Exponential Turnpike Property in Optimal Control

• Exponential Turnpike Property is a fundamental concept in optimal control, stating that optimal trajectories remain exponentially close to a steady-state reference over most of a long time-horizon.
• It leverages Hamiltonian hyperbolicity and Riccati equation frameworks to decompose trajectories into central steady arcs with transient boundary layers.
• This property underpins efficient numerical algorithms, feedback design, and model reduction techniques in nonlinear, infinite-dimensional, and stochastic control problems.

The exponential turnpike property is a central result in optimal control theory, characterizing the asymptotic structure of optimal trajectories on large (finite) time-horizons. It states that—for a wide class of optimal control problems, including nonlinear, linear-quadratic, and infinite-dimensional systems—the optimal state, control, and adjoint variables remain, for most of the time interval, exponentially close to a certain reference (typically steady-state) solution obtained from an associated static problem. Departures from this steady arc occur only in boundary layers near the initial and terminal times. This property provides both deep theoretical insights and practical tools for numerical computation, feedback design, and problem reduction over long horizons.

1. Mathematical Statement of the Exponential Turnpike Property

At the core of the exponential turnpike property is a quantitative estimate of the following form. Consider a finite-dimensional nonlinear control problem

$$\begin{cases} \dot{x}(t) = f(x(t), u(t)), \quad t \in [0, T],\ x(0) = x_0,\ J_T(u(\cdot)) = \int_0^T L(x(t), u(t))\,dt + \Phi(x(T)). \end{cases}$$

Let $(\bar x, \bar u)$ denote an optimal steady-state solution to the associated static problem (minimizing $L(x,u)$ subject to $f(x, u) = 0$). The exponential turnpike property asserts that, under suitable regularity and controllability conditions and for large $T$, the optimal solution $(x^T(t), u^T(t))$, together with the adjoint variable $\lambda^T(t)$ from the Pontryagin maximum principle, satisfy for all $t \in [0, T]$ the bound

$$\|x^T(t) - \bar x\| + \|\lambda^T(t) - \bar \lambda\| + \|u^T(t) - \bar u\| \leq C_1 \left( e^{-C_2 t} + e^{-C_2 (T - t)} \right),$$

where $C_1, C_2 > 0$ are independent of $T$ and $(\bar x, \bar \lambda, \bar u)$ is the extremal triple of the static problem (Trélat et al., 2014). This estimate encapsulates the exponentially small deviation away from steady-state along the whole temporal bulk, with only transient arcs visible near the endpoints.

For infinite-dimensional, linear-quadratic, or periodic systems, analogous estimates hold in suitable function norms (e.g., Hilbert space norms), and the reference can be a periodic trajectory rather than a static point (Trelat et al., 2016, Trélat et al., 5 Feb 2024, Li et al., 2022). In mean-field, stochastic, or PDE settings, Wasserstein or $L^2$ distances may be used (Herty et al., 24 Jun 2024, Bayraktar et al., 13 Feb 2025, Sun et al., 25 Jul 2024).

2. Structural Foundations: Hamiltonian and Riccati Framework

The mathematical mechanism underlying the phenomenon lies in the hyperbolicity of the extremal Hamiltonian system. Upon linearization of the Pontryagin extremals about the steady state, the system

$$\begin{cases} \delta \dot{x}(t) = A \delta x(t) + B \delta u(t),\ \delta \dot{\lambda}(t) = - Q \delta x(t) - A^T \delta \lambda(t), \end{cases}$$

(augmented accordingly for nonlinear cases) splits into stable and unstable subspaces under a symplectic transformation, as encoded by a block-diagonal decomposition of the Hamiltonian matrix (Trélat et al., 2014, Trélat et al., 26 Mar 2025). The exponential rates $C_2$ in the estimate above are exactly the real parts of the eigenvalues of the closed-loop system matrices, which, in turn, are governed by the solution of an algebraic Riccati equation

$XA + A^* X - X B H_{uu}^{-1} B^* X - W = 0.$

Under negative definiteness of $H_{uu}$, positive definiteness of $W$, and Kalman-type controllability, this Riccati equation admits minimal and maximal solutions $E_-, E_+$ that define the invariant subspaces and guarantee exponential decay (Trélat et al., 2014, Li et al., 2022, Trélat et al., 5 Feb 2024). The exponential turnpike property in infinite dimension (Hilbert spaces) is likewise reduced to the spectral properties of related operator Riccati equations (Trelat et al., 2016, Trélat et al., 5 Feb 2024), under exponential stabilizability and detectability of the system (Li et al., 2022, Guglielmi et al., 13 Mar 2024).

