Given a linear system contains $N$ nodes with input $\mathbf{V} = \{v_1,\cdots, v_N\}$. All the numbers are in the real domain.
$\frac{d\mathbf{T}}{dt} = \mathbf{A}\mathbf{T} + \mathbf{B}(\mathbf{V})$
in which $\mathbf{A}$ is an $N\times N$ constant matrix. $\mathbf{T} = [T_i]_{N\times 1}$ and is required to in the range of $[0, T_{max}]$, $\mathbf{B} = [B_i]_{N\times 1}$ and $B_i = v_i^3 + C_i$, where $C_i$ is a given constant.
Objective: max $\frac{\sum_{i=1}^N v_i}{\sum_{i=1}^N v_i^3}$
Constraint: $\mathbf{A}^{-1} \mathbf{B} \leq \mathbf{T}_{max}$ in which $A_i, B_i, C$ are constants. $ 0 \leq v_i \leq 1$. $\sum_{i=1}^N v_i = K$, K is given.
Currently, I have no hint of how to solve it. Or would you please provide some references for a similar problem?