The Independent Administrative Corporation Japan Agency for Marine-Earth Science and Technology (JAMSTEC) has been developing light-and-small autonomous underwater vehicles (AUV), named "Marine Robot Experimental 1" ("MR-X1"). In this paper, the motion control problem of "MR-X1" is considered. Since the dynamics of "MR-X1" depends upon its own speed, the problem in question becomes a non-linear control problem. A new controller design method using linear matrix inequalities (LMIs) is proposed in order to adapt the problem in question. The algorithm, which gives a solution of a linear matrix inequality (LMI), can adapt to solve numerous LMIs simultaneously. LMIs can be obtained by substituting several speeds into the dynamics of "MR-X1". The proposed controller, which can be derived from the solution of LMIs, becomes a stabilizing controller for all substituted speeds.