My question is based on the following passage in Liberzon's "calculus of variations and optimal control theory". I have added it to clarify the context of my question:
My question is:
Liberzon chooses the definitions of the control variables $u_1:=\frac {\dot x}{\sqrt y}$ and $u_2:=\frac {\dot y}{\sqrt y}$.
Firstly, I don't intuitively see what the meaning of $u_1$ and $u_2$ are, conceptually.
More importantly, How do we know that this is the correct choice? Why couldn't I have chosen some other definition? How does this definition help us solve the problem?
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Edit: Here is some context as requested by @Ian:
