On construction of common Lyapunov function for a family of mechanical systems with one degree of freedom

Certain classes of families of nonlinear mechanical systems with one degree of freedom, which is described by second order differential equation are studied. There are two parameters, damping and rigidity coefficients, in these equqtions and we assume that switching can take place in these coefficients. The problem of stability and dissipativity of corresponding hybrid system, which contains a considered family of systems and a switching law, defining which system is active in every moment is investigated. Conditions of the existence of CLFs of the a given form are obtained using second Lyapunov method. Fulfilment of these conditions provides asymptotic stability of equilibrium positions of corresponding switched systems for any switching law. It is proved that for considered families of essentially nonlinear systems we can guarantee the existence of CLFs under weaker assumptions than for linear ones. Thus, in comparison with linear systems, nonlinear ones are “more stable” with respect to switching of parameters values. Theorems 1 and 2 can be used for the design of stabilizing controls for mechanical systems. Challenging direction for further research is extension of the obtained results to the switched nonlinear mechanical systems with several degrees of freedom. Bibliogr. 16.