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The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension

Emmanuel Trélat, Xingwu Zeng, Can Zhang

Abstract

In this paper, we establish an exponential periodic turnpike property for linear quadratic optimal control problems governed by periodic systems in infinite dimension. We show that the optimal trajectory converges exponentially to a periodic orbit when the time horizon tends to infinity. Similar results are obtained for the optimal control and adjoint state. Our proof is based on the large time behavior of solutions of operator differential Riccati equations with periodic coefficients.

The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension

Abstract

In this paper, we establish an exponential periodic turnpike property for linear quadratic optimal control problems governed by periodic systems in infinite dimension. We show that the optimal trajectory converges exponentially to a periodic orbit when the time horizon tends to infinity. Similar results are obtained for the optimal control and adjoint state. Our proof is based on the large time behavior of solutions of operator differential Riccati equations with periodic coefficients.
Paper Structure (16 sections, 8 theorems, 101 equations, 1 figure)

This paper contains 16 sections, 8 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Formulation of the LQ optimal control problem
  4. The periodic turnpike theorem
  5. Examples
  6. Periodic heat equation
  7. Periodic wave equation
  8. A numerical simulation
  9. Auxiliary results
  10. Reminders on differential Riccati equations
  11. Properties of the evolution operator
  12. Exponential convergence estimate
  13. Proof of Theorem \ref{['periodictarget']}
  14. Further comments
  15. Nonlinear case
  16. ...and 1 more sections

Key Result

Theorem 2.1

Assume that $(A(\cdot),B(\cdot))$ is exponentially $\theta$-periodic stabilizable, and that $(A(\cdot),C(\cdot))$ is exponentially $\theta$-periodic detectable. Then the problem $(Per)_\theta$ has a unique solution $(y_\theta(\cdot),u_\theta(\cdot),\lambda_\theta(\cdot))$ (extended by $\theta$-perio for every $t\in[0,T]$.

Figures (1)

  • Figure 1: Example of a periodic turnpike.

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Theorem 2.1
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 7 more