## 3 Analytical and Numerical Solutions
### Case of Constant Density and Incompressible Hull
The solution of Equation (17) taking account of initial conditions was found in the form:  
\[\mathrm{v_{s}}=\sqrt{\frac{2 gl^{*}\eta}{C_{D}}}. \tag{6}\]0  
\[\mathrm{v_{s}}=\sqrt{\frac{2 gl^{*}\eta}{C_{D}}}. \tag{6}\]1  
The calculation of the process of deceleration for the case of an instantaneous change in buoyancy on the basis of an analytical solution taking account of the vehicle's full stop and the subsequently accelerated ascension is presented in Figures 7 and 8 for the case of the cylinder and in Figures 9 and 10 for a sphere of identical volume. The analytical results obtained for the deceleration process are valid on the assumption of an instantaneous change in relative buoyancy from \(-\eta\) to \(\eta\). The finiteness of the time of the buoyancy change can be made through the numerical solution:  
\[\mathrm{v_{s}}=\sqrt{\frac{2 gl^{*}\eta}{C_{D}}}. \tag{6}\]2  
where the control was adopted in the form \(f_{d}(t)=1-t/t_{0}\) for \(0\leq t\leq 2t_{0}\) and \(f_{da}(t)=-1\) for \(t\geq 2t_{0}\), allowing accelerated ascension after the vehicle completely stops. In Figures 11 and 12 are calculated data characterizing the influence of finiteness of time of the buoyancy change upon the process of deceleration and subsequent ascension of the cylinder. The results obtained above enable estimating the full time of submersion \(t_{\Sigma}\). Summed up therewith are the time of acceleration, time of steady motion and time of deceleration until complete stop. The time of acceleration \(t_{a}\) and deceleration \(t_{d}\) are determined through Formulae (10) and (15), and the time of steady submersion is determined by means of the formula  
\[\mathrm{v_{s}}=\sqrt{\frac{2 gl^{*}\eta}{C_{D}}}. \tag{6}\]3  
where \(z_{ma}\)_n_\(z_{md}\) are the lengths of acceleration and deceleration distances. The calculated data are presented in Table 1 for the case of a cylinder with length \(l_{c}=2\) m. The full time of submersion is given in the last column of Table in seconds and minutes (in bold font). \[\mathrm{v_{s}}=\sqrt{\frac{2 gl^{*}\eta}{C_{D}}}. \tag{6}\]4 Speed of submersion of the cylinder from the start of deceleration until complete stop considering the subsequent ascension for an instantaneous change in buoyancy (\(t_{0}=0\)) from \(-\eta\) to \(\eta\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\eta\) & \(t_{a}[\mathbf{s}]\) & \(z_{ma}[\mathbf{m}]\) & \(v_{s}[\mathbf{m}/\mathbf{s}]\) & \(t_{d}[\mathbf{s}]\) & \(z_{md}[\mathbf{m}]\) & \(t_{s}[\mathbf{s}]\) & \(t_{\text{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{\bf{         }}}}}}}}}}}} ]\) \\ \hline
0.001 & 29.7 & 1.5 & 0.21 & 41.4 & 5.83 & 917 & 988/**16** \\ \hline
0.005 & 28.4 & 6.3 & 0.49 & 43.2 & 10.2 & 375 & 447/**7.5** \\ \hline
0.01 & 27.5 & 9.8 & 0.69 & 30.2 & 13.2 & 257 & 315/**5.2** \\ \hline
0.02 & 25.1 & 13.1 & 0.98 & 28.1 & 17.3 & 173 & 226/**3.8** \\ \hline
0.03 & 23.7 & 15.4 & 1.2 & 27.0 & 20.4 & 137 & 188/**3.1** \\ \hline
0.05 & 22.4 & 19.0 & 1.55 & 26.0 & 25.2 & 100 & 148/**2.5** \\ \hline \end{tabular}
\end{table} 
Table 1: Dependence of the full time of submersion of the cylinder on the change in relative volume of replacement for \(H=200\) m, \(l=2\) m, \(t_{0}=20\) s.  
Figure 7: Submersion of the cylinder from the start of deceleration until complete stop considering the subsequent ascension for an instantaneous change in buoyancy (\(t_{0}=0\)) from \(-\eta\) to \(\eta\). The full submersion time was also calculated by means of the direct numerical solution in Equation (20). Used therewith was the control function \(f(t)\), defined in the following way:  
\[f(t)=\left\{ \begin{array}{l}t/t_{0},t\leq t_{0};\\ 1,t\geq t_{0}.\end{array} \right. \tag{22}\]  
for \(t\geq t_{\it as}\),  
\[f(t)=\left\{ \begin{array}{l}1-(t-t_{\it as})/t_{0},t_{\it as}\leq t\leq t_{\it as }+2t_{0};\\ -1,t\geq t_{\it as}+2t_{0},\end{array} \right. \tag{23}\]  
where \(t_{\it as}\) is the moment of time to turn on the BE for deceleration until completely stopping and starting ascension.