Solving Dynamic Constrained Optimization Problems .Please take me out of this doubt

General DSGE Modeling
1

Hi everyone
I have an optimization problem in my DSGE model and I doubt how to solve it. thank you for taking me out of this doubt. Because if I make a mistake, I will face bigger problems in Dynare.I know optimization, but having a dynamic constraint on my optimization problem makes it a bit complicated. Thank you in advance for your kindness
I will not write the expectation operator for simplicity.

My optimization problem in the simplest case is as follows
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Which solution do you think is the best?. The third and fourth solutions seem to be more correct than the others. Please take me out of this doubt. Is there a better way to solve this problem?

2

I am not sure I understand. Why can you not take the derivative with respect to L_t in approach 1? You will simply get that the second constraint is not binding, i.e. \mu_{L,t}=0. But that is expected here, because nothing restricts L_t.

3

Thank you very much, Professor. According to your guidance, the answer should be this:

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Of course, if we simplify the answer obtained from other approaches, we will reach the same result.I realized what I had done wrong