Alexander Aleksandrov

In this paper, the stability problem of trivial equilibrium positions is addressed for two classes of nonlinear mechanical systems with unbounded delays. It should be noticed that the delaydependent terms in the equations considered can be interpreted as integral parts of proportionalintegral- differential (PID) controllers. It is known that such terms can improve the characteristics of transient processes and provide the damping of undesirable vibrations. First, assuming that strongly nonlinear dissipative and positional forces acting on the system are homogeneous of different homogeneity degrees, an original approach to the Lyapunov – Krasovskii functional construction is proposed. With the aid of this functional, it is proved that the asymptotic stability of the auxiliary delay-free system implies the asymptotic stability for the original time-delay system. Next, we study a mechanical system that is subject to linear gyroscopic forces in addition to nonlinear homogeneous dissipative and positional forces. To derive asymptotic stability conditions for such a system, a special technique to the application of the decomposition method is developed. The investigated system composed of the second-order equations is represented as a complex system describing interaction of two isolated subsystems consisting of the first-order equations. This form of the decomposition is an extension of the classical one for delay-free linear gyroscopic systems. However, in the linear case, the asymptotic stability was guaranteed only under an additional restriction on the system. It was assumed that there is a large positive parameter at the vector of the gyroscopic forces. In the present contribution, it is proved that, for systems with strongly nonlinear dissipative and positional forces, such a constraint is not required. The effectiveness of the obtained results is demonstrated on the problem of nonlinear PID controller design providing the triaxial stabilization of a rigid body.