A Boltzmann approach to mean-field sparse feedback control

Abstract We study the synthesis of optimal control policies for large-scale multi-agent systems. The optimal control design induces a parsimonious control intervention by means of l , sparsity-promoting control penalizations. We study instantaneous and infinite horizon sparse optimal feedback controllers. In order to circumvent the dimensionality issues associated to the control of large-scale agent-based models, we follow a Boltzmann approach. We generate (sub)optimal controls signals for the kinetic limit of the multi-agent dynamics, by sampling of the optimal solution of the associated two-agent dynamics. Numerical experiments assess the performance of the proposed sparse design.

[1]  B. Piccoli,et al.  Multiscale Modeling of Pedestrian Dynamics , 2014 .

[2]  Karl Kunisch,et al.  Infinite Horizon Sparse Optimal Control , 2016, J. Optim. Theory Appl..

[3]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[4]  Alfio Borzì,et al.  Modeling and control through leadership of a refined flocking system , 2015 .

[5]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[6]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[7]  Maurizio Falcone,et al.  An Efficient Policy Iteration Algorithm for Dynamic Programming Equations , 2013 .

[8]  G. Toscani,et al.  Kinetic models of opinion formation , 2006 .

[9]  Massimo Fornasier,et al.  Mean-Field Pontryagin Maximum Principle , 2015, J. Optim. Theory Appl..

[10]  Cédric Villani,et al.  On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations , 1998 .

[11]  Lorenzo Pareschi,et al.  Kinetic description of optimal control problems and applications to opinion consensus , 2014, 1401.7798.

[12]  Guy Theraulaz,et al.  Self-Organization in Biological Systems , 2001, Princeton studies in complexity.

[13]  Rainer Hegselmann,et al.  Opinion dynamics and bounded confidence: models, analysis and simulation , 2002, J. Artif. Soc. Soc. Simul..

[14]  Massimo Fornasier,et al.  Mean Field Control Hierarchy , 2016, Applied Mathematics & Optimization.

[15]  M. Fornasier,et al.  Mean-Field Optimal Control , 2013, 1306.5913.

[16]  Giacomo Albi,et al.  Invisible Control of Self-Organizing Agents Leaving Unknown Environments , 2015, SIAM J. Appl. Math..

[17]  Lorenzo Pareschi,et al.  On a Kinetic Model for a Simple Market Economy , 2004, math/0412429.

[18]  Karl Kunisch,et al.  Local Minimization Algorithms for Dynamic Programming Equations , 2015, SIAM J. Sci. Comput..

[19]  Felipe Cucker,et al.  Emergent Behavior in Flocks , 2007, IEEE Transactions on Automatic Control.

[20]  Lorenzo Pareschi,et al.  Binary Interaction Algorithms for the Simulation of Flocking and Swarming Dynamics , 2012, Multiscale Model. Simul..

[21]  M. Fornasier,et al.  Sparse stabilization and optimal control of the Cucker-Smale model , 2013 .

[22]  G Albi,et al.  Boltzmann-type control of opinion consensus through leaders , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  A. Bertozzi,et al.  A Nonlocal Continuum Model for Biological Aggregation , 2005, Bulletin of mathematical biology.

[24]  Lorenzo Pareschi,et al.  Reviews , 2014 .

[25]  P. Lions,et al.  Mean field games , 2007 .

[26]  Jesús Rosado,et al.  Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model , 2010, SIAM J. Math. Anal..

[27]  Massimo Fornasier,et al.  Mean-field sparse optimal control , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[29]  Massimo Fornasier,et al.  Un)conditional consensus emergence under perturbed and decentralized feedback controls , 2015 .