This work investigates informative planning of sensing agents over infinite time horizon when the system of interest is expressed as a continuous-time linear system. The objective of this planning problem, termed persistent monitoring problem, is to maintain the monitoring uncertainty at the minimum. The method reduces the persistent planning problem into a periodic planning problem; it is formulated as a periodic optimal control or optimization problem to determine the optimal periodic sensor plan as well as the period. The plan induces the periodic Riccati equation and is proven to lead an arbitrary initial uncertainty state to the optimized periodic trajectory. It is also proven that any infinite-horizon (non-periodic) sensor plan is able to be approximated arbitrary well by a periodic sensor plan. A suboptimal filtering mechanism is proposed by using the resulting optimal periodic solution. Two numerical examples on (a) a relaxed periodic sensor scheduling for a two dimensional linear system, and (b) persistent monitoring by a mobile sensor of two-dimensional diffusion dynamics show the validity of the proposed approach.