Abstract
A complex system describing interaction of subsystems of the second order with delay in connections between them is studied. Necessary and sufficient conditions of the existence of a diagonal Lyapunov–Krasovskii functional for the considered system are derived. The obtained results are applied for the stability a nalysis of a mechanical system and a model of population dynamics. In addition, it is shown that they can be used in a problem of formation control.
Highlights
1 Introduction Diagonal Lyapunov functions is a powerful tool for the stability analysis of wide classes of systems [Kaszkurewicz and Bhaya, 1999]
It should be noted that diagonal Lyapunov functions are especially often used for the stability investigation of complex systems, neural networks and models of population dynamics [Hofbauer and Sigmund, 1998; Kaszkurewicz and Bhaya, 1999; Arcat and Sontag, 2006; Aleksandrov, Aleksandrova and Platonov, 2013; Talagaev, 2017; Alyshev, Dudarenko and Melnikov, 2018]
To illustrate the effectiveness of the proposed approaches, consider a group consisting of five agents with the integrator dynamics (17)
Summary
Diagonal Lyapunov functions is a powerful tool for the stability analysis of wide classes of systems [Kaszkurewicz and Bhaya, 1999]. In [Aleksandrov and Mason, 2016], a criterion of diagonal Riccati stability was obtained for linear time-invariant difference-differential systems of a general form (not necessary for positive ones). An interesting and important problem is that of finding classes of time-delay systems for which constructively verifiable conditions of diagonal Riccati stability can be derived. Some such classes both linear and nonlinear systems were determined in [Aleksandrov and Mason, 2016; Aleksandrov, Mason and Vorob’eva, 2017; Aleksandrov and Mason, 2018].
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