## Research

## Current Research Projects

EPSRC New Horizons Award, Project: Overcoming the Curse of Dimensionality with Tensor Decompositions, 3/2021-3/2024.

UKRI-EPSRC COVID-19 Fund, Project: Optimal Lockdown, 8/2020-2/2021.

EPSRC Grant, Project: Elastic Manufacturing Systems, 9/2020-9/2023.

Gaspard Monge Program for Optimization, Operations Research and Data Science, Project TIDAL: Taming the Curse of Dimensionality in Dynamic Programming Equations, 7/2019--7/2022.

## High-Dimensional Dynamic Programming & 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:

S. Dolgov, D. Kalise and K. Kunisch.

**Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations**, arXiv:1908.01533, to appear in SIAM Journal on Scientific Computing.B. Azmi, D. Kalise and K. Kunisch.

**Data-Driven Recovery of Optimal Feedback Laws through Optimality Conditions and Sparse Polynomial Regression,**Journal of Machine Learning Research 22(48):1−32, 2021.D. Kalise, K. Kunisch and S. Kundu.

**Robust feedback control of nonlinear PDEs by polynomial approximation of Hamilton-Jacobi-Isaacs equations**, SIAM Journal of Applied Dynamical Systems 19(2): 1496–1524, 2020.D. Kalise, K. Kunisch and Z. Rao (eds.)

**Hamilton-Jacobi-Bellman Equations: Numerical Methods and Applications in Optimal Control**, Vol. 21 De Gruyter - Radon Series on Computational and Applied Mathematics, 2018.D. Kalise and K. Kunisch.

**Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs**, SIAM Journal on Scientific Computing 40(2)(2018):A629--A652.D. Kalise, A. Kroener, and K. Kunisch.

**Local minimization algorithms for dynamic programming equations**, SIAM Journal on Scientific Computing 38(3)(2016):A1587--A1615.A. Alla, M. Falcone, and D. Kalise.

**An efficient policy iteration algorithm for the solution of dynamic programming equations**, SIAM Journal on Scientific Computing 37(1)(2015):A181-A200.O. Bokanowski, M. Falcone, R. Ferretti, L. Gruene, D. Kalise, and H. Zidani. V

**alue iteration convergence of epsilon-monotone schemes for stationary Hamilton-Jacobi equations**, Discrete and Continuous Dynamical Systems - Series A 35(9)(2015):4041--4070.

## Optimal Control of Systems Governed by Partial Differential Equations

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:

S. Dolgov, D. Kalise and K. Kunisch.

**Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations**, arXiv:1908.01533, to appear in SIAM Journal on Scientific Computing.B. Azmi, D. Kalise and K. Kunisch.

**Data-Driven Recovery of Optimal Feedback Laws through Optimality Conditions and Sparse Polynomial Regression,**Journal of Machine Learning Research 22(48):1−32, 2021.D. Kalise, K. Kunisch and S. Kundu.

**Robust feedback control of nonlinear PDEs by polynomial approximation of Hamilton-Jacobi-Isaacs equations**, SIAM Journal of Applied Dynamical Systems 19(2): 1496–1524, 2020.D. Kalise, K. Kunisch and K. Sturm.

**Optimal actuator design based on shape calculus**, Mathematical Models and Methods in Applied Sciences 28(13)(2018): 2667--2717.D. Kalise and K. Kunisch.

**Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs**, SIAM Journal on Scientific Computing 40(2)(2018):A629--A652.D. Kalise and A. Kroener.

**Reduced-order minimum time control of advection-reaction-diffusion systems via dynamic programming**, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems MTNS14, 1196-1202, 2014.E. Hernández, D. Kalise, and E. Otárola.

**Numerical approximation of the LQR problem in a strongly damped wave equation**, Computational Optimization and Applications 47(1)(2010):161-178.

## 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.

Related Publications:

J.A. Carrillo, D. Kalise, F. Rossi and E. Trélat.

