  This paper is devoted to analysing the explicit slow decay rate and turnpike
in the infinite-horizon linear quadratic optimal control problems for
hyperbolic systems. Assume that some weak observability or controllability are
satisfied, by which, the lower and upper bounds of the corresponding algebraic
Riccati operator are estimated, respectively. Then based on these two bounds,
the explicit slow decay rate of the closed-loop system with Riccati-based
optimal feedback control is obtained. The averaged turnpike property for this
problem is also further discussed. We then apply these results to the LQ
optimal control problems constraint to networks of one-dimensional wave
equations and also some multi-dimensional ones with local controls which lack
of GCC(Geometric Control Condition).
