Single-occupancy vehicles (SOVs) are charged to use the high-occupancy-Toll (HOT) lanes, while high-occupancy-vehicles (HOVs) can drive in them at no cost. The pricing scheme for HOT lanes has been extensively studied at local bottlenecks or at the network level through computationally expensive simulations. However, the HOT lane pricing study on a freeway corridor with multiple origins and destinations as well as multiple interacting bottlenecks is a challenging problem for which no analytical results are available. In this paper, we attempt to fill the gap by proposing to study the traffic dynamics in the corridor based on the relative space paradigm. In this new paradigm, the interaction of multiple bottlenecks and trips can be captured with Vickrey's bathtub model by a simple ordinary differential equation. We consider three types of lane choice behavior and analyze their properties. Then, we propose a distance-based dynamic pricing scheme based on a linear combination of I-controllers. This closed-loop controller is independent of the model and feeds back the travel time difference between HOT lanes and general-purpose lanes. Given the mathematical tractability of the system model, we analytically study the performance of the proposed closed-loop control under constant demand and show the existence and stability of the optimal equilibrium. Finally, we verify the results with numerical simulations considering a typical peak period demand pattern. In the future, we are interested in extending this work and testing the performance of the proposed linear combination of I-controllers for other traffic flow models.