In The Tempest by Shakespeare, does Prospero undergo any character development?
It is fair to say that, no, Prospero undergoes no character development. He has had twelve years of exile to contemplate and plan for his removal from the island and reinstatement in his duchy. What we see in The Tempest is the end result of these twelve years. The final result of the carefully orchestrated action--as Prospero employs his magical arts and Ariel carries them out (a role Goethe later gave Ariel in Faust Part II)--is reconciliation, reunion, and reinstatement. Thus, whatever Prospero is at the end of the situation that he has thought about, planned for, and developed for so long and so carefully, we find that he is from the beginning of the play: he is the master magician, the master reconciliator, the master ruler who has learned and developed over twelve years time, but not over the course of the play. You might say the play represents Propero's life's climax and resolution.
Approach, my Ariel, come.
All hail, great master! grave sir, hail! I come
To answer thy best pleasure; be't to fly,
To swim, to dive into the fire, to ride
On the curl'd clouds, to thy strong bidding task
Ariel and all his quality.
Hast thou, spirit,
Perform'd to point the tempest that I bade thee? (I.ii)
Although Prospero is not the most dynamic character Shakespeare ever created, I believe he does undergo some significant character development. The Epilogue alone, which is spoken by Prospero, shows that he is finally able to relinquish some of the power he's valued so much over the course of the play - he is without his magic and explains that he must be validated by the audience and their applause/praise. The Prospero we met at the beginning of the play would not have admitted needing help/validation from others.