Ergodic switching control for diffusion-type processes

Jose-Luis Menaldi; Maurice Robin

doi:10.3934/puqr.2023003

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2023, Volume 8, Issue 1: 53-74. Doi: 10.3934/puqr.2023003

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Ergodic switching control for diffusion-type processes

Jose-Luis Menaldi^1, and
Maurice Robin^2, ,

1.
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
2.
Fondation Campus Paris-Saclay, 91190 Saint-Aubin, France

Corresponding author: Maurice Robin
Corresponding author: Maurice Robin

Received: April 07, 2022

Accepted: October 04, 2022

Early access: November 2022

Published: March 2023

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Abstract

We consider the control of the discrete component $ n_t $ of a switching Markov process $ x_t = ( z_t, n_t) $ when there is a running cost and an immediate cost $ c(i, j) $ for switching $ n_t $ from $ i $ to $ j $. We study the minimization of the ergodic (or long-term average) total cost. Essentially, this paper treats the case where, for $ n_t=n $ fixed, $ z_t $ is a reflected diffusion or a reflected diffusion with jumps, $ n_t $ being, for fixed $ z $, a continuous-time Markov chain. Using the vanishing discount approach, we extend existing results dealing with the situation where $ n_t $ evolves only by the switching control action and the diffusion is non-degenerate. Moreover, we solve the ergodic problem for a class of diffusions which can be degenerate and for an example with absorbing state.

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Mathematics Subject Classification: 49J40, 60J60, 60J75.

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References

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