Strong Turnpike Property in Optimal Control

• Strong Turnpike Property is a phenomenon in optimal control where trajectories exponentially approach a steady or periodic reference, except at the initial and terminal stages.
• It is demonstrated through pointwise-in-time exponential estimates derived from Hamiltonian hyperbolicity and Riccati block diagonalization, ensuring rapid convergence.
• The concept is versatile, applying to finite-dimensional, PDE, stochastic, and shape optimization settings to bridge finite- and infinite-horizon problem analyses.

Searching arXiv for relevant papers on strong/exponential turnpike property. The strong turnpike property, also called the exponential turnpike property, is a long-horizon phenomenon in optimal control whereby an optimal trajectory spends most of the interval close to a distinguished reference solution—typically a steady-state, but in some settings a periodic trajectory, a static optimal shape, or an infinite-horizon stochastic reference pair—and departs from it only through initial and terminal boundary layers. In its standard form, there exist constants $C,\nu>0$, independent of the horizon $T$, such that

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$

with analogous estimates for adjoints, shapes, or feedback laws depending on the model class (Trélat et al., 26 Mar 2025).

1. Canonical definition and equivalent formulations

The term “strong turnpike property” refers to a pointwise-in-time estimate, uniform with respect to the time horizon, and exponentially decaying away from both endpoints. In finite-dimensional nonlinear optimal control, one typical formulation is

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$

for all $t\in[0,T]$, where $(\bar x,\bar u)$ is a steady optimal pair (Sakamoto et al., 2019). In many Pontryagin-based formulations, the estimate also includes the adjoint, yielding exponential closeness of the full extremal $(x_T,u_T,\lambda_T)$ to a steady extremal $(\bar x,\bar u,\bar\lambda)$ (Trélat et al., 2014).

In Hilbert-space optimal control, the reference object may be a steady-state or a periodic solution. For an optimal solution triplet $(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$ and a reference turnpike $(\bar y,\bar u,\bar\lambda)$, the estimate takes the form

$T$0

for almost every $T$1; in the periodic case, $T$2 is replaced by a $T$3-periodic optimal trajectory $T$4 (Trelat et al., 2016).

The same designation is used in settings where the controlled object is not a Euclidean trajectory. In optimal shape design for parabolic PDEs, the strong turnpike can be expressed in Hausdorff distance as

$T$5

in the Mayer case, and the concluding formulation states the symmetric estimate

$T$6

together with exponential convergence of the associated state and adjoint (Lance et al., 2019). In long-horizon portfolio selection, the strong turnpike property is an exponential-rate convergence of the optimal feedback policy $T$7 to the constant Merton proportion $T$8,

$T$9

which is the control-theoretic analogue of a turnpike estimate in policy space (Bian et al., 2014).

A standard consequence is that the boundary layers have width $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$0 in time, while the interior arc remains exponentially close to the turnpike. This makes the strong turnpike strictly stronger than integral or measure variants, which only control time averages or the amount of time spent away from the reference (Trélat et al., 26 Mar 2025).

2. Turnpike references: steady-state, periodic, and constrained

The reference configuration is usually determined by an associated static optimization problem. In the standard finite-dimensional setting, the turnpike $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$1 solves

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$2

and the associated multiplier $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$3 satisfies the stationarity conditions

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$4

so that $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$5 is an equilibrium of the extremal flow (Trélat et al., 26 Mar 2025).

In Hilbert spaces, the steady reference $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$6 is defined by the static first-order system

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$7

while the periodic turnpike is a $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$8-periodic optimal solution $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; C\bigl(e^{-\nu t}+e^{-\nu(T-t)}\bigr), \qquad t\in[0,T],$$9 of the periodic problem (Trelat et al., 2016). In generalized infinite-dimensional LQ problems, the steady-state extremal is characterized algebraically by

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$0

which identifies the turnpike as the equilibrium point of the Hamiltonian dynamics (Guglielmi et al., 2024).

