.
View the PhD thesis
Master’s Thesis. A free boundary problem for the CaCO3 neutralisation of acid waters
Advisors: Prof. Riccardo Ricci and Dr. Angiolo Farina – University of Florence, Italy
Description: In this thesis, we analyze a parabolic free boundary model arising from a problem of neutralization of acid waters via the filtration through calcium carbonate. After having developed the model according to the Physics, we computed an approximate but reliable solution, investigating its properties and its asymptotic behavior. This analysis has been repeated also in cylindrical and spherical geometry, both configurations being relevant in the description of the physical phenomena at the basis of our model.
Research interests
My research interests include the analysis, simulation, and control of dynamical systems and, more recently, the mathematical foundations of Machine Learning and the integration of Artificial Intelligence into control theory and dynamical modeling. Specifically, my expertise covers:
• Control theory for nonlocal PDE models, such as hyperbolic, parabolic, and dispersive fractional equations, memory-type dynamics, and PDEs with integral potentials;
• Numerical analysis for local and nonlocal PDEs with control or optimization purposes;
• Stochastic and data-driven approaches to control and optimization problems;
• Integration of ML with control theory, focusing on learning-based feedback design and hybrid mathematical–AI frameworks;
• Mathematical analysis of learning algorithms, exploring their structure through tools from dynamical systems, optimal control, and variational analysis.
My research activities also include collaborations in electrical and control engineering, particularly on the modeling, stability analysis, and control of power grids, developed jointly with the University of Mondragón and several industrial partners in the Basque Country. More recently, I have become involved in biomedical research projects, where I apply mathematical modeling and AI-based methodologies to the study of healthy aging.
Publications
2025
- Biccari, U. & Zuazua, E. (2025). Gaussian beam ansatz for finite difference wave equations. Found. Comput. Math., 25(1), 1–54. https://doi.org/10.1007/s10208-023-09632-9
- Biccari, U. (2025). Spiking Neural Networks: a theoretical framework for Universal Approximation and training. (Unpublished, Submitted). arXiv: 2509.21920
- Lyu, K., Biccari, U., & Wang, J. (2025). Robust stabilization of hyperbolic PDE-ODE systems via Neural Operator-approximated gain kernels. (Unpublished, Submitted). arXiv: 2508.03242
- Morales, R. & Biccari, U. (2025). A Multi-Objective Optimization framework for Decentralized Learning with coordination constraints. (Unpublished, Submitted). arXiv: 2507.13983
- Biccari, U., Warma, M., & Zuazua, E. (2025). Boundary observation and control for fractional heat and wave equations. (Unpublished, Submitted). arXiv: 2504.17413
2024
- Antil, H., Biccari, U., Ponce, R., Warma, M., & Zamorano, S. (2024). Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation. Evol. Equ. Control Theory, 13(3), 893–924. https://doi.org/10.3934/eect.2024010
2023
- Abita, R. & Biccari, U. (2023). Multiplicity of solutions for fractional \(q(\cdot)\)-Laplacian equations. J. Elliptic Parabol. Equ., 9(2), 1101–1129. https://doi.org/10.1007/s41808-023-00239-3
- Biccari, U., Song, Y., Yuan, X., & Zuazua, E. (2023). A two-stage numerical approach for the sparse initial source identification of a diffusion-advection equation. Inverse Probl., 39(9), 30. https://doi.org/10.1088/1361-6420/ace548
- Biccari, U. & Zuazua, E. (2023). Multilevel control by duality. Syst. Control Lett., 175, 16. https://doi.org/10.1016/j.sysconle.2023.105502
2022
- Biccari, U., Esteve-Yagüe, C., & Oroya-Villalta, D. J. (2022). Multilevel selective harmonic modulation via optimal control. Appl. Math. Optim., 86(3), 30. https://doi.org/10.1007/s00245-022-09917-5
- Biccari, U., Warma, M., & Zuazua, E. (2022). Control and numerical approximation of fractional diffusion equations. Handbook of Numerical Analysis, Vol. 23.
