Current Research Projects

High-Dimensional Hamilton-Jacobi-Bellman PDEs

I'm interested in the design of reliable computational methods for the solution of optimal control problems and differential games using dynamic programming. Here, the optimal value function is globally characterized as the viscosity solution of a first-order fully nonlinear PDE, the Hamilton-Jacobi-Bellman equation, over the state space of the system dynamics. However, this approach is strongly limited to low-dimensional dynamics -the curse of dimensionality-, and therefore an important research topic is the design of dynamic programming-based schemes wich are able to handle high-dimensional problems. This is a fundamental problem in optimal control theory. My research focuses on the analysis and application of computational methods to order to deliver reasonable computation times for high-dimensional online feedback synthesis. In the past, I have worked on high-order monotone iterative schemes for Hamilton-Jacobi-Bellman equations related to optimal control and differential games, as well as on policy iteration algorithms for dynamic programming, and the use of semismooth Newton methods for an accurate control approximation.  Recently, I have started exploring techniques related to global polynomial approximation, tensor decomposition methods, and casaulity-free approaches in the context of nonlinear regression and artificial neural networks.

Selected Publications:

Optimal Control of Systems Governed by PDEs

I'm interested in computational methods for the design of optimal feedback controllers for systems governed by partial differential equations. Only simpler cases such as the linear quadratic regulator problem are well-understood from both theoretical and computational perspectives. I focus on the design and analysis of control schemes for the optimal control of PDE's via dynamic programming-based methods. In particular, by applying techniques stemming from modern linear systems theory, such as Riccati equations for control synthesis and model reduction, we recover finite-dimensional controllers and study its convergence and performance. I also work on feasible implementations of the HJB synthesis, including minimum time and control-constrained problems.  In the recent years, we have started to study design problems related to optimal shape and location of actuators and sensors and their impact on the closed-loop performance.

Selected Publications:

Controlling Agent-based Dynamics Across Scales

Over the last years, the study of multi-agent systems has become a topic of increasing interest in mathematics, biology, sociology, and engineering, among many other disciplines. Multi-agent systems are usually modelled as a large-scale set of particles interacting under simple binary rules, such as attraction, repulsion, and alignment forces. The wide applicability of this setting ranges from modelling the collective behaviour of bird flocks, to the study of data transmission over communication networks, including the description of opinion dynamics in human societies, and the formation control of platoon systems. Borrowing a leaf from statistical mechanics, many of these applications admit a mulsticale descrption, i.e., the system can be described in terms of its microscopic/particle dynamics, or through the evolution of meso/macroscopic state such as the density of agents. I'm interested in designing control algorithms for such systems having in mind a mulsticale descrption of the dynamics -if possible-. In particular, we aim at optimality-based formulations, which allow the policy maker to make a rational use of resources and to impose strong constraints on the problem. This latter enforced by the inclusion of non-smooth ℓ-1 costs and constraints. Recently, such a multiscale design approach has led us to the study of mean field optimal control and mean field games, as well as to the study of control strategies at the kinetic level.

Selected Publications:

Applied Mathematics in Engineering and Natural Sciences

I'm always interested in working side by side with engineering, scientists, and practitioners in general who can benefit from the methods we have developed over the years. I have effectively collaborated with people in control engineering, power electronics, behavioral ecologists, biologists, atmospheric scientists, and experimental physicists.

Selected publications: