Turnpike Theorems in Optimal Control

• Turnpike theorems are principles in optimal control that demonstrate optimal trajectories stay close to a stationary or periodic state over long horizons.
• They leverage static optimization and KKT structures to reveal boundary layers and interior arcs with exponential, linear, or polynomial convergence behaviors.
• These results apply across diverse settings—including LQ control, PDE-constrained problems, stochastic games, and mean-field formulations—offering practical insights for system design.

Turnpike theorems are long-horizon asymptotic results asserting that optimal trajectories, controls, adjoints, strategies, or state distributions spend most of the time near a distinguished stationary object. In control-theoretic formulations this object is often an optimal steady state of an associated static problem; in broader settings it may be a turnpike set, a periodic orbit, a stationary strategy pair, an invariant distribution, or an affine turnpike trajectory (Kolokoltsov et al., 2012, Trelat et al., 2017). The modern theory spans finite- and infinite-dimensional optimal control, PDE-constrained optimization, stochastic games, sparse control, portfolio choice, mean-field control, and Wasserstein-space formulations, with quantitative regimes ranging from exponential boundary-layer estimates to measure, integral, linear, polynomial, and ideal-convergence statements (Trélat, 2020, Mammadov et al., 2022).

1. Classical formulation and principal variants

The classical formulation compares a long-horizon dynamic problem with an associated static optimization problem. In the deterministic finite-horizon language emphasized in the Markov-game survey, a middle turnpike states that for large $T$, the optimal state remains close to a stationary object on an interior interval $[T_0,T-T_0]$, while an early turnpike states that, once sufficiently far from the terminal date, optimal behavior remains close to the stationary object (Kolokoltsov et al., 2012). In infinite-dimensional optimal control, the turnpike object can be a closed set $\mathcal T\subset X$, not necessarily a singleton, and the basic conclusion can be expressed as vanishing time-average distance to $\mathcal T$ (Trelat et al., 2017).

Variant	Representative formulation	Typical setting
Exponential turnpike	$C(e^{-\nu t}+e^{-\nu(T-t)})$ decay from both endpoints	LQ control, meanfield Pontryagin triples
Measure-turnpike	$|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$	Strictly dissipative constrained systems
Integral-turnpike	$$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$	Non-autonomous infinite-dimensional systems
Linear or polynomial turnpike	$O(1/T)$ or $O((t+1)^{-\beta}+(T-t+1)^{-\beta})$	Partial steady-states, oscillatory systems
Ideal turnpike	$\mathcal I$-convergence to a stationary point	Discrete-time non-convex asymptotics

The turnpike set need not be a steady point. In the “linear turnpike theorem,” some coordinates remain at a partial steady-state while other coordinates evolve monotonically along an affine-in-time path forced by endpoint conditions, so the turnpike object is a trajectory $[T_0,T-T_0]$0 rather than a singleton (Trélat, 2020). In infinite-dimensional oscillatory systems with collapsing high-frequency spectral gap, the turnpike remains pointwise but becomes polynomial and is only available in a weaker weighted topology $[T_0,T-T_0]$1 (Zuyev et al., 2 Mar 2026). In discrete time, turnpike conclusions can be phrased through $[T_0,T-T_0]$2-convergence, which includes ordinary convergence, statistical convergence, and convergence modulo logarithmic-density-zero sets (Mammadov et al., 2022).

This range of formulations suggests that “turnpike theorem” is not a single estimate but a family of asymptotic selection principles. What remains invariant is the separation between a short transient structure near the endpoints and a long interior arc governed by a stationary or stationary-like optimal object.

2. Static optimization, KKT structure, and the linear-quadratic archetype

A central model class is linear-quadratic optimal control. In the fixed-endpoint problem studied in finite and infinite dimension, the dynamics are

$[T_0,T-T_0]$3

with quadratic tracking cost

$[T_0,T-T_0]$4

and value function

$[T_0,T-T_0]$5

The associated static problem is

$[T_0,T-T_0]$6

with unique minimizer $[T_0,T-T_0]$7 and KKT relations

$[T_0,T-T_0]$8

The corresponding turnpike is the steady optimizer $[T_0,T-T_0]$9 of the stationary Hamiltonian system (Askovic et al., 2023).

For this LQ class, the turnpike notion is an exponential state/control/costate turnpike. The optimal triple satisfies

$\mathcal T\subset X$0

with constants independent of $\mathcal T\subset X$1 (Askovic et al., 2023). In infinite-dimensional Hilbert-space LQ problems with terminal cost, the state and adjoint satisfy the refined estimate

$\mathcal T\subset X$2

where $\mathcal T\subset X$3; this isolates the initial and terminal mismatches driving the two boundary layers (Breiten et al., 2018).

