  Motivated by singular limits for long-time optimal control problems, we
investigate a class of parameter-dependent parabolic equations. First, we prove
a turnpike result, uniform with respect to the parameters within a suitable
regularity class and under appropriate bounds. The main ingredient of our proof
is the justification of the uniform exponential stabilization of the
corresponding Riccati equations, which is derived from the uniform null control
properties of the model. Then, we focus on a heat equation with rapidly
oscillating coefficients. In the one-dimensional setting, we obtain a uniform
turnpike property with respect to the highly oscillatory heterogeneous medium.
Afterward, we establish the homogenization of the turnpike property. Finally,
our results are validated by numerical experiments.
