Who was the Tetrarch of Abilene during the time of Jesus?

According to Luke 3:1 it was Lysanias. But this may be a mistake because Lysanias was executed by Mark Anthony in 36BC. see link Lysanias

As usual Luke is clearly demonstrated to be a reliable historian without error.

Schurer put forward a compelling argument which showed a number of reasons why the 'early Lysanias -Luke in error' hypothesis does not hold water.

Firstly, the earlier Lysanias, son of Ptolemy was known as a king and not a Tetrarch. He thus had a much bigger kingdom and a more senior title.

Secondly Schurer demonstrated that it is impossible that the small tetrarchy under the later Lysanias was merely named after the earlier one who had died some 60 years earlier.

"From these passages we see that the tetrarchy of Abila had belonged previously to A.D. 37 to a certain Lysanias, and seeing that Josephus nowhere previously makes any mention of another Lysanias, except the contemporary of Anthony and Cleopatra, 40-36 B.C. . . . criticismhas endeavoured in various ways to show that there had not afterwards been any other, and that the tetrarchy of Abilene had its name from the older Lysanias. But this is impossible" (Schürer, 337). (emphasis mine)

Source: The New Catholic Encyclopedia

Thirdly, Schurer points out that an inscription found at Abila (the source of the name Abilene) contains a a statement which verifies the veracity of Luke. The inscription uses terminology which was not used until after the AD 14 death of Augustus. It refers to a certain 'freedman of Lysanias' who 'built a street and erected a temple in the time of the "Augusti (Sebastoi)".

Summary:

Thus, despite the contentions of earlier scholars that Luke was in error, it has since been concluded that here, as in so many other places, he speaks with complete accuracy. It is always important to have the latest data.

Lysanias was thus the Tetrarch of Abilene during Jesus' time.