The reference need not lie in the interior of the admissible set. In constrained discrete-time LQ control, the steady-state pair $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$1 is defined by the constrained steady program

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$2

If $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$3 with a positive Karush–Kuhn–Tucker multiplier, the turnpike reference lies on the boundary $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$4, yet the turnpike conclusion remains valid at the level of measure-turnpike estimates (Li et al., 2023). This shows that “turnpike reference” is a variational notion rather than an interior equilibrium requirement.

Periodic and nonstationary analogues also fit the same pattern. For periodic tracking in PDE control, the reference object is a unique $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$5-periodic optimal solution (Trelat et al., 2016). In stochastic regime-switching and mean-field LQ problems, the turnpike is an infinite-horizon reference pair built from stationary Riccati solutions and backward equations rather than from a deterministic static point (Mei et al., 7 Aug 2025, Mei et al., 3 Nov 2025). This suggests that the unifying ingredient is not time-independence per se, but the existence of a preferred long-run optimal regime.

3. Hyperbolicity, Riccati structure, and other proof mechanisms

A central proof mechanism is the hyperbolic structure of the Hamiltonian system arising from the Pontryagin maximum principle. In finite-dimensional nonlinear control, linearization around the steady extremal produces a Hamiltonian matrix, and under the strong Legendre, quasi-concavity, and Kalman hypotheses one obtains a saddle structure. A symplectic change of variables based on algebraic Riccati equations splits the dynamics into stable and unstable components, leading to forward decay on one block and backward decay on the other, and hence to the two-sided estimate (Trélat et al., 2014).

In Hilbert spaces, the same logic survives in operator form. The first-order optimality system is written as a state–adjoint evolution

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$6

with $$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$7 hyperbolic under stabilizability, detectability, and Legendre assumptions. The operator Riccati equation

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$8

and the Lyapunov operator

$$\|x_T(t)-\bar x\|+\|u_T(t)-\bar u\| \;\le\; K\Bigl(e^{-\mu t}+e^{-\mu(T-t)}\Bigr),$$9

are then used to define the dichotomy transformation

$t\in[0,T]$0

which block-diagonalizes the linearized Hamiltonian system into

$t\in[0,T]$1

with $t\in[0,T]$2 exponentially stable (Trelat et al., 2016). The exponential turnpike estimate follows by integrating $t\in[0,T]$3 forward and $t\in[0,T]$4 backward.

A geometric reformulation replaces Riccati-based block diagonalization by invariant manifold theory. For nonlinear Hamiltonian flows with a hyperbolic equilibrium, stable and unstable manifolds $t\in[0,T]$5 and $t\in[0,T]$6 carry forward and backward exponential decay. The $t\in[0,T]$7-lemma is then used to show that, for large $t\in[0,T]$8, there exists a trajectory connecting suitable transverse discs near $t\in[0,T]$9 and $(\bar x,\bar u)$0, and this trajectory satisfies the two-sided exponential estimate (Sakamoto et al., 2019). In this approach, the turnpike is a manifestation of the geometry of the stable and unstable manifolds rather than primarily of explicit Riccati algebra.

A distinct route avoids the optimality system entirely. In globally Lipschitz control-affine dynamics, the strong turnpike can be proved by constructing quasi-turnpike controls using controllability, deriving a uniform bound on the optimal cost, obtaining a uniform $(\bar x,\bar u)$1-bound from Grönwall estimates, and then bootstrapping local bounds over shrinking subintervals. This produces the same exponential profile

$(\bar x,\bar u)$2

without linearization or small-data fixed-point arguments (Esteve-Yagüe et al., 2020).

Dissipativity furnishes yet another mechanism, typically first yielding weaker turnpike notions. In PDE shape control, strict dissipativity with storage $(\bar x,\bar u)$3 and supply rate $(\bar x,\bar u)$4 produces measure-turnpike for state and adjoint; in the Mayer case, this is sharpened to exponential turnpike for the shape variable by combining eigenfunction expansion of the backward adjoint, level-set characterization via the bathtub principle, analyticity of level sets, and geometric control of level-set perturbations in Hausdorff distance (Lance et al., 2019).

