## 3 Main Results  
### Stability of AUV System with Time-Varying Delays  
In this subsection, we present the stability proof of AUV systems I (4) and II (6). **Theorem 1**. : _For given positive scalars \(\epsilon_{i},i=1,2,3,...,6\), \(\rho\), \(0\leq\tau_{m}<\tau_{M}\), the uncertain time-varying delay system (4) 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:_  
\[[\hat{\Omega}_{s}]_{24\times 24}= \begin{bmatrix}[\hat{\Omega}_{0s}]_{20 \times 20}&\hat{\beta}B&0&\hat{\gamma}&0\\ *&-\kappa^{-1}&0&0&0\\ *&*&-\kappa\hat{\gamma}_{a}^{-1}&0&0\\ *&*&*&-\kappa^{-1}&0\\ *&*&*&*&-\kappa\hat{\gamma}_{s}^{-1}\end{bmatrix} <0, \tag{12}\]  
_where_  
\[[\hat{\Omega}_{0s}]_{20\times 20}= \begin{bmatrix}[\hat{\Omega}_{1}]_{8\times 8 }&[\hat{\Omega}_{2}]_{8\times 4}&[\hat{\Omega}_{3}]_{8\times 8}\\ *&[\hat{\Omega}_{4}]_{4\times 4}&0\\ *&*&[\hat{\Omega}_{5}]_{8\times 8}\end{bmatrix} ,\]\[[\hat{\Omega}_{1}]_{8\times 8}= \begin{bmatrix}\hat{O}_{1,1}&\hat{O}_{1,2}&\hat{O}_{1,3}&0& \hat{O}_{1,5}&\hat{O}_{1,6}&\hat{O}_{1,7}&\hat{O}_{1,8}\\ *&\hat{O}_{2,2}&\hat{O}_{2,3}&\hat{O}_{2,4}&0&\hat{O}_{2,6}&\hat{O}_{2,7}&0\\ *&*&\hat{O}_{3,3}&\hat{S}&0&0&0&0\\ *&*&*&\hat{O}_{4,4}&0&0&0&0\\ *&*&*&*&-\gamma^{2}&\tau_{m}B_{w}^{T}&\delta B_{w}^{T}&0\\ *&*&*&*&*&\hat{O}_{6,6}&0&0\\ *&*&*&*&*&\hat{O}_{7,7}&0\\ *&*&*&*&*&*&-I\end{bmatrix} ,\]  
\[\hat{O}_{1,1}=A\hat{P}+\hat{P}A^{T}+B\hat{K}+\hat{K}^{T}B^{T}+\hat{T}_{1}+\hat {T}_{2}+\hat{T}_{3}-\hat{T}_{4},\,\hat{O}_{1,2}=A_{d}\hat{P},\,\hat{O}_{1,3}= \hat{T}_{4},\,\hat{O}_{1,5}=B_{w},\]  
\(\hat{O}_{1,6}=\tau_{m}(\hat{P}A^{T}+\hat{K}^{T}B^{T}),\,\hat{O}_{1,7}=\delta( \hat{P}A^{T}+\hat{K}^{T}B^{T}),\,\hat{O}_  
where  
\[-\tau_{m}\int_{t-\tau_{m}}^{t}\dot{x}^{T}(s)T_{4}\dot{x}(s)\mathrm{d}s \leq -[x(t)-x(t-\tau_{m})]^{T}T_{4}[x(t)-x(t-\tau_{m})] \tag{17}\] \[-\delta\int_{t-\tau_{M}}^{t-\tau_{m}}\dot{x}^{T}(s)T_{5}\dot{x}(s) \mathrm{d}s = -\delta\int_{t-\tau(t)}^{t-\tau_{m}}\dot{x}^{T}(s)T_{5}\dot{x}(s) \mathrm{d}s\] (18) \[-\delta\int_{t-\tau_{M}}^{t-\tau(t)}\dot{x}^{T}(s)T_{5}\dot{x}(s) \mathrm{d}s.\]  
Each term on the right-hand side of (18) can be written as:  
\[-\delta\int_{t-\tau(t)}^{t-\tau_{m}}\dot{x}^{T}(s)T_{5}\dot{x}(s) \mathrm{d}s\leq-\frac{\delta}{\tau(t)-\tau_{m}}[x(t-\tau_{m})-x(t-\tau(t))]^{T }T_{5} \tag{19}\] \[\times[x(t-\tau_{m})-x(t-\tau(t))]\] \[-\delta\int_{t-\tau_{M}}^{t-\tau_{m}}\dot{x}^{T}(s)T_{5}\dot{x}(s) \mathrm{d}s\leq-\frac{\delta}{\tau_{M}-\tau(t)}[x(t-\tau(t))-x(t-\tau_{M})]^{T }T_{5}\] (20) \[\times[x(t-\tau(t))-x(t-\tau_{M})].\]  
Applying the inequalities (17), (19), and (20) in (16), we have:  
\[\dot{V}_{3}(t,x(t))\leq\dot{x}^{T}(t)(\tau_{m}^{2}T_{4}+\delta^{2}T_{5})\dot{ x}(t)-[x(t)-x(t-\tau_{m})]^{T}T_{4}[x(t)-x(t-\tau_{m})] \tag{21}\] \[- \begin{bmatrix}x(t-\tau_{m})-x(t-\tau(t))\\ x(t-\tau(t)-x(t-\tau_{M})\end{bmatrix} ^{T} \begin{bmatrix}T_{5}&S\\ S^{T}&T_{5}\end{bmatrix}\begin{bmatrix}x(t-\tau_{m})-x(t-\tau(t))\\ x(t-\tau(t)-x(t-\tau_{M})\end{bmatrix} .\end{bmatrix}\]  
It should be noted that the inequalities (17), (19), and (20) come from [PERSON]'s inequality (Lemma 2), and the inequality (21) comes from Theorem 1 of [29]. By using Definition 1 and combining it with (14), (16) and (21), we have:  
\[\dot{V}(t,x(t))+y^{T}(t)y(t)-\gamma^{2}w^{T}(t)w(t)\leq\zeta^{T}(t)[\Omega] \zeta(t) \tag{22}\]  
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  
\[[\Omega]_{7\times 7}=