  The turnpike phenomenon stipulates that the solution of an optimal control
problem in large time, remains essentially close to a steady-state of the
dynamics, itself being the optimal solution of an associated static optimal
control problem. Under general assumptions, it is known that not only the
optimal state and the optimal control, but also the adjoint state coming from
the application of the Pontryagin maximum principle, are exponentially close to
a steady-state, except at the beginning and at the end of the time frame. In
such results, the turnpike set is a singleton, which is a steady-state. In this
paper, we establish a turnpike result for finite-dimensional optimal control
problems in which some of the coordinates evolve in a monotone way, and some
others are partial steady-states of the dynamics. We prove that the discrepancy
between the optimal trajectory and the turnpike set is then linear, but not
exponential: we thus speak of a linear turnpike theorem.