4. Recurring hypotheses and system-theoretic characterizations

Although the formulations vary widely, several hypotheses recur. A first family is second-order coercivity. In the Hilbert-space framework, one assumes a Legendre condition: $(\bar x,\bar u)$5 is negative-definite and invertible at the reference (Trelat et al., 2016). In the finite-dimensional nonlinear theorem, the strong Legendre condition requires $(\bar x,\bar u)$6 to be symmetric negative-definite, while the reduced Hessian term

$(\bar x,\bar u)$7

must be symmetric positive-definite (Trélat et al., 2014). These conditions enforce local saddle geometry for the Hamiltonian flow.

A second family is controllability, stabilizability, and detectability. In finite-dimensional LQ theory and its nonlinear perturbations, the Kalman rank condition or stabilizability/detectability ensures hyperbolicity of the Hamiltonian matrix [(Trélat et al., 2014); (Trélat et al., 26 Mar 2025)]. In Hilbert spaces, the linearization block must satisfy exponential stabilizability and exponential detectability, and in the LQ case these reduce to the classical requirements that $(\bar x,\bar u)$8 be stabilizable and $(\bar x,\bar u)$9 detectable (Trelat et al., 2016). In generalized infinite-dimensional LQ control, the same properties appear as the key necessary and sufficient system-theoretic conditions, together with the spectral assumption $(x_T,u_T,\lambda_T)$0, where

$(x_T,u_T,\lambda_T)$1

so that $(x_T,u_T,\lambda_T)$2 has a spectral gap around the imaginary axis and exhibits an exponential dichotomy on $(x_T,u_T,\lambda_T)$3 (Guglielmi et al., 2024).

A third family is problem-specific regularity. In parabolic shape design, the exponential shape turnpike requires uniform ellipticity, the maximum principle, analytic hypoellipticity of $(x_T,u_T,\lambda_T)$4, and analyticity of adjoint level sets, because the proof converts $(x_T,u_T,\lambda_T)$5-control of the adjoint into Hausdorff-distance control of shape level sets (Lance et al., 2019). In semilinear heat control with tracking target $(x_T,u_T,\lambda_T)$6, the strong turnpike theorem for the actual minimizer holds under a small-target assumption $(x_T,u_T,\lambda_T)$7, even though no smallness is imposed on the initial datum (Pighin, 2020). By contrast, for globally Lipschitz control-affine systems and their semilinear wave and heat counterparts, the controllability-based proof yields exponential turnpike without smallness assumptions on the initial data or the running target, provided the required local controllability estimates are available (Esteve-Yagüe et al., 2020).

The literature also contains explicit equivalence results. For generalized infinite-dimensional LQ problems with linear and quadratic terms in the running cost, an invertible change of variables transforms the problem into a standard quadratic form, and exponential turnpike for the generalized problem is equivalent to exponential turnpike for the associated standard LQ problem (Guglielmi et al., 2024). This places the strong turnpike property squarely within system theory: it is not merely a variational curiosity, but is closely tied to the dichotomy structure of the underlying linearized dynamics.

5. Representative settings and model classes

The strong turnpike property has been proved in a broad range of settings, with the reference object and the norm of closeness adapted to the model.

