## 2 Problem Formulation
### Dynamic Model of Surface Vessels  
The dynamic model of each surface vehicle in the direction of its movement in a group of N ASVs, described by [30]:  
\[M_{i}\dot{V}_{i}+C_{i}(V_{i})V_{i}+D_{i}V_{i}+\tau_{i\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny {I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}} \text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I }}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I }}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I }}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I }}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I }}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}} \text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I }}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}} \text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}} \text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}} \text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}} \text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{ \tiny{I}}\text{\tiny{I}}\text{\tiny{I}}\text{\Therefore, considering the new torque vector as \(\tau_{ai}(t)=,\left[\tau_{u},\tau_{t}\right]^{T}\), the dynamic model of the ASV can be written as follows:  
\[M_{1i}\dot{v}_{i}+C_{1i}(v_{i})v_{i}+D_{1i}v_{i}+\tau_{w1\ i}(t)=\tau_{a\ i}(t) \tag{7}\]  
where we have as follows:  
\[M_{1i}=\left[ \begin{array}{cc}m_{11i}&0\\ 0&m_{33i}\end{array} \right] \tag{8}\]  
\[C_{1i}(v_{i})=\left[ \begin{array}{cc}0&-m_{22i}v_{i}\\ (m_{22i}-m_{11i})v_{i}&0\end{array} \right] \tag{9}\]  
\[D_{1i}=\left[ \begin{array}{cc}d_{11i}&0\\ 0&d_{33i}\end{array} \right] \tag{10}\]  
In Equation (7), \(\tau_{w1i}(t)\in\ \mathcal{R}^{2}\) denotes the vector of instant disturbances that caused by the environment (sea or ocean). \[\tau_{w1i}(t)=\left[ \begin{array}{c}\tau_{wu\ i}(t)\\ \tau_{w\ r}(t)\end{array} \right] \tag{11}\]  
This vector is bounded as \(\left\|\tau_{w1i}(t)\right\|\leq\lambda_{\rm r1i}\) where \(\lambda_{\rm r1i}\) is a positive constant. Further, \(\tau_{ai}(t)=\ \left[\tau_{ui}(t),\tau_{n}(t)\right]^{T}\) denotes the input vector that is exposed to saturation constraint:  
\[\left|\tau_{ai}(t)\right|\leq\tau_{ai,max} \tag{12}\]  
Property 1. The following properties can be proven for model (7).