The control plane functions and essential responsibilities of the 5G core network include:

1. Access and Mobility Management Function (AMF):
   1. Responsible for managing user access, mobility, and authentication.
2. Session Management Function (SMF)
   1. Manages the establishment, modification, and release of sessions for user data.
3. Policy Control Function (PCF)
   1. Provides policy decisions and rules for network resources, charging and quality of service (QoS).
4. Application Function (AF)
   1. Interacts with external applications and services to provide 5G network capabilities.
5. Authentication Server Function (AUSF)
   1. Authenticates and authorizes users for network access.
6. Security Edge Protection Proxy (SEPP)
   1. Protects the 5G core network from external security threats.
7. Network Exposure Function (NEF)
   1. Exposes network capabilities and services to external entities.
8. Network Function Repository Function (NRF)
   1. Manages the registration, discovery, and authorization of network functions.
9. Unified Data Management (UDM)
   1. Manages user data, including subscription, authentication and authorization information.
10. Unified Data Repository (UDR)
    1. Stores and manages user data.

Bibliography:

1. https://mobilepacketcore.com/glossary/service-based-interfaces-sbis-5g-network/