Averaging of differential inclusions with unbounded right-hand sides
The author studies the differential inclusion problem (1): \(\dot x\in \epsilon X(t,x)\), \(x(0)=x_ 0\) where \(x\in {\mathbb{R}}^ n\), \(\epsilon >0\) is a small parameter and X(t,x) is a \(2\pi\)-periodic multivalued function relative to \(t\in {\mathbb{R}}\), with values in the set of all nonempty closed subsets of \({\mathbb{R}}^ n\). It is assumed that \(0\in X(t,x)\) for every t and x. With the above problem (1) he associates the problem (2): \(^{\dot-}\in \bar X(\bar x)\), \(\bar x(0)=x_ 0\) where \(\bar X(x)\) is obtained by the averaging process \(\bar X(x)=1/2\pi \int^{2\pi}_{0}X(t,x)dt.\) Here the integral is to be understood in the following sense: If F(t) is a multivalued function defined on some interval [a,b] then the integral \(\int^{b}_{a}F(t)dt\) is the closure in \({\mathbb{R}}^ n\) of the set \(\{y: y=\int^{b}_{a}f(t)dt\), \(f(t)\in F(t)\}\). He shows that under certain conditions, which are too involved to be reproduced here the problem (2) approximates the problem (1) in a certain sense.
scientific article; zbMATH DE number 4132582