# Differential Games and Distributed Energy-Based Control

Differential game theory is the study of problems in which a dynamical system is influenced by a set of players via their control strategies. In the case of nonzero-sum games each player selects its strategy to optimise an individual cost functional. Solving a nonzero-sum differential games involves obtaining solutions to coupled partial differential equations (PDEs). Since closed-form solutions cannot, in general, be found, it is often necessary to seek approximate solutions.  For certain classes of differential games two methods of constructing approximate solutions, which require solving algebraic equations in place of the PDEs, have been developed [1, 2, 3, 4].  The methods have been applied to different applications, amongst others the multi-agent collision avoidance problem, where the problem of steering a team of agents from their initial positions to predefined goals is considered [4, 5].  Ideas from differential game theory have also been applied to the problem of continuously monitoring a region using a team of agents [6, 7].

Mean-field games consider differential games with infinitely many, indistinguishable players. These games are characterised by two PDEs, with a particular forward-backward structure. Again, obtaining closed-form solutions is not possible in general. Hence, approximate solutions for some mean-field games have been developed [8, 9, 10]. Possible applications for the theory include population dynamics and power systems.

On a somewhat different front, distributed energy-based control is of interest to the STABLE-NET project, which is a collaborative project between several universities located in the UK and China.  One part of this project concerns distributed control. In particular it is of interest to develop a distributed approach to energy-based control, possibly incorporating ideas from game theory. Applications for the work include distributed power networks in which renewable energy sources are integrated into the power grid.

[1]  T. Mylvaganam, M. Sassano, A. Astolfi, ” Approximate solutions to a class of nonlinear differential games”, Conference on Decision and Control, 2012

[2]  T. Mylvaganam, M. Sassano, A. Astolfi, ” Approximate solutions to a class of nonlinear differential games using a shared dynamic extension”, European Control Conference, 2013

[3] T. Mylvaganam, M. Sassano, A. Astolfi, “Constructive epsilon-Nash Equilibria for Nonzero-Sum Differential Games”, To appear in Transactions on Automatic Control

[4] T. Mylvaganam, A. Astolfi,  “Approximate Solutions to a Class of Nonlinear Stackelberg Differential Games”, Conference on Decision and Control, 2014

[5] T. Mylvaganam, M. Sassano, A. Astolfi, “A Constructive Differential Game Approach to Collision Avoidance in Multi-Agent Systems”, American Control Conference, 2014

[6] T. Mylvaganam, A. Astolfi, “Approximate Optimal Monitoring: Preliminary Results”, American Control Conference 2012

[7] T. Mylvaganam, A. Astolfi, “Approximate Optimal Monitoring”, European Control Conference 2014

[8] D. Bauso, T. Mylvaganam, A. Astolfi, “Approximate Solutions for Crowd-Averse Robust Mean-Field Games”, European Control Conference 2014

[9] D. Bauso, T. Mylvaganam, A. Astolfi, “A two-point boundary value formulation of a mean-field crowd-averse game”, IFAC World Congress, 2014

[10] T. Mylvaganam, D. Bauso, A. Astolfi, “Mean-Field Games and Two-Point Boundary Value Problems”, Conference on Decision and Control, 2014

People involved in the game theory part:

• Thulasi Mylvaganam (thulasi.mylvaganam06@imperial.ac.uk)
• Mario Sassano (Università di Roma "Tor Vergata", Facoltà di Ingegneria Dipartimento di Ingegneria Civile e Ingegneria Informatica (D.I.C.I.I.) )
• Alessandro Astolfi  (a.astolfi@imperial.ac.uk)

People involved in the mean field games:

• Thulasi Mylvaganam, Dario Bauso (Universitá di Palermo, Dipartimento di Ingegneria Informatica),
• Alessandro Astolfi

Internal people involved in STABLE-NETS

• Thulasi Mylvaganam
• Alessandro Astolfi
• Francesca Boem (f.boem@imperial.ac.uk)
• Thomas Parisini (t.parisini@imperial.ac.uk)
• Stefanie Kuenzel (stefanie.kuenzel06@imperial.ac.uk)
• Bikash Pal (b.pal@imperial.ac.uk)