  In this work we derive an interval turnpike result for adjoints of finite-
and infinite-dimensional nonlinear optimal control problems under the
assumption of an interval turnpike on states and controls. We consider
stabilizable dynamics governed by a generator of a semigroup with
finite-dimensional unstable part satisfying a spectral decomposition condition
and show the desired turnpike property under continuity assumptions on the
first-order optimality conditions. We further give stronger estimates for
analytic semigroups and provide a numerical example with a boundary controlled
semilinear heat equation to illustrate the results.