3. Extensions to Nonlinear, Infinite-Dimensional, and Stochastic Systems

The property is not confined to finite-dimensional or linear systems. For dissipative nonlinear PDEs such as the two-dimensional Navier–Stokes equations, semilinear heat equations, and fractional parabolic equations, under appropriate smallness of the target or control region, an exponential turnpike holds for both state and adjoint variables (Zamorano, 2016, Pighin, 2020, Warma et al., 2020). The central insight is that strong dissipation and stabilizability channel the temporal dynamics towards a stationary or periodic attractor, and the influence of initial data (or terminal constraints) becomes negligible in sufficiently large-time regimes.

In mean-field, particle, or multi-agent systems, the exponential turnpike persists uniformly in the number of interacting entities, and is retained in the kinetic and hydrodynamic (macroscopic) limits (Herty et al., 24 Jun 2024, Herty et al., 17 Oct 2025). For stochastic systems and control of mean-field stochastic differential equations, the Riccati equation analysis extends with suitable stochastic generalization, and the exponential turnpike property holds in expectation as well as in the distributional (e.g., Wasserstein) sense (Bayraktar et al., 13 Feb 2025, Sun et al., 4 Jun 2024, Sun et al., 25 Jul 2024).

Generalized cost structures, including linear and quadratic running cost components, can be recast in a Riccati framework that ensures the property remains intact (equivalent in both the classical LQ and generalized LQ problems) under detectability and stabilizability (Guglielmi et al., 13 Mar 2024).

4. Practical Implications for Numerical Methods and Feedback Design

The exponential turnpike property strongly influences the design and performance of numerical algorithms for optimal control. For long-horizon direct transcription and indirect (Pontryagin/shooting) methods, leveraging the property allows for:

• Initialization of boundary value algorithm states (including adjoint variables) at $t = T/2$ with the steady-state value, yielding robust and rapidly converging Newton iterations, especially when standard shooting becomes unstable due to sensitivity to boundary data (Trélat et al., 2014, Geshkovski et al., 2022, Trélat et al., 26 Mar 2025).
• Concentration of temporal discretization points ("turnpike-adapted grids") near the endpoints, using coarser meshes in the central interval where the solution is nearly stationary, leading to computational savings and increased accuracy (Geshkovski et al., 2022).
• Efficient initialization and dimension reduction of receding horizon (model predictive) controllers by using the steady-state or periodic turnpike as a base, with only local adjustment for endpoint constraints (Breiten et al., 2018, Trélat et al., 5 Feb 2024).

In multi-scale systems or multi-agent settings, the uniformity of turnpike rates across scales suggests that steady-state-based controllers can be designed consistently at different model resolutions (Herty et al., 17 Oct 2025).

For mean-field game and McKean-Vlasov control problems, convergence of the underlying (forward-backward) value functions, state distributions, and controls to the ergodic solution exponentially away from boundaries enables one to treat large-time–average and stationary ergodic problems as surrogates of the original long-horizon finite time-horizon control (Cecchin et al., 13 Sep 2024, Bayraktar et al., 13 Feb 2025).

