Research
My main research area is Control Theory. The control of systems governed by PDE is a modern research area of applied mathematics, leading to rich, innovative research and to powerful ideas and methods. In particular, my study focuses on controlling parabolic and dispersive equations, as well as neural ordinary differential equations.
My work on the parabolic and dispersive equations focusses on the proof of the existence (or of the non-existence) of controls and on obtaining estimates for the cost of such control. A general mathematical formulation of the question I am interested in is the following: given a PDE system, I want to find a force that allows to pass from a first state to a second one or, alternatively, to give a proof of what are the limitations for the existence of such force. When the answer is positive, I will try to estimate the amount of energy that such a force requires. I have answered these questions both in my PhD thesis and I continue to do research on this topic.
I have worked with a great variety of PDEs that model many physical phenomena. In particular: the heat equation, which governs heat diffusion, as well as other processes, such as particle diffusion or propagation of action potential in nerve cells; the penalized Navier-Stokes system, a PDE which behaves well numerically and approximates the incompressible Navier-Stokes system, which models the motion of Newtonian viscous fluids; and the KdV equation, which describes waves on shallow water surfaces.
To know more about this topic and my contributions to this topic, you may watch any of these talks about the controllability of parabolic equations:
My talk about parabolic equations with vanishing for the Bilbao Analysis and PDE group.
My talk about average controllability of the heat equation for Control in Times of Crisis.
My talk explaining results about the controllability of the heat equation for the Chair of Dynamics, Control, and Numerics of the Friedrich-Alexander-Universität Erlangen-Nürnberg.
Moreover, you can also consult the following blog-post summarising some of my papers:
The following post is about the average controllability of the heat equation.
The following post is about the controllability of the heat equation with vanishing diffusion in networks.
My work on neural ODE is based on determining the properties of the minimizer of the risk minimization functional for non-linear ODE with some specific structure. It is a work in the context of long time horizon control problems, which is a branch of Optimal Control which studies the properties satisfied by the optimal control and optimal trajectories for a long time horizon. This new branch of control theory has many relevant interesting problems. In fact, it has been the second Working Package of the ERC-Dycon. To know more about the controllability properties on neural ODE for a long time horizon, I recommend taking a look at the slides and notes of Borjan Geshkovski, to read the introduction and Part III of his PhD thesis, or to read my paper.
Additionally, I am currently interested in the existence and uniqueness of solutions of partial differential equations and in their asymptotic properties. Finally, I would like to add that my potential interests include any topic of Mathematical Analysis.
For further details, my list of publications can be consulted.