# LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: The Non-Ergodic Case

## TL;DR

This paper develops an LP framework for deterministic discrete-time long-run average optimal control problems, especially addressing cases where the optimal value depends on initial conditions, expanding the theoretical understanding of such problems.

## Contribution

It introduces a novel LP formulation and duality approach for non-ergodic long-run average control problems with initial condition dependence.

## Key findings

- LP formulation characterizes the optimal value
- Dual problem provides optimality conditions
- Addresses non-ergodic cases with initial dependence

## Abstract

We formulate and study the infinite dimensional linear programming (LP) problem associated with the deterministic discrete time long-run average criterion optimal control problem. Along with its dual, this LP problem allows one to characterize the optimal value of the optimal control problem. The novelty of our approach is that we focus on the general case wherein the optimal value may depend on the initial condition of the system.