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Pathwise turnpike and dissipativity results for discrete-time stochastic linear-quadratic optimal control problems

Jonas Schießl, Ruchuan Ou, Timm Faulwasser, Michael Heinrich Baumann, Lars Grüne

Abstract

We investigate pathwise turnpike behavior of discrete-time stochastic linear-quadratic optimal control problems. Our analysis is based on a novel strict dissipativity notion for such problems, in which a stationary stochastic process replaces the optimal steady state of the deterministic setting. The analytical findings are illustrated by a numerical example.

Pathwise turnpike and dissipativity results for discrete-time stochastic linear-quadratic optimal control problems

Abstract

We investigate pathwise turnpike behavior of discrete-time stochastic linear-quadratic optimal control problems. Our analysis is based on a novel strict dissipativity notion for such problems, in which a stationary stochastic process replaces the optimal steady state of the deterministic setting. The analytical findings are illustrated by a numerical example.
Paper Structure (8 sections, 9 theorems, 55 equations, 1 figure, 1 table)

This paper contains 8 sections, 9 theorems, 55 equations, 1 figure, 1 table.

Table of Contents

  1. INTRODUCTION
  2. SETTING AND PRELIMINARIES
  3. Problem formulation
  4. Deterministic Dissipativity and Turnpike
  5. STATIONARY SOLUTIONS AND OVERTAKING OPTIMALITY
  6. STOCHASTIC DISSIPATIVITY AND TURNPIKE
  7. NUMERICAL EXAMPLE
  8. SUMMARY

Key Result

Lemma 1

Assume strict $(x,u)$-dissipativity at $(x^s,u^s)$. Then for each $x_0 \in \mathbb{R}^n$ there exists a constant $C \in \mathbb{R}$ such that for each $\delta > 0$, each control sequence $u(\cdot)$ satisfying $J_N(x_0,u) \leq N\ell(x^s,u^s) + \delta$ and each $\varepsilon > 0$ the value $Q_{\varepsi

Figures (1)

  • Figure 1: Fixed realization $w$ of the noise and corresponding realizations of the optimal solutions $(X_N^*(\cdot),U_N^*(\cdot))$ from \ref{['eq_exampleOCP']} on different horizons $N$ (dashed) and the optimal stationary pair $(X^s_*(\cdot),U^s_*(\cdot))$ (solid red).

Theorems & Definitions (21)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Remark 1: Stationarity in probability measures
  • Lemma 2
  • proof
  • Definition 3
  • Theorem 1
  • proof
  • Lemma 3
  • ...and 11 more