- Biccari, U. (2022). Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evol. Equ. Control Theory, 11(1), 301–324. https://doi.org/10.3934/eect.2021014
- Biccari, U., Hernández-Santamar\’\ia, V., & Vancostenoble, J. (2022). Existence and cost of boundary controls for a degenerate/singular parabolic equation. Math. Control Relat. Fields, 12(2), 495–530. https://doi.org/10.3934/mcrf.2021032
- Biccari, U. & Zuazua, E. (2022). Multilevel selective harmonic modulation by duality. IFAC-PapersOnLine, 55(16), 56–61.
2020
- Biccari, U., Marica, A., & Zuazua, E. (2020). Propagation of one- and two-dimensional discrete waves under finite difference approximation. Found. Comput. Math., 20(6), 1401–1438. https://doi.org/10.1007/s10208-020-09445-0
- Biccari, U. & Warma, M. (2020). Null-controllability properties of a fractional wave equation with a memory term. Evol. Equ. Control Theory, 9(2), 399–430. https://doi.org/10.3934/eect.2020011
- Biccari, U., Warma, M., & Zuazua, E. (2020). Controllability of the one-dimensional fractional heat equation under positivity constraints. Commun. Pure Appl. Anal., 19(4), 1949–1978. https://doi.org/10.3934/cpaa.2020086
- Biccari, U. & Zuazua, E. (2020). A stochastic approach to the synchronization of coupled oscillators. Frontiers in Energy Research, 8, 115.
2019
- Biccari, U. & Hernández-Santamar\’\ia, V. (2019). Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects. IMA J. Math. Control Inf., 36(4), 1199–1235. https://doi.org/10.1093/imamci/dny025
- Biccari, U., Ko, D., & Zuazua, E. (2019). Dynamics and control for multi-agent networked systems: a finite-difference approach. Math. Models Methods Appl. Sci., 29(4), 755–790. https://doi.org/10.1142/S0218202519400050
- Biccari, U. (2019). Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Math. Control Relat. Fields, 9(1), 191–219. https://doi.org/10.3934/mcrf.2019011
- Biccari, U. & Hernández-Santamar\’\ia, V. (2019). Null controllability of Linear and semilinear nonlocal heat equations with an additive integral kernel. SIAM J. Control Optim., 57(4), 2924–2938. https://doi.org/10.1137/18M1218431
- Biccari, U. & Micu, S. (2019). Null-controllability properties of the wave equation with a second order memory term. J. Differ. Equations, 267(2), 1376–1422. https://doi.org/10.1016/j.jde.2019.02.009
2018
- Biccari, U., Warma, M., & Zuazua, E. (2018). Local regularity for fractional heat equations. Recent advances in PDEs: analysis, numerics and control. In honor of Prof. Fernández-Cara’s 60th birthday. Based on talks given at the workshop, Sevilla, Spain, January 25–27, 2017. https://doi.org/10.1007/978-3-319-97613-6_12
- Biccari, U. & Hernandez-Santamaria, V. (2018). The Poisson equation from non-local to local. Electron. J. Differ. Equ., 2018, 13.
