Is it permissible to give a portion of one's Zakat al-Mal to a young man who is psychologically and physically ill, abandoned by his family despite their financial ability to care for him?

Zakat is disbursed only to those eligible, as mentioned in Surah At-Tawbah, verse 60. Mental illness alone does not make its owner eligible for zakat. If the young man suffering from mental illness is among those eligible for zakat, then it is permissible to give it to him, even if his family is able to spend on him but has not done so, for the criterion is his own condition, not his family's. If he is not among those eligible for zakat, then it is not permissible to give it to him.