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Stochastic Optimal Control of Interacting Particle Systems

Friday, 10th January 2020

Location: Technische Universität Berlin
Main building, Room H 3005
Straße des 17. Juni 135, 10623 Berlin

Guests are welcome!

Programme

Friday, 10th January 2020

Programme
15:00	Stochastic Optimal Control of Ferromagnetic Spin Systems in a Heat Bath Andreas Prohl, Universität Tübingen, Germany
15:50	Coffee Break
16:10	An Existence Result for a Class of Potential Mean Field Games of Controls Laurent Pfeiffer, Inria-Saclay, Center of Applied Mathematics, Ecole Polytechnique, Paris, France
16:35	Optimal control of mean field equations with monotone coefficients and applications in neuroscience Antoine Hocquet, Technische Universität Berlin, Germany
17:00	Informal get-together ("Stammtisch")

Abstracts

Stochastic Optimal Control of Ferromagnetic Spin Systems in a Heat Bath

Andreas Prohl, Universität Tübingen, Germany

I motivate a model to control ferromagnetic spin systems (both, finite and infinite) in the presence of thermal fluctuations by exterior forcing. For finite spin systems, I compare the use of Pontryagin's maximum principle for a stochastic gradient method (which requires to solve high-dimensional BSDEs) with the use of the dynamic programming principle (where to probabilistically solve the linearized, transformed HJB equation avoids to compute BSDEs) to numerically solve the optimization problem.

This is joint work with M. Jensen (U Sussex, Brighton), T. Dunst, C. Schellnegger (U Tübingen), A.K. Majee (IIT Delhi), and G. Vallet (U Pau).

An Existence Result for a Class of Potential Mean Field Games of Controls

Laurent Pfeiffer, Inria-Saclay, Center of Applied Mathematics, Ecole Polytechnique, Paris, France

Mean field game theory aims at describing a Nash equilibrium between a very large number of agents, each of them solving an optimal control problem. I will present an existence result for a model where the cost functional to be minimized by each agent involves a price variable depending on the average control (with respect to all agents). This situation typically arises in Cournot models, where the price of some raw material is an increasing function of the total demand.
At a mathematical level, the problem is formulated as a set of coupled forward and backward PDEs, which coincides with the optimality conditions of a mean field type optimal control problem.

Reference: J. Frédéric Bonnans, Saeed Hadikhanloo, Laurent Pfeiffer. Schauder Estimates for a Class of Potential Mean Field Games of Controls. Appl. Math. Optim., online first, 2019.

Optimal Control of Mean Field Equations with Monotone Coefficients and Applications in Neuroscience

Antoine Hocquet, Technische Universität Berlin, Germany

We are interested in the optimal control problem associated with an appropriate
cost functional under a mean-field type constraint of the form

dX_t = b(t, X_t, L(X_t), a_t) dt + σ(t, X_t, L(X_t) ,a_t) dW_t    (*)

with monotonicity assumptions on the coefficients, and where for practical
purposes the control a_t is deterministic. Under stronger assumptions,
the mathematical treatment of (*) was investigated in the past decade by Carmona,
Delarue (among others). However, these works do not seem to cover
the monotone case, which was suggested as a model for neuron networks by
Fitz-Hugh and Nagumo, with a first mathematical investigation by
Baladron and co-authors in 2012. After addressing the existence of minimizers via
a martingale approach, we show a maximum principle for (*), and investigate a
gradient algorithm for the approximation of the optimal control. This is joint work with Alexander Vogler (TU Berlin).