2017
- Biccari, U., Warma, M., & Zuazua, E. (2017). Addendum: “Local elliptic regularity for the Dirichlet fractional Laplacian''. Adv. Nonlinear Stud., 17(4), 837–839. https://doi.org/10.1515/ans-2017-6020
- Biccari, U., Warma, M., & Zuazua, E. (2017). Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud., 17(2), 387–409. https://doi.org/10.1515/ans-2017-0014
2016
- Biccari, U. & Zuazua, E. (2016). Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function. J. Differ. Equations, 261(5), 2809–2853. https://doi.org/10.1016/j.jde.2016.05.019
Projects
• CoDeFeL: Control and Deep and Federated Learning. ERC Advanced PI: Enrique Zuazua (2024 – 2029)
• DyCMaMod, Dinámica y control para el aprendizaje automático y la Modelización (2024 – 2027)
Academy
Academic Courses
• 01.09.2024 – 31.01.2025.Calculus, Bachelor degree in Computer Engineering, (6 ECTS), University of Deusto, Bilbao campus, Spain
• 01.09.2023 – 31.01.2024.Algebra, Bachelor degree in Industrial Engineering, (6 ECTS), University of Deusto, Bilbao campus, Spain
• 01.09.2021 – 31.01.2022. Algebra, Bachelor degree in Computer Engineering, (6 ECTS), University of Deusto, Bilbao campus, Spain
Other Courses
• 01.08.2022 – 12.08.2022.Course on Numerical methods for fractional equation control and optimization, University of Shangai (online)
Certificates
• IKERTRAMOS call 2019: positive evaluation of the Agencia de Calidad del Sistema Universitario Vasco (UNIBASQ) for the research activity in the 6 years period 2013-2018
• PROFESOR DE UNIVERSIDAD PRIVADA: positive evaluation of Aneca for the role of professor of private university” y “CONTRATADO DOCTOR: positive evaluation of Aneca for the role of tenured professor
News
• 01-02.12.2025 Spiking Neural Networks: theoretical framework for universal approximation and training, COPI2A 3rd. meeting, PDF Slides
• 20.11.2025 Deep Operator Networks in Control Theory: Concepts and Applications, FAU MoD/GMU workshop: The Mathematics of Scientific Machine Learning and Digital Twins, PDF Slides
• 20.06.2022 FAU DCN-AvH Mini workshop: Recent advances: (Multilevel Control), FAU – Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany, PDF Slides
• 31.01.2020 Controllability of Fractional Heat Equations under Positivity Constraints, 5th. Congress of young researchers RSME, Castelló, Spain, PDF Slides
• 21.08.2019 Controllability of the 1d fractional heat equation under positivity constraints, 8th workshop on Partial Differential Equations, optimal design and numerics, Benasque, Spain, PDF Slides
• 03.04.2019 Dynamics and control for multi-agent networked systems, Universidad de Cantabria, Santander, Spain, PDF Slides
• 14.03.2019 Dynamics and control for multi-agent networked systems, Friedrich-Alexander Universität, FAU Erlangen-Nürnberg, Germany, PDF Slides
• 05.12.2018 Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects, 1st workshop on dynamics, control and numerics for fractional PDE’s, San Juan, Puerto Rico, PDF Slides
• 30.08.2018 Propagation of one and two-dimensional discrete waves under finite difference approximation, 14th French-Romanian conference in applied mathematics, Bordeaux, France, PDF Slides
• 06.07.2018 Propagation of one and two-dimensional discrete waves under finite difference approximation, University of Craiova, Romania, PDF Slides
• 01.03.2018 Controllability of Partial Differential Equations with integral kernels, MINAKE 2018 – Microlocal and Numerical Analysis, Kinetic Equations Control Conference, Madrid, Spain, PDF Slides
• 29.08.2017 Finite Element approximation of the one-dimensional fractional Poisson equation with applications to numerical control, 7th workshop on Partial Differential Equations, optimal design and numerics, Benasque, Spain, PDF Slides
• 25.08.2017 Control of partial differential equations involving the fractional Laplacian, 7th workshop on Partial Differential Equations, optimal design and numerics, Benasque, Spain, PDF Slides
• 08.03.2017 Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, Universidad Autónoma de Madrid, Spain.
• DyCon post: Moving control strategy for memory-type equations
• DyCon post: Stochastic optimization for simultaneous control
• DyCon post: Synchronized Oscillators
• DyCon post: An eulerian-lagrangian scheme for the problem of the inverse design of hyperbolic transport equations
• DyCon post: Simulation of Fractional Heat Equation
• DyCon post: Propagation of one and two-dimensional discrete waves under finite difference approximation
• DyCon post: Finite Element approximation of the one-dimensional fractional Laplacian
• DyCon post: Controllability of the one-dimensional fractional heat equation under positivity constraints
• DyCon post: WKB expansion for a fractional Schrödinger equation with applications to controllability
• DyCon post: LQR control of a fractional reaction diffusion equation