On the Stability and Stabilization Problems of Volterra Integro-Differential Equations

In this paper, the stability and stabilization problems for nonlinear Volterra integrodifferential equations with unlimited delay are considered. The development of the direct Lyapunov method in the study of the limiting properties of the solutions of these equations is carried out by using Lyapunov functionals with a semidefinite time derivative. The topological dynamics of these equations has been constructed revealing the limiting properties of their solutions. The assumption of the existence of a Lyapunov functional with a semidefinite time derivative gives a more complete solution to the positive limit set localization problem. On this basis new theorems on sufficient conditions for the asymptotic stability and instability of the zero solution of nonlinear Volterra integro-differential equations are proved. These theorems are applied to the problem of the equilibrium position stability of the hereditary mechanical systems as well as the regulation problem of the controlled mechanical systems using a proportional-integro-differential controller. As an example, the regulation problem of a mobile robot with three omnidirectional wheels and a displaced mass center is solved using the nonlinear integral controllers without velocity measurements.