Upvote:3
This kind of question (why do empires not control the desert) features heavily in Ibn Khaldun's famous Muqaddimah. His conclusion is that those desert arreas are much less productive than other areas and the population is considerably harder to control. Life in those parts is tough and so people are also tough. In fact his observation is that desert tribes will often seize power for themselves if, or rather when, the (previous) ruling dynasty declines.
What is fascinating is that somewhat similar dynamics are at play on China's northern border. Many dynasties were not able to reach north of the Gobi desert. Those that were (e.g. Liao, Jin, Yuan, Qing) usually originated from north of the Great Wall themselves.