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

Abstract

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.

On uniform asymptotic stability for nonlinear integro-differential equations of Volterra type

“…Such a simplification is possible, including due to the structure of the right side of the system. A significant advantage is the fact that the structure of the functional space used for constructing limiting functionals is simplified compared to the known constructions for the infinite delay case [Andreev and Peregudova, 2018a;Andreev and Peregudova, 2018b].…”

“…The use of Lyapunov-Krasovskii functionals was considered in [Andreev and Peregudova, 2018a;Andreev and Peregudova, 2018b;Burton, 1982;Burton, 1980] and others. In this case, the use of the phase space BC turns out to be natural and convenient from the point of view of presentation and verification of stability conditions.…”

“…Assumption 3 is a generalization of the assumptions from [Hino and Murakami, 1991] where precompact conditions for the linear Volterra IDE with an almost periodic right-hand side are considered. The conditions of Assumption 3 also differ from those from the papers [Andreev and Peregudova, 2018a;Andreev and Peregudova, 2018b;Andreev and Peregudova, 2017] offering precompact results for a non-periodic right-hand side.…”

“…In particular, the method of Lyapunov-Krasovskii functionals is used for stability analysis. In recent works using this method, it has been possible to increase the efficiency of this method by using the technique of limiting equations [Andreev and Peregudova, 2018a;Andreev and Peregudova, 2018b]. A similar approach is applicable to the other branch of the direct method -the Razumikhin technique [Haddock and Terjéki, 1990;Hara et al, 1992;Seifert, 1974;Seifert, 1973].…”

Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method

“…In the relevant literature three methods, which are called the second Lyapunov method, Lyapunov-Krasovskiȋ method and Lyapunov-Razumikhin method, come to the forefront to investigate qualitative properties of solutions of linear and nonlinear integro-differential equations both without and with retardation. Among these methods, the second Lyapunov method and Lyapunov-Krasovskiȋ method are extensively used to study various qualitative behaviors of solutions of integro-differential equations of integer order (see, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]). To the best of our knowledge, the Lyapunov-Razumikhin method is less used during that kind of investigation [23,25,26].…”