Question: What is lasting and stable happiness? How does one knows if one has reached this stage?
Answer: Lasting and stable happiness is best embodied and expressed by the Buddha. It is the peaceful and blissful state of mind that is unperturbed by any internal or external conditions, that runs on wisdom (seeing the way things really are) and compassion (responding kindly to others who lack the same wisdom).
To know if this has been attained, one can note how one responds in unfavourable conditions. Does one become frustrated easily and suffer? When we are able to make the best of each situation with some compassion and wisdom, we are to that extent doing well at that moment.
Different folks have different magnitudes of compassion and wisdom, but the Buddha teaches us that we can increase them until we perfect them, to realise True Happiness as he did in Buddhahood. When we advance towards it, we will know it to be so as our character and behaviour will increasingly resemble the Buddha's, as reflected in the scriptures. In the mean time, with every bit of diligent Dharma practice, we get closer to this goal.