Averaging of multivalued differential equations
Here, the following nonlinear perturbed differential inclusion is investigated \[ z'(t)\in\varepsilon F(z(t),y(t)),\quad y'(t)\in G(y(t)), \] where \(\varepsilon >0\) is the perturbation parameter, \(t\in [0,T/\varepsilon]\), \(z\) is the slow motion and \(y\) the fast motion. Additionally, \(F\) and \(G\) are assumed to be Lipschitz continuous in Hausdorff sense, with compact, convex and nonempty values. The author shows that, under a transitivity condition on the fast subsystem, there exists an averaged system whose trajectories approximate, uniformly in Hausdorff sense, the slow trajectories of the original one. Moreover, the approximation is of order \(O(\varepsilon^{\frac{1}{3}})\), as \(\varepsilon\rightarrow 0\).
scientific article; zbMATH DE number 1930050