What does a function as a subscript mean?

Question

I stumbled across an interesting formula while reading a paper:

I'm put off by the $Q(i)$ and $S(i)$ functions as subscripts. I've never seen that notation before, and I'm wondering what it means.

Additionally, does it change the meaning of the 2 that's above each subscript, which i would normally take to mean squaring?

Edit: The Paper is "Sampling Based Model Predictive Control with Application to Autonomous Vehicle Guidance", by Dunlap, Collins, and Caldwell. The formula is on page two.

Unless this link expires, you can find the paper here

Answer 1

It means weighted norm using the subscripted notation:

$$\sum_{i=1}^N ||\boldsymbol{r}(k+i)-\boldsymbol{y}(k+i)||^2_{\boldsymbol{Q}(i)}+\sum_{i=0}^{M-1} ||\boldsymbol{u}(k+i)||^2_{\boldsymbol{S}(i)}$$

is equivalent to

$$\sum_{i=1}^N (\boldsymbol{r}(k+i)-\boldsymbol{y}(k+i))^T\boldsymbol{Q}(i)(\boldsymbol{r}(k+i)-\boldsymbol{y}(k+i))+\sum_{i=0}^{M-1} (\boldsymbol{u}(k+i))^T\boldsymbol{S}(i)\boldsymbol{u}(k+i)$$

where $\square ^T$ is the matrix transpose symbol.