Aggregation of a class of large-scale, interconnected, nonlinear dynamical systems
Given, in general, a countable collection of large-scale, interconnected nonlinear dynamical systems, the aim of this paper is the construction of a ``meaningful'' average in terms of a function \(w(t)\) whose evolution is proved to be described by a ``reduced order'' differential equation. In this respect, a statement of the problem along with auxiliary results are presented in Section 2, so that the main results are then shown in Section 3. More precisely, given the string of equations, \[ \dot X(i)= f\bigl( X(i), X(i-1),\dots, X(i-r+1), u(t)\bigr);\;X(j)(t)= X(1)(t),\;j\leq 1; \;i=1,2, \dots, \] a function \(w(t)\) is a ``meaningful'' average for the given collection of interconnected systems if and only if it satisfies the differential equation \[ \dot w=f(w,w, \dots,w,u) \] provided \(f\) satisfies certain appropriate conditions. Finally, some concluding remarks are also given in Section 4.
large-scale, interconnected nonlinear dynamical systems
scientific article; zbMATH DE number 1790999