Asymptotic methods in the theory of differential equations with discontinuous and multi-valued right-hand sides (Q676815)

The averaging Bogolyubov theorem is generalized for differential inclusions. It is shown that for every solution of the averaged differential inclusion \(\xi'\in\varepsilon \overline{X}(\xi)\), \[ \overline{X}(x)= 1/T\lim_{T\to\infty} \int_0^T X(t,x)dt \] there exists a solution of the original differential inclusion \(x'\in X(t,x)\) such that the difference between these two solutions is small on the time scale \(1/\varepsilon\). This implies that the attainable set \(R(t,\varepsilon)\) of the original differential inclusion is upper semicontinuous with respect to \(\varepsilon\) at \(\varepsilon=0\), if \(R(t,0)\) denotes the attainable set of the averaged differential inclusion. If, for example, \(F(t,x)\) is piecewise continuous, then it is proved that the map \(R(t,\varepsilon)\) is continuous at \(\varepsilon=0\). If the differential inclusion \(x'\in F(t,x)\) is obtained as a Filippov regularization of a discontinuous differential equation, then solutions may move along the discontinuity manifold. An averaging principle for such sliding regime is also discussed in the present paper.