Requesting references about recursive functions where the variables are continuous

Question

I have a recursive function that looks something like this. The variable x is a continuous variable. Do anyone have a reference that looks into a similar problem? $$f_i(y)=\min_{0\le x\le\overline{X_i}}\{f_{i-1}(y+B_i-x)+c_i(y+x)\}$$ The base case is $f_0(y)=0\quad\forall y\in[0,B]$.

Answer 1

If the intent is to optimize $f_N(y)$ for some fixed $N,$ it might be a form of dynamic programming.