Granting the user that runs PHP/web server permissions to restart any service is generally a bed idea from security perspective not even taking about permission to manage all system daemons or full sudo.
Once you have been warned, you have to make sure the user that is running the PHP script has the permission to run the restart command. As already stated, the issue has likely little to do with PHP. There are several ways from you to proceed here:
You can grant the PHP user the sudo permission to run the desired command and nothing else (how to do that is covered elsewhere) and invoke the command directly from PHP:
sudo systemctl restart mysql.
rst.shfile, get it owned by root, writeable by noone else but root and set SUID bit on that file. This way you can invoke the script without sudo but the script will be run as root thanks to the SUID bit.
#1 feels safer and simpler.