Does God control "demons"? It is my understanding that a “demon” (either Greek δαιμόνιον or δαίμων) is an “evil spirit” (Greek πνεῦμα πονηρόν).

In Acts 19:13, it is written,


  Then certain of the vagabond Jews, exorcists, took upon them to call over those who had evil spirits [τὰ πνεύματα τὰ πονηρὰ] the name of the Lord Jesus, saying, “We adjure you by Jesus whom Paul preaches.”


I think most agree that these “evil spirits” are equivalent to “demons.” Furthermore, I did a phrase search on the phrase “evil spirit” in the Old Testament, which produced the following results:


  Jdg. 9:23
  Then God sent an evil spirit between Abimelech and the men of Shechem; and the men of Shechem dealt treacherously with Abimelech:
  
  1 Sam. 16:14
  But the Spirit of the LORD departed from Saul, and an evil spirit from the LORD troubled him.
  
  1 Sam. 16:15
  And Saul’s servants said unto him, Behold now, an evil spirit from God troubleth thee.
  
  1 Sam. 16:16
  Let our lord now command thy servants, [which are] before thee, to seek out a man, [who is] a cunning player on an harp: and it shall come to pass, when the evil spirit from God is upon thee, that he shall play with his hand, and thou shalt be well.
  
  1 Sam. 16:23
  And it came to pass, when the [evil] spirit from God was upon Saul, that David took an harp, and played with his hand: so Saul was refreshed, and was well, and the evil spirit departed from him.
  
  1 Sam. 18:10
  And it came to pass on the morrow, that the evil spirit from God came upon Saul, and he prophesied in the midst of the house: and David played with his hand, as at other times: and [there was] a javelin in Saul's hand.
  
  1 Sam. 19:9
  And the evil spirit from the LORD was upon Saul, as he sat in his house with his javelin in his hand: and David played with [his] hand.


Each of those use the same Greek expression in the LXX that is found for “evil spirit” in the NT (πνεῦμα πονηρόν or an inflection thereof). Also, the only other time we see the phrase “evil spirit(s)” is in Luke 7:21, 8:2 and Acts 19:12–13, 19:15–16. If we examine those passages in the Old Testament, most if not all say something to the effect of “evil spirit of/ from Yahveh/God.”

So, does God have control over demons? It seems so in the Old Testament. He is the one said to be sending the evil spirit. If this be the case, does God allow people to become possessed by demons, including those said to be possessed in the New Testament?
If God is omnipotent, then God is ultimately in charge of everyone, even demons. No one does anything apart from the will of God.

Many accept the existence of two wills in God - a hidden will of election (hidden or decretal will), and a revealed will. Following that idea, we could say that demons are obedient to God's hidden will (as indeed, everything perfectly is), while disobedient to God's revealed will. But it is his decretal will that they be disobedient to his revealed will, so it is his will that they disobey his will, and their disobedience is itself thus a form of obedience to God.

My answer is heavily influenced by Reformed theology, although Luther likewise believed in two wills. (I myself am a universalist rather than Reformed, but TULIP is only one step away from universalism - replacing the L with a U.) As I see it, the two wills doctrine is the only way to maintain the omnipotence of God.

I guess the alternative is to embrace metaphysical libertarianism, and maybe the sovereignty of God can be salvaged through middle knowledge. Personally, I find metaphysical libertarianism to be incoherent, and middle knowledge even more so. (I'd also note that both notions are, in my appraisal, alien to the Bible.) But there is no denying this is an alternative possible answer to your question.