Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - DYCON (Dynamic Control and Numerics of Partial Differential Equations)

Teaser

DyCon: Dynamic ControlThe overall objective of the DyCon project is to make a breakthrough contribution in the broad area of Control of Partial Differential Equations (PDE) and their numerical approximation methods by addressing key unsolved issues appearing systematically in...

Summary

DyCon: Dynamic Control
The overall objective of the DyCon project is to make a breakthrough contribution in the broad area of Control of Partial Differential Equations (PDE) and their numerical approximation methods by addressing key unsolved issues appearing systematically in real life applications.
Partial Differential Equations (PDEs) are one of the main modelling ingredients in a diverse array of fields, from the management of natural resources, to meteorology, aeronautics, biomedicine or social behaviour, to name only a few. It constitutes a classical field in Mathematics in which significant progress has been done in the last decades. Despite of this, some of the key mathematical issues remain unsolved, particularly when addressing their dynamic control, aiming to understand how the dynamics of solutions can be driven to a desired target by means of external actuators or the tuning of some of the relevant parameters entering in the system. This wide class of open problems is amplified when addressing the challenging issue of developing the numerical techniques allowing to implement these control strategies computationally.
DyCon pursues three objectives: 1) to contribute with new key theoretical methods and results, 2) to develop the corresponding numerical tools, and 3) to build up new computational software, the DyCon-COMP computational platform, thereby bridging the gap to applications.
One of the problems that the DyCon project addresses from a mathematical perspective is the development of optimal control strategies in large temporal horizons, particularly important in those challenges that society faces in the medium and long term: energy, climate change, epidemiology, neurodegenerative diseases, waste management, wealth distribution, multiculturalism and immigration.
The strength of the mathematical approach to these challenges, in symbiosis with Scientific Computing, lies on the fact that the main analytical and computational discoveries can then be implemented on a variety of models, addressing different applications, as those described above.
DyCon also addresses relevant problems such as those related to: a) models involving memory effects, in which the dynamics is of an integro-differential nature, making the purely PDE approach insufficient; b) the need in practical applications to consider natural constraints and controls and states, which makes some of the ideal mathematical control strategies impossible to be implemented; and c) the interplay between discrete and continuous modelling, particularly important when addressing societal challenges.

The global objective of DyCon is not only to provide the mathematical answer to some of these key issues, but also to develop a computational platform allowing for a systematic numerical treatment of these problems, aiming to develop a valuable instrument for technological transfer.

Work performed

• The problem of identifying the diffusivity coefficient on elliptic problems out of the resolvent was formulated and completely solved in one space dimension, describing its use in the development of greedy algorithms to obtain optimal approximations (in the sense of the Kolmogorov width) of parameter dependent problems.
• The turnpike property was re-interpreted in the context of hyperbolic dynamical systems, showing that the stable and unstable manifolds in optimality systems provide important insight for the appearance of the turnpike. The notion of turnpike has also been extended to simplified versions of optimal shape design problems for parabolic dynamics.
• A “waiting time” phenomenon was exhibited for state-constrained controllability problems, both for finite and infinite-dimensional dynamics. This was achieved following a novel recursive technique, combining dissipative effects (when present) and following a continuous path of steady states.
• At present we are developing a novel computational method allowing to recover all possible inversions for hyperbolic scalar conservations laws in one space dimension, complementing in a significant manner the previous existing theoretical results in the field.
• We have extended the moving control strategy proposed previously by our team to more general memory-type equations, including finite-dimensional, and parabolic and hyperbolic infinite-dimensional systems, possibly involving non-local terms.
• In the context of population dynamics, the so-called “shadow system”, arising as a singular limit when the diffusivity in one of the populations tends to infinity, was also satisfactorily solved form the controllability and numerics viewpoint.
• We analysed the dynamics and the control theoretical properties of multi-agent networked models with respect to the number of agents entering in the system. We first established a link with the spatial semi-discretization of parabolic equations, showing that, in general, the time and cost of control are unbounded in the infinite-agents limit. Our results establish a bridge among the existing ones in the literature, where finite and infinite-dimensional dynamics were considered separately
• Intensive analytical and computational research has also been developed for the guidance by repulsion model in which a finite number of drivers actively interact so to drive a crowd of evaders to a given final gate. These results reproduce in a promising manner well known driver-evader competition dynamics in animals and humans.
• The DyCon Computational Platform has been developed in a twofold manner combining the DyCon Blog and the DyCon Computational Toolbox, an open source MATLAB library built upon the code created by our team. It aims to unify all the computational developments obtained throughout the DyCon Project to create a unique tool for solving problems of Optimal Control through the adjoint methodology.

Final results

DyCon has been conceived and run in a holistic manner, integrating from the very early the research efforts in the mathematical aspects and the software development. This has been done, with the aim of increasing the potential of interaction of the team with researchers in other fields and other players in the broad European space of Research and Innovation.
The managing and computational engineers’ members of the team collaborate very actively with the researchers, not only on the regular administrative duties but also on the identification and organization of their potential contribution to the DyCon agenda.
The DyCon Computational Platform, together with its Blog, are evolving in a very positive manner and are becoming a very visible source of inspiration and software for many researchers all around the world. The numerous algorithms, tutorials, sample codes, software and simulations developed within the project are available via the DyCon Computational Laboratory on the web of the project. Freely available via the project webpage, these methods and tools have also been released via Zenodo, GitHub and MathWorks. The DyCon Computational Toolbox is an open source MATLAB library built upon the code created by our team.
As an example of the successful output of this combined analytical-computational efforts, the simulation work developed in the modelling and control of interacting agents within the “guidance by repulsion” paradigm, has produced very interesting new scenarios of computational dynamics that have led to deep analysis questions, and allowing the team members to interact with researchers in other areas, including Social Sciences.
The DyCon research achievements obtained so far are diverse. In the context of the control of nonlinear reaction of diffusion arising in Social Sciences and Biology the team has achieved sharp results on the control of these systems, preserving the natural constraints, either towards travelling waves or to steady states. Significant progress has been achieved also in the context of non-local models, and fractional diffusion where it has been established that a critical minimal amount of diffusivity is necessary to assure the controllability of those systems.
DyCon project has evolved according to the original plan and the complete development of its research program is expected. Special emphasis will be done in the context of free boundary problems, a grand challenge in this area in which very little is known.

Website & more info

More info: http://cmc.deusto.eus/.