Is it permissible in Islamic law to completely abandon one's children due to an inability to raise them properly?

The fundamental principle is that a disbelieving mother has no right of custody over her Muslim children. However, if you are prevented from having custody of your children by an oppressive power, do not abandon them; for you will be held accountable for them on the Day of Judgment. Utilize your bi-weekly visits to nurture them in Islam and instill its values in their hearts. If you do everything within your capacity, then there is no sin upon you, for Allah Almighty says: "Allah does not charge a soul except [with that within] its capacity," and "So fear Allah as much as you are able."