Optimal Control example from Wikipedia

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I am interested in Optimal Control:

https://en.wikipedia.org/wiki/Optimal_control

and I am reading the discrete example at the bottom of the Wikipedia page:

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where they give the solution as: enter image description here

I have some questions (from top red annotations to bottom):

  1. Where does ($\lambda_{t+1} u_t$) come from?
  2. How did they equate negative the differential, to $-(\frac{u_t}{x_t})^2$ ?
  3. What does it mean using the initial and turn T conditions?