The turnpike property in optimal control says that over a long time horizon, the optimal trajectory spends most of its time near a single steady state — the turnpike. Short deviations occur at the beginning and end, where the system transitions to and from the desired state, but the middle is dominated by proximity to the static optimum. For dissipative systems, the convergence to the turnpike is exponential: the trajectory approaches and departs the steady state on timescales much shorter than the total horizon.
Zuyev and Trélat show that for a class of infinite-dimensional oscillating systems — wave equations, beam equations, systems that conserve energy rather than dissipating it — the turnpike property still holds but with polynomial rather than exponential convergence. The trajectory approaches the steady state, but slowly. The departure is equally gradual. The time spent near the turnpike shrinks as a polynomial fraction of the total horizon, not an exponential one.
The difference matters because polynomial convergence means the transient phases consume a larger fraction of the total time. For a dissipative system with a 100-time-unit horizon, the transients might occupy units 1-5 and 95-100. For an oscillating system, they might occupy units 1-30 and 70-100. The turnpike still exists, but the system visits it reluctantly, spending less of its time in the neighborhood of the optimal steady state.
The through-claim is about what dissipation buys. Dissipation is usually framed as a loss — energy leaving the system, entropy increasing. In optimal control, dissipation is a gift: it makes the system forget its initial condition quickly and approach the optimal steady state exponentially fast. Without dissipation, the system remembers its oscillatory structure and reaches the turnpike only polynomially. The cost of conservation is slow convergence. The benefit of loss is fast optimality.