ASM instance are started and stopped in a similar way to normal database instances.

The options for the STARTUP command are:
FORCE - Performs a SHUTDOWN ABORT before restarting the ASM instance.
MOUNT - Starts the ASM instance and mounts the disk groups specified by the ASM_DISKGROUPS parameter.
NOMOUNT - Starts the ASM instance without mounting any disk groups.
OPEN - This is not a valid option for an ASM instance.

The options for the SHUTDOWN command are:
NORMAL - The ASM instance waits for all connected ASM instances and SQL sessions to exit then shuts down.
IMMEDIATE - The ASM instance waits for any SQL transactions to complete then shuts down. It doesn't wait for sessions to exit.
SHUTDOWN NORMAL and SHUTDOWN IMMEDIATE would return an error if a database instance is UP (ie. using the diskgroups managed by the ASM Instance)

SQL> shutdown immediate;
ORA-15097: cannot SHUTDOWN ASM instance with connected RDBMS instance

ABORT - The ASM instance shuts down instantly.

srvctl can be used to start all the ASM instances which are a part of the cluster.
Please refer to Srvctl-Major-Options-and-Usage