  In this work, we study the steady-state (or periodic) exponential turnpike
property of optimal control problems in Hilbert spaces. The turnpike property,
which is essentially due to the hyperbolic feature of the Hamiltonian system
resulting from the Pontryagin maximum principle, reflects the fact that, in
large time, the optimal state, control and adjoint vector remain most of the
time close to an optimal steady-state. A similar statement holds true as well
when replacing an optimal steady-state by an optimal periodic trajectory. To
establish the result, we design an appropriate dichotomy transformation, based
on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our
results with examples including linear heat and wave equations with periodic
tracking terms.
