This work may be viewed as a contribution to the study of control of the dynamics of physical systems in which the potential have a catastrophic single well 6 configuration. We demonstrated that The mathematical model of various non-linear structures (inverted pendulum, articulated beam, elastic beam fixed at its base and free at the top) is that of a particle moving in a catastrophic single well 6 potential. control by sandwich and the one using piezoelectric absorber are investigated. The range of coupling or control parameters that can produce an e ective control (reduction of amplitude, avoidance of unbounded motion and horseshoes chaos) are determined. Emphasis is laid on the e ects of time-delay between the detection of the structure's motion and the restoring action of the control device. Numerical simulation has been done to con rm and complement the analytical prediction.