<p>The following mechanical system has been considered: an unstable rigid body (the second body) has an arbitrary axisymmetric cavity completely filled with an ideal incompressible fluid, and it is connected to the first and third rigid bodies by means of elastic restoring spherical hinges. Based on the well-known P.V. Kharlamov’s equations and S.L. Sobolev’s state function, the problem of stabilizing the unstable rotation of the second rigid body (with fluid) in a medium with resistance by rotating two other rigid body has been considered. Dissipative and constant moments act on all rigid bodies. The equations of perturbed motion were obtained in the case where the centers of mass of the rigid bodies and the fluid are located on the third principal axis of inertia. In the case of three consecutively connected Lagrangian gyroscopes, with the intermediate one with fluid, these equations are presented in the form of a countable system of ordinary differential equations, and the corresponding transcendental characteristic equation is obtained. Taking into account the fundamental oscillation tone of ideal fluid, a seventh-order characteristic equation was derived, and conditions for asymptotic stability of uniform rotations of the Lagrangian gyroscopes and the fluid were obtained in the form of a system of six inequalities. The stabilization of unstable rotation of a Lagrangian gyroscope with ideal fluid by rotating its solid parts and using the elasticity of the joints was considered. The stabilization conditions were written for the kinetic moments of the first and third gyroscopes and the elasticity coefficients of the joints. These conditions are exact for an ellipsoidal cavity and a cavity formed by confocal ellipsoids, but they are only approximate for all other parameters so that additional oscillation tones of ideal fluid have to be taken into account for their refinement. It was shown that if the center of mass of the gyroscope with fluid and the center of mass of the first or third gyroscope coincide with their common point, the stabilization of unstable rotation of the Lagrangian gyroscope with fluid is impossible. The mechanical system under consideration is multiparametric; therefore, for further analytical studies, some special cases of the initial problem were analyzed: the equality of the absolute values of the kinetic moments of the first and third gyroscopes, the equality of the elasticity coefficients of the spherical hinges, and the equality of the first and third gyroscopes and dissipation in them. Conditions were found under which the leading coefficients in the obtained inequalities are positive, which means that the stabilization of the unstable rotation of the Lagrangian gyroscope with ideal fluid is always possible if the angular velocity of the first or third gyroscopes or the elasticity coefficients of the spherical hinge increase. The main stabilization condition is that the first oscillation tone of ideal fluid has to exceed unity. For an ellipsoidal cavity, this means that it must be squeezed along the rotation axis.</p>