The system-theoretic content of these estimates is explicit. Generalized LQ problems in infinite dimension, with both quadratic and linear terms in the running cost, are shown to have turnpike properties strongly connected to detectability and stabilizability, and the exponential turnpike property for generalized LQ and LQ optimal control problems is equivalent (Guglielmi et al., 2024). This places turnpike behavior within the same structural regime as Riccati solvability and exponentially stable closed-loop dynamics.

3. Value-function asymptotics and receding-horizon refinements

A major refinement of classical trajectory turnpikes is the asymptotic expansion of the value function itself. For the fixed-endpoint LQ problem, the long-horizon value satisfies

$\mathcal T\subset X$4

or equivalently

$\mathcal T\subset X$5

where $\mathcal T\subset X$6 is the static optimal value and $\mathcal T\subset X$7 are infinite-horizon stabilization costs for entering and leaving the turnpike (Askovic et al., 2023). In the free-terminal-state case,

$\mathcal T\subset X$8

with $\mathcal T\subset X$9 an ergodic constant (Askovic et al., 2023).

This expansion makes the classical three-arc picture quantitative. The bulk term $\mathcal T$0 records the long stay near the steady optimizer; the $\mathcal T$1 correction is exactly the sum of transient stabilization energies; the remainder is exponentially small. In the same framework, the optimal state and control admit an explicit superposition of a forward stable manifold arc and a backward stable manifold arc, propagated by the exponentially stable operators $\mathcal T$2 and $\mathcal T$3 (Askovic et al., 2023).

Turnpike structure also feeds directly into receding-horizon control. For infinite-dimensional autonomous LQ problems, the derivative of the finite-horizon value function has the form

$\mathcal T$4

so far from the terminal time it is well approximated by the static multiplier $\mathcal T$5 (Breiten et al., 2018). This motivates a terminal penalty $\mathcal T$6 in receding-horizon control. The resulting control satisfies an exponential error estimate with respect to the exact finite-horizon optimum, and in the simplest choice $\mathcal T$7, $\mathcal T$8, $\mathcal T$9,

$C(e^{-\nu t}+e^{-\nu(T-t)})$0

(Breiten et al., 2018).

4. Dissipativity, strong duality, and constrained infinite-dimensional turnpikes

A broad abstract framework for infinite-dimensional optimal control systems is built on integral-turnpike and measure-turnpike properties. For non-autonomous problems on reflexive Banach spaces, with very general terminal constraints $C(e^{-\nu t}+e^{-\nu(T-t)})$1, the integral-turnpike property around a closed turnpike set $C(e^{-\nu t}+e^{-\nu(T-t)})$2 takes the form

$C(e^{-\nu t}+e^{-\nu(T-t)})$3

under assumptions of viability, controllability to and from $C(e^{-\nu t}+e^{-\nu(T-t)})$4, and a coercive value-function gap (Trelat et al., 2017). In the autonomous free-endpoint case, strict dissipativity yields the stronger measure-turnpike property: for every $C(e^{-\nu t}+e^{-\nu(T-t)})$5, the set

$C(e^{-\nu t}+e^{-\nu(T-t)})$6

satisfies $C(e^{-\nu t}+e^{-\nu(T-t)})$7 uniformly in $C(e^{-\nu t}+e^{-\nu(T-t)})$8 (Trelat et al., 2017).

The decisive structural implication is

$C(e^{-\nu t}+e^{-\nu(T-t)})$9

with storage function $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$0 built from a Lagrange multiplier of the static problem (Trelat et al., 2017). This identifies a direct bridge from static constrained optimization to long-time dynamic optimal control.

PDE-constrained shape optimization provides a geometric version of the same paradigm. For second-order linear parabolic equations with a time-varying shape $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$1 acting as the support of a source term, strict dissipativity yields a state-adjoint measure-turnpike in the Lagrange case, while in the Mayer case the optimal shape satisfies an exponential turnpike in Hausdorff distance: $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$2 (Lance et al., 2019). Here the turnpike object is no longer a control amplitude but a moving superlevel set of the adjoint. This suggests that the turnpike principle is compatible with highly nonconvex and geometric control classes, provided dissipativity and level-set regularity are available.

5. Nonclassical regimes: linear, polynomial, sparse, ideal, stochastic, and portfolio turnpikes

Not all turnpikes are exponentially hyperbolic. In finite-dimensional problems with a state split into steady coordinates $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$3 and monotone coordinates $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$4, the optimal long-time behavior is a partial steady-state $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$5 with $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$6 affine in time. The resulting estimate is

$|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$7

while $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$8 and $|Q_{\varepsilon,T}|\le \Lambda(\varepsilon)$9 (Trélat, 2020). The paper accordingly speaks of a linear turnpike theorem.

