I want to ask a question about how to find an optimal solution.
Here are my equations:
\begin{align} &GB_2 + GB_3 + GB_4 + \cdots + GB_r = 0 \nonumber \\ &GB_1 + GB_3 + GB_4 + \cdots + GB_r = 0 \nonumber \\ &GB_1 + GB_2 + GB_4 + \cdots + GB_r = 0 \nonumber \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \vdots \nonumber \\ &GB_1 + GB_2 + GB_3 + \cdots + GB_{r-1} = 0 \nonumber \end{align} where $G\in \mathbf{R}^{1 \times 2}$ is a unknown constant matrix, and $B_1, B_2, \cdots, B_r$ are constant matrices as follows: \begin{align} B_1 = B_3 = \begin{bmatrix} 0 \\ -0.1395 \end{bmatrix} , B_2 = B_4 = \begin{bmatrix} 0 \\ -0.0395 \end{bmatrix}, \end{align}
First, I have approached this problem by arranging the above equations into a state-space form. However, I don't know how to figure out the optimal solution of $G$. Is there a way to find the optimal solution? Thank you so much.