## 3 Main Results
### Stability of AUV System with Time-Varying Delays
.\]  
Using Lemma 2 of [20], we have:  
\[[\hat{\Omega}]_{20\times 20} = [\hat{\Omega}_{1}]_{8\times 8}+\sum_{i=1}^{6}(\epsilon_{i}^{-1}[ \mathcal{M}_{i}]_{8\times 1}[\mathcal{M}_{i}]_{8\times 1}^{T}+\epsilon_{i}[ \mathcal{N}_{i}]_{8\times 1}[\mathcal{N}_{i}]_{8\times 1}^{T}).\]  
By applying Schur Complement Lemma 1 to \(\hat{\Omega}\), we can guarantee that LMI (12) holds, that is, \(\hat{\Omega}_{5}<0\). Thus, we concludes that the system (4) is asymptotically stable and satisfies the \(H_{\infty}\) performance level \(\gamma\). This completes the proof. Theorem 3.1: _For given positive scalars \(\epsilon_{i},i=1,2,3,...,6\), \(\rho\), \(0\leq\tau_{m}<\tau_{M}\), the uncertain time-varying delays system (6) under controller (2) is asymptotically stable and satisfies the \(H_{\infty}\) performance level \(\gamma\), if there exist symmetric positive definite matrices \(\hat{P}\), \(\hat{T}_{1}\), \(\hat{T}_{2}\), \(\hat{T}_{3}\), \(\hat{T}_{4}\), \(\hat{T}_{5}\), and for any matrix \(\hat{S}\), \(\hat{K}\) with compatible dimensions, one satisfies \( \begin{bmatrix}\hat{T}_{5}&\hat{S}\\ \hat{S}^{T}&\hat{T}_{5}\end{bmatrix} \geq 0\), such that the following LMI holds:_  
\[[\hat{\Xi}_{s}]_{24\times 24}= \begin{bmatrix}[\hat{\Xi}_{\infty}]_{20\times 20}& \hat{\beta}B&0&\bar{\gamma}&0\\ *&-\kappa^{-1}I&0&0&0\\ *&*&-\kappa\bar{\chi}_{a}^{-1}&0&0\\ *&*&*&-\kappa^{-1}I&0\\ *&*&*&*&-\kappa\bar{\chi}_{s}^{-1}\end{bmatrix} <0, \tag{24}\]_where_  
\[[\hat{\Xi}_{\infty}]_{20\times 20}= \begin{bmatrix}[\hat{\Xi}_{1}]_{8\times 8}&[\hat{\Omega}_{ 2}]_{8\times 4}&[\hat{\Omega}_{3}]_{8\times 8}\\ *&[\hat{\Omega}_{4}]_{4\times 4}&0\\ *&*&[\hat{\Omega}_{5}]_{8\times 8}\end{bmatrix} ,\]  
\(\hat{\Xi}_{1,1}=A\hat{P}+\hat{P}A^{T}+B\hat{K}+\hat{K}^{T}B^{T}+\hat{T}_{1}+ \hat{T}_{2}+\hat{T}_{3}-\hat{T}_{4},\hat{\Xi}_{1,2}=A_{d}\hat{P}+B_{d}\hat{K}, \hat{\[PERSON]}_{1,3}=\hat{T}_{4},\hat{\[PERSON]}_{1,5}=B_{w},\hat{\[PERSON]}_{1,6}=\tau_{m}( \hat{P}A^{T}+\hat{K}^{T}B^{T}),\hat{\[PERSON]}_{1,7}=\delta(\hat{P}A^{T}+\hat{K}^{T}B ^{T}),\hat{\[PERSON]}_{1,8}=\hat{P}C^{T},\hat{\[PERSON]}_{2,2}=-(1-\mu)\hat{T}_{3}-2\hat {T}_{5}+\hat{S}^{T},\hat{\[PERSON]}_{2,3}=\hat{T}_{5}-\hat{S}^{T},\hat{\[PERSON]}_{2,4}= \hat{T}_{5}-\hat{S},\hat{\[PERSON]}_{2,6}=\tau_{m}(\hat{P}A^{T}_{d}+\hat{P}B^{T}_{d}),\hat{\[PERSON]}_{2,7}=\delta(\hat{P}A^{T}_{d}+\hat{P}B^{T}_{d}),\hat{\[PERSON]}_{3,3}=- \hat{T}_{1}-\hat{T}_{4}-\hat{T}_{5},\hat{\[PERSON]}_{3,4}=\hat{S},\hat{\[PERSON]}_{44}=- \hat{T}_{2}-\hat{T}_{5},\hat{\[PERSON]}_{5,5}=-\gamma^{2},\hat{\[PERSON]}_{5,6}=\tau_{m}B^{ T}_{w},\hat{\[PERSON]}_{5,7}=\delta B^{T}_{w},\hat{\Xi}_{6,6}=-2p\hat{P}+\rho^{2}\hat{T}_{4}, \hat{\Xi}_{2,7}=-2\rho\hat{P}+\rho^{2}\hat{T}_{5},\hat{\Xi}_{8,8}=-1\) _and all other parameters are defined in Theorem 1. Furthermore, the controller gain matrix is given by \(K=\hat{K}\hat{P}^{-1}\)._  
Let us consider the Lyapunov-Krasovskii functional (13) as in Theorem 1 and take its derivative along the trajectories of system (6) yields:  
\[\dot{V}_{1}(t,x(t)) = 2x^{T}(t)P\dot{x}(t)\] \[= 2x^{T}(t)P[\left(A+\alpha(t)\Delta A\right)x(t)+\left(A_{d}+\alpha (t)\Delta A_{d}\right)x(t-\tau(t))\] \[+\left.B[u(t)+\beta(t)\chi_{a}(t)]+B_{d}u(t-\tau(t))+B_{w}w(t) \right].\]  
By utilizing Equations (16)-(21) and Definition 1, we obtain:  
\[\dot{V}(t,x(t))+y^{T}(t)y(t)-\gamma^{2}w^{T}(t)w(t)\leq\zeta^{T}(t)[\Xi]\zeta(t) \tag{26}\]  
where \(\zeta^{T}(t)=[x^{T}(t)\;\;x^{T}(t-\tau(t))\;\;x^{T}(t-\tau_{m})\;\;x^{T}(t- \tau_{M})\;\;w^{T}(t)\;\;\dot{x}^{T}(t)\;\;\dot{x}^{T}(t)]\) and  
\[[\Xi]_{7\times 7}= \begin{bmatrix}C^{c}_{1,1}&C^{\Xi}_{1,2}&O_{1,3}&0&O_{1,5}& O_{1,6}&O_{1,7}\\ *&O_{2,2}&O_{2,3}&O_{2,4}&0&O^{\Xi}_{2,6}&O^{\Xi}_{2,7}\\ *&*&O_{3,3}&\hat{S}&0&0&0\\ *&*&*&O_{4,4}&0&0&0\\ *&*&*&*&-\gamma^{2}&\tau_{m}B^{T}_{w}&\delta B^{T}_{w}\\ *&*&*&*&*&-T^{4}_{1}&0\\ *&*&*&*&*&-T^{4}_{5}\end{bmatrix}