I have experienced this rising pitch among young American teachers (though I haven't really noticed it elsewhere). They refer to it as "up-speak."
Among them, I've been told that such a tone is used to convey friendliness and a positive tone.
Furthermore, it is used to keep the listener's interest, to clue them in to important keywords, or at the end of a phrase to show that the sentence is not yet over.
The following are some examples in which a teacher might use this:
When I say go^^, I want you to quickly and quietly^^, line up at the back of the room^^, and wait for me to tell you to go.
When I read^^, I constantly ask questions^^ and make predictions^^ about the story^^, so that I'm actively interacting with the text instead of just identifying words.
I see Jonah^^ working quietly on his project^^. Mary^^ is also working quietly on her project^^. John^^, I need to see you working quietly as well.
This site provides some additional explanation and examples: Statement Rising Pitch Boundaries
I can definitely see this rising pitch conveying weakness or lack of authority in situations which require sternness, but it can also be used to further contrast a general positive tone with instances that require sternness (and would therefore not use up-speak, especially to end the sentence).