The 'Controlling a Stage' process in PRINCE2 is used to manage and control each 'delivery stage' of a project. This process focuses on the day-to-day management of the project stages, ensuring that the project remains within its agreed tolerances and effectively addressing issues as they arise.
PRINCE2 Foundation Manual

During the Initiating a Project process in PRINCE2, the Project Plan provides the Business Case with necessary details, including cost estimates. This plan outlines the overall scope and timing of the project and sets the baseline for project management activities, including cost control. References: PRINCE2 Foundation manual, PRINCE2 official resources

In PRINCE2, the "Closing a Project" process has the objective of confirming that the project's products are acceptable and meet the established acceptance criteria. This process ensures that the project has delivered the required outputs as agreed upon, and these outputs are approved and accepted by the customer. References:
PRINCE2 Foundation Guide, "Closing a Project" section