Problem Description
My equations are very long, they are not going from one column to the next, that is disturbing the layout of the paper, too much undesired space. What should I do? I tried \allowdisplayreaks
. It is not working.
MWE:
\allowdisplaybreaks
\begin{equation} \tag{41} \label{41}
\begin{split}
\mathbb{E}\{\bigtriangleup{V}(\tilde{\mathrm{e}}_{i,k})\}&= \mathbb{E}\{ \displaystyle{\sum \limits_{i=1}^{N}}
\{\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k} +\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tilde{\mathrm{e}}_k \\&\mspace{20mu}+\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}- \tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\mspace{20mu}+\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k
- \tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}+ \tilde{\mathrm{e}}_{k}^{T}\Phi_1^{T}P_iS_i\tilde{\mathrm{e}}_{i,k}
+ \tilde{\mathrm{e}}_k^T\Phi_1^TP_i\Phi_1\tilde{\mathrm{e}}_k
\\&\mspace{20mu}+ \tilde{\mathrm{e}}_k^T\Phi_1^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}- \tilde{\mathrm{e}}_k^T\Phi_1^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\mspace{20mu}+\tilde{\mathrm{e}}_k^T\Phi_1^TP_i\bar{\mathrm{B}}\mathrm{w}_k
-\tilde{\mathrm{e}}_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\Phi_1\tilde{\mathrm{e}}_k
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}- \Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\bar{\mathrm{B}}\mathrm{w}_k
\\&\mspace{20mu}-\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
\\&\mspace{20mu}-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Phi_1\tilde{\mathrm{e}}_k
\\&\mspace{20mu}-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}+\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\mspace{20mu}- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k
\\&\mspace{20mu}+ \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tilde{\mathrm{e}}_k
\\&\mspace{20mu}+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}- \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\mspace{20mu}+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k
-\mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{\mathrm{e}}_k
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}+\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k
+\mathrm{v}_{i,k}D_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}\}
\\&\mspace{20mu}- \displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_{i,k}^TP_i\tilde{\mathrm{e}}_{i,k}\}.
\end{split}
\end{equation}
Substituting (\ref{41}) into (\ref{40}) and manipulating it similar to (\ref{19}) leads to
\allowdisplaybreaks
\begin{equation*} \tag{42} \label{42}
\begin{split}
J&= \mathbb{E}\{\displaystyle{\sum \limits_{i=1}^{N}}
\{\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tilde{e}_{i,k} +\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tilde{\mathrm{e}}_k
+\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k
\\&\mspace{20mu}- \tilde{\mathrm{e}}_{i,k}^TS_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
+ \tilde{\mathrm{e}}_{k}^{T}\Phi_1^{T}P_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
+ \tilde{\mathrm{e}}_k^T\Phi_1^TP_i \Phi_1\tilde{\mathrm{e}}_k
\\&\mspace{20mu}+\tilde{\mathrm{e}}_k^T\Phi_1^TP_i\bar{\mathrm{B}}\Phi_1\mathrm{w}_k
-\tilde{\mathrm{e}}_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
\\&\mspace{20mu}+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tilde{\mathrm{e}}_k
+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k
-\mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{e}_k
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}w_k
+\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}- \displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_{i,k}^TP_i\tilde{\mathrm{e}}_{i,k} + \sigma N\displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_{i,k}^{T}\tilde{\mathrm{e}}_{i,k} - \sigma\displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_k^{T}\tilde{\mathrm{e}}_k
\\&\mspace{20mu}+ f'^{T}(x_k,\vartheta, \rho_k)P_if'(x_k,\vartheta, \rho_k)
\\&\mspace{20mu}+ \lambda_{max}(L_i^TP_iL_i)\zeta_i^T(\tilde{x}_{i,k}, \tau_k)\zeta_i(\tilde{x}_{i,k}, \tau_k)\}
\\&\mspace{20mu}+ \displaystyle{\sum \limits_{i=1}^{N}} \dfrac{1}{\xi N} \tilde{\mathrm{e}}_{i,k}^{T}\tilde{\mathrm{e}}_{i,k}
- \gamma_1 \mathrm{w}_k^{T}\mathrm{w}_k
- \displaystyle{\sum \limits_{i=1}^{N}} \gamma_2 \mathrm{v}_{i,k}^{T}\mathrm{v}_{i,k}\}.
\end{split}
\end{equation*}