# Fault Estimation for Semi-Markov Jump Neural Networks Based on the Extended State Method
## 3 Main Results
Consequently, we can derive the following from (40):  
\[\text{E}\{\Delta(\mathbb{V}(\mathcal{V}(g),\sigma(g),p)-(\rho-1)\mathbb{V}( \mathcal{V}(g),\sigma(g),p)-\mathcal{D}(g)\}\leq 0 \tag{43}\]  
Assume that when the dissipative supply rate \(\mathcal{D}(g)=0\) and disturbance \(\widetilde{\omega}(g)=0\), it has  
\[\text{E}\{\Delta(\mathbb{V}(\mathcal{V}(g),\sigma(g),p)-(\rho-1)\mathbb{V}( \mathcal{V}(g),\sigma(g),p)\}\leq 0 \tag{44}\]  
Now, it can be deduced that  
\[\text{E}\{\mathbb{V}(\mathcal{V}(g),\sigma(g),p)\}\leq\rho^{s}\text{E}\{ \mathbb{V}(\mathcal{V}(0),\sigma(0),p)\} \tag{45}\]  
where \(g=0,1,2,3,\ldots\)  
Based on the Lyapunov-Krasovskii function, combined with (28), then \[\mathbb{V}(\mathcal{V}(0),\sigma(0),p)=\sum\limits_{j=1}^{3}\mathbb{V}_{j}( \mathcal{V}(0),\sigma(0),p) \tag{46}\] \[=\mathcal{V}^{\mathrm{T}}(0)\mathrm{P}_{p,\sigma(0)}\mathcal{V}(0)+ \sum\limits_{s=-d_{1}}^{-1}\rho^{-s-1}\mathcal{V}^{\mathrm{T}}(s)\mathsf{X}_{ 1}\mathcal{V}(s)+\sum\limits_{s=-d(0)}^{-1}\rho^{-s-1}\mathcal{V}^{\mathrm{T}} (s)\mathsf{X}_{2}\mathcal{V}(s)\] \[+\sum\limits_{s=-d_{2}}^{-1}\rho^{-s-1}\mathcal{P}^{\mathrm{T}}( s)\mathsf{X}_{3}\mathcal{V}(s)+\sum\limits_{h=-d_{2}+1}^{-d_{1}}\sum\limits_{s=0+h}^{-1 }\rho^{-s-1}\mathcal{V}^{\mathrm{T}}(s)\mathsf{X}_{2}\mathcal{V}(s)\] \[+(d_{2}-d_{1})\sum\limits_{h=-d_{2}=s=0+h}^{-d_{1}-1}\rho^{-s-1} \theta^{\mathrm{T}}(s)\mathsf{X}_{2}\theta(s)\] \[\leq\lambda_{1}\rho^{s}\mathrm{sup}_{s\in\zeta[-d_{2},0]}\mathrm{ E}\{\|\;\overline{\sigma}(g)\;\|^{2}\}\]  
where  
\[\lambda_{1}=max\Big{\{}\eta_{max}\Big{(}\mathrm{P}_{p,\sigma(0)},p\Big{)}\in M \Big{\}}+d_{1}\eta_{max}(\mathsf{X}_{1})+d_{2}\eta_{max}(\mathsf{X}_{2})+d_{2 }\eta_{max}(\mathsf{X}_{3})+d_{2}\eta_{max}(\mathsf{X}_{2})+d_{12}\eta_{max}( \mathsf{X}_{4})\]  
From (28) and (28), we obtain  
\[\mathrm{E}\{\mathbb{V}(\mathcal{V}(g),\sigma(g),p)\}\leq\lambda_{2}\mathrm{E} \Big{\{}\|\;\mathcal{V}(g)\;\|^{2}\Big{\}} \tag{47}\]  
where \(\lambda_{2}=min\Big{\{}\eta_{min}\Big{(}\mathrm{P}_{p,\sigma(g)},p\Big{)}\in M \Big{\}}\). Substituting \(g=0\), we obtain  
\[\text{E}\{\Delta(\mathbb{V}(\mathcal{V}(g),\sigma(g),p)-(\rho-1)\mathbb{V}( \mathcal{V}(g),\sigma(g),p)\}\leq 0 \tag{44}\]0  
From Equation (46), we obtain an upper bound on the Lyapunov function variation, while Equation (48) provides a lower bound. By combining these two results and eliminating the gradient term \(\Delta(\mathbb{V}(\mathcal{V}(0),\sigma(0),p)\), we derive the following:  
\[\text{E}\{\Delta(\mathbb{V}(\mathcal{V}(g),\sigma(g),p)-(\rho-1)\mathbb{V}( \mathcal{V}(g),\sigma(g),p)\}\leq 0 \tag{44}\]1  
where \(\overline{\lambda}=\frac{\lambda_{1}}{\lambda_{2}}\) is a stability coefficient ensuring convergence, and \(0<\rho<1\) is the decay rate controlling the exponential stability. \(\Box\)  
Therefore, based on Definition 1, it can be inferred that the augmented s-MJNNs (22) achieve mean square exponential stability (MSES). However, when the control gain parameters are unknown, Theorem 2 further demonstrates that the augmented s-MJNNs (22) maintain strict \((\chi_{1},\chi_{2},\chi_{3})-\varpi\) dissipation performance, ensuring robustness in the presence of parametric uncertainties. **Theorem 2**. : _It is known that \(\delta\in[0,1]\), \(d_{1}>0,d_{2}>0,0<\rho<1\), and the gain matrix \(\overset{\rightarrow}{K}_{p,\sigma(g)}\), in order to make the augmented s-MJNN (22), have a strict \((\chi_{1},\chi_{2},\chi_{3})-\varpi\) dissipative performance. If a symmetric matrix can be found \(P_{p,\sigma(g)}>0(p\in\mathcal{Q})\), \(X_{a}(a=1,2,3,4)\) and an arbitrary matrix \(\mathcal{L}_{1}>0\), \(\mathcal{L}_{2}>0\), \(\mathfrak{L}_{p,\sigma(g)}\), \(\varphi_{p,\sigma(g)}(b=1,2)\), the following LMI holds:_  
\[