Setting	Turnpike object	Representative strong result
Finite-dimensional nonlinear OCP	Steady extremal $(x_T,u_T,\lambda_T)$8	State, control, and adjoint satisfy two-sided exponential estimates; midpoint-shooting is motivated by the adjoint turnpike (Trélat et al., 2014)
Hilbert-space and PDE control	Steady-state or $(x_T,u_T,\lambda_T)$9-periodic state–control–adjoint	Exponential estimates follow from operator Riccati and Lyapunov equations; examples include linear heat and wave equations (Trelat et al., 2016)
Geometric nonlinear control	Hyperbolic steady equilibrium of the Hamiltonian flow	Stable/unstable manifolds and the $(\bar x,\bar u,\bar\lambda)$0-lemma yield local exponential turnpike for locally optimal solutions (Sakamoto et al., 2019)
Lipschitz nonlinear control and semilinear PDEs	Running target steady pair $(\bar x,\bar u,\bar\lambda)$1 or $(\bar x,\bar u,\bar\lambda)$2	Quasi-turnpike controls and bootstrap produce exponential decay without Pontryagin linearization (Esteve-Yagüe et al., 2020)
Parabolic shape design	Static optimal shape $(\bar x,\bar u,\bar\lambda)$3	In the Mayer case, $(\bar x,\bar u,\bar\lambda)$4 decays exponentially; the shape remains nearly constant in the middle of the horizon (Lance et al., 2019)
Investment with general utility	Merton-optimal proportion $(\bar x,\bar u,\bar\lambda)$5	The optimal feedback policy converges exponentially to $(\bar x,\bar u,\bar\lambda)$6, with explicit error bounds under dual utility assumptions (Bian et al., 2014)
Regime-switching stochastic LQ and mean-field LQ	Infinite-horizon reference pair from stationary Riccati/backward equations	$(\bar x,\bar u,\bar\lambda)$7-type exponential estimates compare finite-horizon optimal pairs with the infinite-horizon reference (Mei et al., 7 Aug 2025, Mei et al., 3 Nov 2025)
Hierarchical particle/mean-field/hydrodynamic control	Consensus velocity $(\bar x,\bar u,\bar\lambda)$8 or steady pair	Exponential decay is established uniformly across microscopic, kinetic, and macroscopic levels (Herty et al., 17 Oct 2025)

This range shows that the strong turnpike property is not confined to deterministic steady tracking. It applies to periodic references, optimal shapes, feedback portfolios, regime-switching stochastic systems, and multiscale alignment models. A plausible implication is that the property is best viewed as a structural consequence of long-horizon optimality combined with a suitable stabilizing mechanism, rather than as a phenomenon restricted to one particular class of dynamics.

6. Relation to weaker turnpike notions, limitations, and open directions

The strong turnpike property is strictly stronger than integral-turnpike and measure-turnpike statements. The latter assert, respectively, a uniform bound on integrated deviations or a uniform bound on the measure of the set of times at which the optimal pair leaves an $(\bar x,\bar u,\bar\lambda)$9-tube around the turnpike. In the survey formulation, strong turnpike provides pointwise-in-time exponential rates, whereas weak versions only bound the fraction of time spent away from the turnpike (Trélat et al., 26 Mar 2025). The discrete-time constrained LQ estimate

$(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$0

is a measure-turnpike statement: apart from at most $(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$1 instants, the optimal pair remains $(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$2-close to the steady reference (Li et al., 2023). In parabolic shape design, strict dissipativity first yields measure-turnpike for state and adjoint in the Lagrange case, while the stronger Hausdorff exponential estimate is obtained only in the Mayer case through additional spectral and geometric arguments (Lance et al., 2019).

Several limitations remain explicit in the literature. In generalized infinite-dimensional LQ theory, the boundedness of $(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$3 and $(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$4 is a key restriction; the unbounded-control or unbounded-observation case, including boundary control for PDEs, is left open because the Riccati equation may fail to generate a bounded feedback and the Hamiltonian may fail to have the domain properties needed for a classical dichotomy argument (Guglielmi et al., 2024). In semilinear heat control, strong exponential turnpike for the true minimizer is proved under a small-target assumption, whereas for general large targets one only obtains convergence of averaged costs and a uniform bound on $(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$5 when the control acts everywhere, $(y^T(\cdot),u^T(\cdot),\lambda^T(\cdot))$6 (Pighin, 2020). In geometric nonlinear control, local theorems can sometimes be globalized by analyzing the full stable and unstable manifolds, but such global manifold information is highly problem-dependent (Sakamoto et al., 2019).

Open problems identified in the survey include global exponential turnpike beyond local neighborhoods in nonlinear problems without global dissipativity or uniqueness of minimizers, the characterization of turnpike sets when the center manifold around an extremal equilibrium is non-trivial, and turnpike properties under minimal assumptions on state/control constraints, lack of smoothness, or non-polyhedral cost structures (Trélat et al., 26 Mar 2025). These questions indicate that the strong turnpike property is by now well understood in several hyperbolic and LQ regimes, but its full scope in constrained, nonsmooth, or nonhyperbolic settings is still unsettled.