A solution to the problem of structurally variable control synthesis for time-invariant Lurie systems is of both theoretical and engineering significance. A theorem on absolute stability of a set applied to Lurie systems solves the problem. The theorem is basic for proving control algorithms that ensure asymptotic stability of a sliding subspace, or its stability with finite time reachability. The selection of the sliding subspace can be such that the system nonlinearity does not influence the sliding motion. The proposed stabilising control algorithms have been tested applying them to a mathematical system and a real system, DC servo motor.