This article is cited in
8 papers
Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side
E. V. Sokolovskaya,
O. P. Filatov Samara State University
Abstract:
Suppose that
$\mathbb R^n$ is the
$p$-dimensional space with Euclidean norm
${\|\cdot\|}$,
$K(\mathbb R^p)$ is the set of nonempty compact sets in
$\mathbb R^p$,
$\mathbb R_+=0,+\infty)$, $D=\mathbb R_+\times\mathbb R^m\times\mathbb R^n\times[0,a]$,
$D_0=\mathbb R_+\times\mathbb R^m$,
$F_0\colon D_0\to K(\mathbb R^m)$, and
$\operatorname{co}F_0$ is the convex cover of the mapping
$F_0$. We consider the Cauchy problem for the system of differential inclusions
$$
\dot x\in\mu F(t,x,y,\mu),\quad \dot y\in G(t,x,y,\mu),\qquad x(0)=x_0,\quad y(0)=y_0
$$
with slow
$x$ and fast
$y$ variables; here
$F\colon D\to K(\mathbb R^m)$,
$G\colon D\to K(\mathbb R^n)$, and
$\mu\in[0,a]$ is a small parameter. It is assumed that this problem has at least one solution on
$[0,1/\mu]$ for all sufficiently small
$\mu\in[0,a]$. Under certain conditions on
$F$,
$G$, and
$F_0$, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any
$\varepsilon>0$, there is a
$\mu_0>0$ such that for any
$\mu\in(0,\mu_0]$ and any solution
$(x_\mu(t),y_\mu(t))$ of the problem under consideration, there exists a solution
$u_\mu(t)$ of the problem
$\dot u\in\mu\operatorname{co}F_0(t,u)$,
$u(0)=x_0$ for which the inequality
$\|x_\mu(t)-u_\mu(t)\|<\varepsilon$ holds for each
$t\in[0,1/\mu]$.
UDC:
517.928 Received: 04.06.2004
DOI:
10.4213/mzm2632
Fulltext:
PDF file (222 kB)
