# Discrete-Time Control for Systems of Interacting Objects with Unknown Random Disturbance Distributions: A Mean Field Approach

Discrete-Time Control for Systems of Interacting Objects with Unknown Random Disturbance... We are concerned with stochastic control systems composed of a large number of N interacting objects sharing a common environment. The evolution of each object is determined by a stochastic difference equation where the random disturbance density $$\rho$$ ρ is unknown for the controller. We present the Markov control model (N-model) associated to the proportions of objects in each state, which is analyzed according to the mean field theory. Thus, combining convergence results as $$N\rightarrow \infty$$ N → ∞ (the mean field limit) with a suitable statistical estimation method for $$\rho$$ ρ , we construct the so-named eventually asymptotically optimal policies for the N-model under a discounted optimality criterion. A consumption-investment problem is analyzed to illustrate our results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Discrete-Time Control for Systems of Interacting Objects with Unknown Random Disturbance Distributions: A Mean Field Approach

, Volume 74 (1) – Sep 7, 2015
31 pages

/lp/springer_journal/discrete-time-control-for-systems-of-interacting-objects-with-unknown-YrdOZ6Gqzv
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9312-6
Publisher site
See Article on Publisher Site

### Abstract

We are concerned with stochastic control systems composed of a large number of N interacting objects sharing a common environment. The evolution of each object is determined by a stochastic difference equation where the random disturbance density $$\rho$$ ρ is unknown for the controller. We present the Markov control model (N-model) associated to the proportions of objects in each state, which is analyzed according to the mean field theory. Thus, combining convergence results as $$N\rightarrow \infty$$ N → ∞ (the mean field limit) with a suitable statistical estimation method for $$\rho$$ ρ , we construct the so-named eventually asymptotically optimal policies for the N-model under a discounted optimality criterion. A consumption-investment problem is analyzed to illustrate our results.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Sep 7, 2015

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