Long time horizons in optimal control: The turnpike property and asymptotic behavior of trajectories
Résumé
This paper investigates the dynamic turnpike phenomenon in Linear-Quadratic optimal control of abstract conservative systems governed by skew-adjoint operators with compact resolvents and bounded control operators. While classical turnpike theory emphasizes asymptotic convergence to a static steady state, this work focuses on characterizing the transient dynamics. We identify the Initial Limit Arc as the unique restriction of the infinite-horizon optimal trajectory to the stable invariant manifold of the Hamiltonian system. Exponential convergence of finite-horizon solutions to this limit arc is established in the energy norm, uniformly with respect to the time horizon. A spectral analysis of the decay rates reveals a fundamental dichotomy: the convergence exhibits either a control-dominated oscillatory regime or a saturation-limited monotonic regime, depending on the interplay between the regularization parameter and the principal eigenvalues. Numerical experiments on the wave equation and the Timoshenko beam system illustrate these regimes and verify the sharpness of the theoretical decay bounds.
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