    \begin{subequations}
        \begin{align}
            (I_{l_j}-\rho  \hat{\Phi} ) \boldsymbol{P}(t) &\geq -\frac{H_0T}{m}  (\mathrm {1}_{l_j} + \rho \hat{\Psi} \cdot \mathrm {1}_{n-l_j}) \ , \\
            (I_{l_j}-\rho  \tilde{\Phi} ) \boldsymbol{P}(t) &\leq \frac{H_0T}{m} ( \mathrm {1}_{l_j} + \rho \tilde{\Psi} \cdot \mathrm {1}_{n-l_j}) \ ,
        \end{align}
    \end{subequations}
