How could I shutdown a remote host, in my network thru ssh, with a local host?

For the following I am assuming that the user you are going to use in remote-host is the same you use in local-host.

In order to do what you want, you have to first authorize your local-host to connect to you remote-host with no password. To do that you have to (as described here):

1. Install ssh:

   sudo apt-get install ssh
   

2. Create public and private keys using ssh-key-gen on local-host by entering this command in your localhost:

   ssh-keygen
   

   You should save the generated key in:

   /home/yourusername/.ssh/id_rsa
   

   Press enter twice to leave the passphrase empty.

   Your identification has been saved in /home/yourusername/.ssh/id_rsa.
   Your public key has been saved in /home/yourusername/.ssh/id_rsa.pub.
   The key fingerprint is:
   XX:XX:XX:xX:XX:xX:XX:XX:XX:XX:XX:XX:XX:XX yourusername@local-host
   

3. Copy the public key to the remote-host using ssh-copy-id:

   yourusername@local-host$ ssh-copy-id -i ~/.ssh/id_rsa.pub remote-host
   yourusername@remote-host's password:
   
   Now try logging into the machine, with:
   
   ssh remote-host
   

   and check in .ssh/authorized_keys to make sure we haven't added extra keys that you weren't expecting.

   Note: ssh-copy-id appends the keys to the remote-host’s /home/yourusername/.ssh/authorized_key.

4. Login to remote-host without entering the password:

   ssh remote-host
   yourusername@remote-host:~$
   

   Access to remote-host with no password. Success!

Now you have to be able to execute sudo shutdown -P 0 with no password. You can do that by modifying /etc/sudoers on remote-host with visudo. That way, user yourusername can execute the shutdown command with no password asked.

1. Login to the remote-host:

   ssh remote.host
   

2. Run:

   sudo visudo
   

   By running visudo, you edit /etc/sudoers in a safe manner.

3. Add this line to the file:

   yourusername ALL = NOPASSWD: /sbin/shutdown
   

4. After doing that, get back to your local-host, create a new empty file and paste this line, modifying the remote-host's name:

   ssh remote.host sudo shutdown -P 0
   

5. Save and close the file, right-click on it to go to its Properties → Permissions, and tick Execute this file as a program.

Script done!