## Explicit solution for varying a using Lagrange multipliers

In the unconstraint case a time-varying a can be handled by using an extended thrust representation similar to S0rdalen (1997). Consider the ROV/ AUV model in Example 1 where, t = T fe

= leKeue

where ue = [ulx,u^,u2x,u2y,U3,u4]T and fe = [/1,f2,f3,f4,f5,f6]T. The optimal azimuth angles and thrust commands are then found as

The main problem is that the optimal solution for «1 and «2 can jump at each sample which requires proper filtering. In the next sections, we propose other solutions to this problem.

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