5. Key Assumptions: Structure and Requirements

The rigorous demonstration of the exponential turnpike property in general nonlinear or infinite-dimensional contexts requires:

• Hyperbolicity (no eigenvalues on the imaginary axis) of the linearized Hamiltonian system.
• Invertibility and negative definiteness of the second derivative of the Hamiltonian with respect to the control variable ($H_{uu}$).
• Positivity and coercivity (dissipativity) of the cost functional; in PDE problems, strict dissipativity is formalized via

$$S(y(\tau)) + \int_0^{\tau} \alpha(\|y(t) - \bar y\|) dt \leq S(y(0)) + \int_0^{\tau} w(y(t), \omega(t)) dt,$$

where $S$ is a storage function and $w$ is the supply rate (Lance et al., 2019, Trélat et al., 26 Mar 2025).

• Structural conditions on the system operators: exponential stabilizability of the dynamics via feedback, exponential detectability of the observation operator (in state-space or Hilbert-space settings) (Trelat et al., 2016, Li et al., 2022, Guglielmi et al., 13 Mar 2024).
• (For collocated or parameter-dependent systems) uniform versions of detectability/stabilizability on average (Hernández et al., 26 Apr 2024).
• For certain mean-field, stochastic, and game-theoretic control frameworks, smallness of the interaction terms and weak forms of monotonicity or dissipativity suffice for the turnpike property, with the coupling approach yielding sharp exponential estimates for both states and duals over large time intervals (Cecchin et al., 13 Sep 2024, Herty et al., 24 Jun 2024).

6. Applications and Broader Impact

The exponential turnpike property has implications for a variety of areas:

• Engineering systems governed by PDEs (fluid flows, heat and wave propagation, shape optimization), where optimal feedback can be designed based on the static or periodic reference (Zamorano, 2016, Lance et al., 2019, Trelat et al., 2016).
• Multi-agent and mean-field systems (crowd dynamics, opinion dynamics, swarms, traffic) for the design of scalable feedback policies and learning-based reduced-order models (Herty et al., 24 Jun 2024, Herty et al., 17 Oct 2025).
• Model predictive and receding horizon control in both deterministic and stochastic settings, allowing for efficient and robust design of long-horizon controllers (Breiten et al., 2018, Trélat et al., 5 Feb 2024).
• Data-driven and machine learning formulations of control, including deep learning architectures interpreted as discretized control flows ("Neural ODEs," ResNets), where the layered dynamics exhibit a turnpike-like behavior, thus suggesting strategies for network compression, parameter initialization, and improved generalization (Geshkovski et al., 2022, Trélat et al., 26 Mar 2025).
• Potential for reducing computational cost by replacing long-time dynamical optimization with static or periodic optimization, particularly in systems with large time constants or significant dissipative effects.

7. Summary Table: Representative Turnpike Estimates

Setting	Turnpike Estimate Type	Key Assumptions
Finite-dim. nonlinear control (Trélat et al., 2014, Trélat et al., 26 Mar 2025)	$$\|x(t) - \bar x\| + \|\lambda(t) - \bar\lambda\| + \|u(t) - \bar u\| \le C (e^{-\nu t} + e^{-\nu(T-t)})$$	Hamiltonian hyperbolicity, Riccati solvability
PDEs (Hilbert space) (Trelat et al., 2016, Li et al., 2022)	$$\|y(t) - y_s\| + \|u(t) - u_s\| + \|\lambda(t) - \lambda_s\| \le C (e^{-\nu t} + e^{-\nu(T-t)})$$	Stabilizability, detectability
Mean-field/Particle/Hydrodynamic (Herty et al., 24 Jun 2024, Herty et al., 17 Oct 2025)	$\mathcal{E}(t) \leq C e^{-\alpha t} \mathcal{E}(0)$	Strict dissipativity, quadratic growth
Stochastic/Game/MFG (Bayraktar et al., 13 Feb 2025, Cecchin et al., 13 Sep 2024, Sun et al., 4 Jun 2024)	$$E[|X_T(t)-\bar X(t)|^2 + |u_T(t)-\bar u(t)|^2] \le C (e^{-\lambda t}+e^{-\lambda(T-t)})$$	Riccati exponential stability, ergodicity

The exponential turnpike property thus not only reveals a universal three-arc temporal structure (transient–steady–transient) in optimal trajectories for controlled dynamical systems, but also forms the backbone for efficient algorithms and analytic reductions in diverse optimal control contexts.