The following is an excerpt from the book ‘Annamalai Swami Final Talks’ (bold added by me):
Question: One can have a temporary experience of the Self, the underlying reality, but then it goes away. Can you offer any guidance on how to stabilise in that state?
Annamalai Swami: A lamp that is lit may blow out if the wind is strong. If you want to see it again, you have to relight it. But Self is not like this. It is not a flame that can be blown out by the passing winds of thoughts and desires. It is always bright, always shining, always there.
If you are not aware of it, it means that you have put a curtain or a veil in front of it that blocks your view. Self does not hide itself behind a curtain. You are the one who puts the curtain there by believing in ideas that are not true. If the curtain parts and then closes again, it means that you are still believing in wrong ideas.
If you have eradicated them completely, they will not reappear. While these ideas are covering up the Self, you still need to do constant sadhana. So, going back to your question, the Self does not need to stabilise itself. It is full and complete in itself. The mind can be stabilised or destabilised, but not the Self.