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September 2018 Controlled equilibrium selection in stochastically perturbed dynamics
Ari Arapostathis, Anup Biswas, Vivek S. Borkar
Ann. Probab. 46(5): 2749-2799 (September 2018). DOI: 10.1214/17-AOP1238

Abstract

We consider a dynamical system with finitely many equilibria and perturbed by small noise, in addition to being controlled by an “expensive” control. The controlled process is optimal for an ergodic criterion with a running cost that consists of the sum of the control effort and a penalty function on the state space. We study the optimal stationary distribution of the controlled process as the variance of the noise becomes vanishingly small. It is shown that depending on the relative magnitudes of the noise variance and the “running cost” for control, one can identify three regimes, in each of which the optimal control forces the invariant distribution of the process to concentrate near equilibria that can be characterized according to the regime. We also obtain moment bounds for the optimal stationary distribution. Moreover, we show that in the vicinity of the points of concentration the density of optimal stationary distribution approximates the density of a Gaussian, and we explicitly solve for its covariance matrix.

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Ari Arapostathis. Anup Biswas. Vivek S. Borkar. "Controlled equilibrium selection in stochastically perturbed dynamics." Ann. Probab. 46 (5) 2749 - 2799, September 2018. https://doi.org/10.1214/17-AOP1238

Information

Received: 1 April 2015; Revised: 1 October 2017; Published: September 2018
First available in Project Euclid: 24 August 2018

zbMATH: 06964348
MathSciNet: MR3846838
Digital Object Identifier: 10.1214/17-AOP1238

Subjects:
Primary: 35R60
Secondary: 93E20

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 5 • September 2018
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