$$\begin{aligned} \Delta V(k) & = V(k + 1) - V(k) \\ & \; = \sum\limits_{{i = 1}}^{{N - 1}} {\sum\limits_{{j = i + 1}}^{N} {[x_{{ij}}^{T} (k)A^{T} P_{{\sigma (k_{{0,x}} )}} Ax_{{ij}} (k)} } \\ \quad + x_{{ij}}^{T} (k)L^{T} B^{T} P_{{\sigma (k_{{0,x}} )}} BLx_{{ij}} (k) \\ \quad + x_{{ij}}^{T} (k - \tau )L^{T} D^{T} P_{{\sigma (k_{{0,x}} )}} DLx_{{ij}} (k - \tau ) \\ \quad + x_{{ij}}^{T} (k - \tau )NC_{{\sigma (k_{{0,x}} )}}^{2} \Gamma ^{T} P_{{\sigma (k_{{0,x}} )}} \Gamma x_{{ij}} (k - \tau ) \\ \quad + x_{{ij}}^{T} (k)A^{T} Q_{{1,\sigma (k_{{0,x}} )}}^{{ - 1}} Ax_{{ij}} (k) \\ \quad + x_{{ij}}^{T} (k)L^{T} B^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{1,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} BLx_{{ij}} (k) \\ \quad + x_{{ij}}^{T} (k)A^{T} Q_{{2,\sigma (k_{{0,x}} )}}^{{ - 1}} Ax_{{ij}} (k) \\ \quad + x_{{ij}}^{T} (k - \tau )L^{T} D^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{2,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} DLx_{{ij}} (k - \tau ) \\ \quad - x_{{ij}}^{T} (k)NC_{{\sigma (k_{{0,x}} )}} A^{T} Q_{{3,\sigma (k_{{0,x}} )}}^{{ - 1}} AC_{{\sigma (k_{{0,x}} )}} x_{{ij}} (k) \\ \quad - x_{{ij}}^{T} (k - \tau )\Gamma ^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{3,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} \Gamma x_{{ij}} (k - \tau ) \\ \quad + x_{{ij}}^{T} (k)L^{T} B^{T} Q_{{4,\sigma (k_{{0,x}} )}}^{{ - 1}} BLx_{{ij}} (k) \\ \quad + x_{{ij}}^{T} (k - \tau )L^{T} D^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{4,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} DLx_{{ij}} (k - \tau ) \\ \quad - x_{{ij}}^{T} (k)L^{T} NC_{{\sigma (k_{{0,x}} )}} B^{T} Q_{{5,\sigma (k_{{0,x}} )}}^{{ - 1}} BC_{{\sigma (k_{{0,x}} )}} Lx_{{ij}} (k) \\ \quad - x_{{ij}}^{T} (k - \tau )\Gamma ^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{5,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} \Gamma x_{{ij}} (k - \tau ) \\ \quad - x_{{ij}}^{T} (k - \tau )L^{T} NC_{{\sigma (k_{{0,x}} )}} D^{T} Q_{{6,\sigma (k_{{0,x}} )}}^{{ - 1}} DC_{{\sigma (k_{{0,x}} )}} Lx_{{ij}} (k - \tau ) \\ \quad - x_{{ij}}^{T} (k - \tau )\Gamma ^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{6,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} \Gamma x_{{ij}} (k - \tau ) \\ \quad - x_{{ij}}^{T} (k)P_{{\sigma (k_{{0,x}} )}} x_{{ij}} (k)] \\ & \; = \sum\limits_{{i = 1}}^{{N - 1}} {\sum\limits_{{j = i + 1}}^{N} {\{ x_{{ij}}^{T} (} } k)[A^{T} P_{{\sigma (k_{{0,x}} )}} A + A^{T} Q_{{1,\sigma (k_{{0,x}} )}} A \\ \quad + L^{T} B^{T} P_{{\sigma (k_{{0,x}} )}} BL + A^{T} Q_{{2,\sigma (k_{{m,x}} )}}^{{ - 1}} A \\ \quad + L^{T} B^{T} P_{{\sigma (k_{{m,x}} )}}^{T} Q_{{1,\sigma (k_{{m,x}} )}} P_{{\sigma (k_{{m,x}} )}} BL \\ \quad - NC_{{ij,\sigma (k_{{0,x}} )}} A^{T} Q_{{3,\sigma (k_{{0,x}} )}}^{{ - 1}} AC_{{ij,\sigma (k_{{0,x}} )}} \\ \quad + L^{T} B^{T} Q_{{4,\sigma (k_{{0,x}} )}} BL \\ \quad - L^{T} NC_{{\sigma (k_{{0,x}} )}} B^{T} Q_{{5,\sigma (k_{{0,x}} )}} BC_{{\sigma (k_{{0,x}} )}} L - P_{{\sigma (k_{{0,x}} )}} ]x_{{ij}} (k) \\ \quad + x_{{ij}}^{T} (k - \tau )[L^{T} D^{T} P_{{\sigma (k_{{0,x}} )}} DL - NC_{{\sigma (k_{{0,x}} )}}^{{(2)}} \Gamma ^{T} P_{{\sigma (k_{{0,x}} )}} \Gamma \\ \quad + L^{T} D^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{2,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} DL \\ \quad - \Gamma ^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{3,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} \Gamma \\ \quad + L^{T} D^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{4,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} DL \\ \quad - \Gamma ^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{5,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} \Gamma \\ \quad - L^{T} NC_{{\sigma (k_{{0,x}} )}} D^{T} Q_{{6,\sigma (k_{{0,x}} )}}^{{ - 1}} DC_{{\sigma (k_{{0,x}} )}} L \\ \quad - \Gamma ^{T} P_{{\sigma (k_{{0,x}} )}}^{T} Q_{{6,\sigma (k_{{0,x}} )}} P_{{\sigma (k_{{0,x}} )}} \Gamma ]x_{{ij}} (k - \tau )\} \\ & \; = \left[ {\frac{{\lambda _{{\max }} (\Pi _{{\sigma (k_{{0,x}} )}} )}}{{\lambda _{{\min }} (P_{{\sigma (k_{{0,x}} )}}^{{ - 1}} )}} - 1} \right]V(k) + \frac{{\lambda _{{\max }} (\Omega _{{\sigma (k_{{0,x}} )}} )}}{{\lambda _{{\min }} (P_{{\sigma (k_{{0,x}} )}}^{{ - 1}} )}}V(k - \tau ) \\ & \; = \left[ {\frac{{\lambda _{{\max }} (\Pi _{{\sigma (k_{{0,x}} )}} )}}{{\lambda _{{\min }} (P_{{\sigma (k_{{0,x}} )}}^{{ - 1}} )}} + \mu q_{{\sigma (k_{{0,x}} )}} \frac{{\lambda _{{\max }} (\Omega _{{\sigma (k_{{0,x}} )}} )}}{{\lambda _{{\min }} (P_{{\sigma (k_{{0,x}} )}}^{{ - 1}} )}} - 1} \right]V(k) \\ \end{aligned}$$
(36)