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Turnpike in optimal control and beyond: a survey

Emmanuel Trélat, Enrique Zuazua

TL;DR

This survey synthesizes the turnpike phenomenon in optimal control across finite and infinite dimensions, detailing how long-horizon optimal trajectories spend most of their time near a stationary turnpike. It spans linear-quadratic settings with explicit exponential turnpike bounds, extends to local nonlinear problems via the Pontryagin framework, and discusses global, chaotic, and multistage behaviors. The work surveys generalizations to PDEs, mean-field games, dissipativity and weak KAM theory, and nontraditional turnpike sets (periodic, partial, shape-based), highlighting numerical and design applications. Overall, it clarifies the mechanisms (notably Hamiltonian saddle structure and dissipativity) that drive turnpike behavior and outlines practical implications for computation and design in long-horizon optimization problems.

Abstract

The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide class of long-horizon optimal control problems, the optimal trajectory spends most of its time near a steady-state solution (the ''turnpike'') rather than being influenced by the initial or final conditions. In this article, we provide a survey on the turnpike property in optimal control, adding several recent and novel considerations. After some historical insights, we present an elementary proof of the exponential turnpike property for linear-quadratic optimal control problems in finite dimension. Next, we show an extension to nonlinear optimal control problems, with a local exponential turnpike property. On simple but meaningful examples, we illustrate the local and global aspects of the turnpike theory, clarifying the global picture and raising new questions. We discuss key generalizations, in infinite dimension and other various settings, and review several applications of the turnpike theory across different fields.

Turnpike in optimal control and beyond: a survey

TL;DR

This survey synthesizes the turnpike phenomenon in optimal control across finite and infinite dimensions, detailing how long-horizon optimal trajectories spend most of their time near a stationary turnpike. It spans linear-quadratic settings with explicit exponential turnpike bounds, extends to local nonlinear problems via the Pontryagin framework, and discusses global, chaotic, and multistage behaviors. The work surveys generalizations to PDEs, mean-field games, dissipativity and weak KAM theory, and nontraditional turnpike sets (periodic, partial, shape-based), highlighting numerical and design applications. Overall, it clarifies the mechanisms (notably Hamiltonian saddle structure and dissipativity) that drive turnpike behavior and outlines practical implications for computation and design in long-horizon optimization problems.

Abstract

The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide class of long-horizon optimal control problems, the optimal trajectory spends most of its time near a steady-state solution (the ''turnpike'') rather than being influenced by the initial or final conditions. In this article, we provide a survey on the turnpike property in optimal control, adding several recent and novel considerations. After some historical insights, we present an elementary proof of the exponential turnpike property for linear-quadratic optimal control problems in finite dimension. Next, we show an extension to nonlinear optimal control problems, with a local exponential turnpike property. On simple but meaningful examples, we illustrate the local and global aspects of the turnpike theory, clarifying the global picture and raising new questions. We discuss key generalizations, in infinite dimension and other various settings, and review several applications of the turnpike theory across different fields.

Paper Structure

This paper contains 36 sections, 2 theorems, 72 equations, 10 figures.

Table of Contents

  1. A bit of history: discovery of the turnpike phenomenon
  2. Linear-quadratic optimal control problems
  3. Context and preliminary considerations
  4. Main result: exponential turnpike property
  5. Proof of Theorem \ref{['thm_turnpike_LQ']}
  6. Finite-dimensional nonlinear optimal control problems
  7. Optimal control problem
  8. Setting.
  9. Application of the Pontryagin maximum principle.
  10. Static optimization problem
  11. Application of the Lagrange multiplier rule.
  12. Main result: local exponential turnpike property
  13. Heuristic explanation of the turnpike property.
  14. Main result.
  15. Global considerations
  16. ...and 21 more sections

Key Result

Theorem 1

There exist constants $C>0$ and $\nu>0$, depending on $A,B,Q,U$ but not on $T,x_0,x_1$, such that, for every $T>0$, for all $x_0,x_1\in\mathrm{IemR}^n$, for every $t\in[0,T]$,

Figures (10)

  • Figure 1: Illustration of the turnpike phenomenon, as described by Samuelson et al. While the blue trajectory passes through the turnpike, spending most of the time on it, because the initial and final states are far enough, the optimal control strategy in red evolves directly from the initial state towards the final one, because they are close enough.
  • Figure 2: Illustration of the turnpike phenomenon: optimal strategy in dashed blue; quasi-optimal strategy in plain.
  • Figure 3: Graph of $s\mapsto(s-1)^2+(s-2)^2+(4s-s^3)^2$.
  • Figure 4: Locally optimal trajectories near the global static minimizer (in blue), and near the two local static minimizers (in red and black).
  • Figure 5: Locally optimal trajectory steering $(x_0,y_0)$ to $(x_1,y_1)$ in time $T$, of cost $\simeq 52.3983$.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Remark 1
  • Example 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Example 2