**Controlling swarms towards flocks and mills**, arXiv:2103.07304, 2021.Y.P. Choi, D. Kalise and A. Peters.

**Collisionless and Decentralized Formation Control for Strings**, arXiv:2102.13621, 2021.G. ALbi, M. Herty, D. Kalise and C. Segala

**. Moment-Driven Predictive Control of Mean-Field Collective Dynamics,**arXiv:2101.01970, 2021.Y.P. Choi, D. Kalise, A. Peters and J. Peszek.

**A collisionless singular Cucker-Smale model with decentralized forcing and applications to formation control for UAVs**, SIAM Journal on Applied Dynamical Systems 18(4)(2019):1954--1981.L. Briceño-Arias, D. Kalise, Z. Kobeisi, M. Lauriere, A. Mateos-Gonzalez and F.J. Silva.

**On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings**, ESAIM: Proceedings and Surveys 65(2019):330--348.J.A. Carrillo, M. Bongini, D. Kalise and R. Bailo.

**Optimal consensus control of the Cucker-Smale model**, IFAC-PapersOnLine 51(3)(2018):1--6.L. Briceño-Arias, D. Kalise, and F.J. Silva.

**Proximal methods for stationary Mean Field Games with local couplings**, SIAM Journal on Control and Optimization 56(2)(2018):801--836.G. Albi, Y.P. Choi, M. Fornasier and D. Kalise.

**Mean field control hierarchy**, Applied Mathematics & Optimization 76(1)(2017):93--135.G. Albi, M. Bongini, E. Cristiani, and D. Kalise.

**Invisible control of self-organizing agents leaving unknown environments**, SIAM Journal on Applied Mathematics 76(4)(2016):1683--1710.M. Bongini, M. Fornasier, and D. Kalise.

**(Un)conditional consensus emergence under perturbed and decentralized feedback controls**, Discrete and Continuous Dynamical Systems - Series A 35(9)(2015): 4071--4094.

## 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:

R. Dutta, S.N. Gomes, D. Kalise and L. Pacchiardi.

**Using mobility data in the design of optimal lockdown strategies for the COVID-19 pandemic,**[arxiv]D. Kalise, N. Nichols, I. Poll, C. Wells and P. Williams.

**Reducing transatlantic flight emissions by fuel-optimised routing****,**Environmental Research Letters 16:025002, 2021.

Y.P. Choi, D. Kalise, A. Peters and J. Peszek.

**A collisionless singular Cucker-Smale model with decentralized formation control**, SIAM Journal on Applied Dynamical Systems 18(4)(2019):1954--1981.G. Albi, M. Bongini, E. Cristiani, and D. Kalise.

**Invisible control of self-organizing agents leaving unknown environments**, SIAM Journal on Applied Mathematics 76(4)(2016):1683--1710.E. Fuentes, D. Kalise, and R. Kennel.

**Smoothened quasi-time-optimal control for the torsional torque in a two-mass system**, IEEE Transactions on Industrial Electronics 63(6)(2016):3954--3963.E. Fuentes, D. Kalise, R.M. Kennel, and J. Rodríguez.

**Cascade-free predictive speed control for electrical drives**, IEEE Transactions on Industrial Electronics 61(5)(2014):2176 - 2184.D. Kalise and I. Lie.

**Modelling and numerical approximation of a 2.5D set of equations for mesoscale atmospheric processes**, Journal of Computational Physics 231(2012):7274-7298.F. Barros, D. Kalise, and C. Martínez.

**General requirement for harvesting antennae at Ca2+ and H+ sinks**, Frontiers in Neuroenergetics, 2:27(2010).J. Fernández, E. Hernández, D. Kalise, V. Muñoz, and M. Zambra. C

**urrent Sheet Thickness in the Plasma Focus Snowplow Model**, Journal of Plasma and Fusion Research Series 8(2009):879-882.