Did Théoden recognize Aragorn as Thorongil?

[Alcuin]

Disguised as “Thorongil”, Aragorn served Thengel King of Rohan (Théoden’s father) and Ecthelion II Ruling Steward of Gondor (Denethor’s father) from 2957 until 2980, about 23 years. Appendix A to the Lord of the Rings strongly hints that Denethor discerned that “Thorongil” was in fact the Chieftain of the Dúnedain and thus the Heir of Isildur and rightful King of Gondor; this fed Denethor’s distrust of Gandalf.

Denethor was two years older than Aragorn. Théoden was 17 years younger. He was about 9 when Thorongil arrived in Rohan. If Thorongil spent about 10 of the 23 years in Rohan, then Théoden was about 19 when Thorongil went to Gondor. However you slice it, he was probably a teenager when Thorongil left. They must have known one another well since Thorongil probably lived in Thengel’s household.

It appears that Thorongil was rather fond of Théoden: when Théoden leaves with Éomer and Merry to ride to Dunharrow after Aragorn has looked into the Stone of Orthanc, Aragorn tells his cousin Halbarad, “There go three that I love, and the smallest not the least.”

Tolkien presents Théoden as intelligent, wise, and insightful. Did Théoden recognize Aragorn as Thorongil? Might Gandalf have told him who Aragorn was when he removed Wormtongue’s bewitchment on the stairs before Meduseld? Did Théoden figure it out on his own? (In the Peter Jackson films, Théoden tells Éowyn that Aragorn served her grandfather, something she has trouble believing.) Might this have influenced his decisions to go to war himself, and to trust Éomer again?