An even weaker rate occurs for an infinite family of controlled oscillators. There the optimal state and costate satisfy a pointwise polynomial turnpike estimate

$$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$0

because the stable/unstable spectral rates satisfy $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$1 as $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$2 (Zuyev et al., 2 Mar 2026). The theorem is pointwise in time, but only in the weaker weighted topology $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$3.

Turnpike structure also appears for nonsmooth sparse objectives. In maximum hands-off control for bounded-input LTI systems, the optimal sparse control and state satisfy

$$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$4

so the long-horizon optimum is mostly zero-input and near the origin, with only short initial and terminal active intervals (Sakamoto et al., 2020). The mechanism there is geometric: the $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$5-equivalent Pontryagin system is hyperbolic, and invariant-manifold theory produces the boundary-layer estimate.

Discrete-time turnpikes admit still more general asymptotic languages. For non-convex infinite-horizon problems with objective $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$6, every optimal process converges to the unique optimal stationary point in the sense of $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$7-convergence, provided the ideal $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$8 is invariant under translations and a Lyapunov-type decrease condition holds (Mammadov et al., 2022). A parallel abstract theorem treats normed spaces, set-valued dynamics $$\frac{1}{T}\int \beta(\operatorname{dist}(y(t),\mathcal T))\,dt\to 0$$9, and paths maximizing the smallest $O(1/T)$0-cluster point of utility, again concluding $O(1/T)$1-convergence to the unique utility-maximizing fixed point (Leonetti et al., 2020).

In stochastic dynamic games, the Bellman/Shapley operator on $O(1/T)$2 yields two distinct turnpikes under a uniform minorization assumption: a stochastic early turnpike on the set of optimal strategies and a stochastic middle turnpike on the distribution of states for finite-horizon two-person zero-sum Markov games on a general Borel state space (Kolokoltsov et al., 2012). In stochastic portfolio choice, turnpike theorems take yet another form: if utility becomes power-like at high wealth, the optimal feedback rule converges to the CRRA benchmark, with rate

$O(1/T)$3

determined by the decay of the zero-coupon bond price and the speed at which utility becomes power-like (Yamamichi, 29 Nov 2025).

6. Mean-field, Wasserstein, and multiscale extensions

Mean-field control introduces turnpikes for distributions rather than finite-dimensional states. For deterministic mean-field optimal control problems obtained as limits of interacting ODE systems, a strict dissipativity inequality and a cheap-control bound imply an integral turnpike with interior decay. At the continuum level,

$O(1/T)$4

satisfies

$O(1/T)$5

showing that the turnpike structure of the finite-$O(1/T)$6 problem is preserved under the mean-field limit (Gugat et al., 2023).

A hierarchical extension proves exponential turnpike behavior consistently across interacting particle systems, kinetic mean-field equations, and two hydrodynamic closures. For sufficiently large $O(1/T)$7, the particle, mean-field, pressureless hydrodynamic, and full Euler optimal trajectories satisfy exponential decay estimates toward the aligned velocity state $O(1/T)$8, with the same constants $O(1/T)$9 across the hierarchy (Herty et al., 17 Oct 2025). This suggests that, at least for controlled alignment systems, turnpike behavior can be stable under passage from microscopic to macroscopic descriptions.

The most intrinsic Wasserstein-space formulation is developed for nonlinear deterministic meanfield optimal control. In the lifted Hilbert-space problem, every optimal Pontryagin triple $O((t+1)^{-\beta}+(T-t+1)^{-\beta})$0 satisfies

$O((t+1)^{-\beta}+(T-t+1)^{-\beta})$1

while the Eulerian Wasserstein-space version yields

$O((t+1)^{-\beta}+(T-t+1)^{-\beta})$2

for optimal Eulerian Pontryagin triples (Bonnet-Weill et al., 24 Mar 2026). The proof separates the Eulerian second-order hypotheses into a horizontal part, transferred by unitary conjugation to the lifted space, and a vertical part, reduced to uniform pointwise stabilizability and detectability conditions on multiplication operators (Bonnet-Weill et al., 24 Mar 2026). This is a particularly explicit instance of how classical Riccati hyperbolicity reappears in mean-field and Wasserstein geometry.

Across these developments, turnpike theorems function as a unifying long-time asymptotic principle rather than a single theorem. They organize a broad class of optimality phenomena: static versus dynamic decomposition, transient versus interior arcs, stabilization versus detectability, and finite-dimensional versus distributional state descriptions. The strongest results remain exponential and hyperbolic; weaker regimes—measure, integral, linear, polynomial, or ideal—emerge when the turnpike object is nonisolated, the spectral gap collapses, or the asymptotic notion itself is generalized.