Averaging method for impulsive differential inclusions with fuzzy right-hand side

Abstract

In this paper the substantiation of the partial scheme of the averaging method for impulsive differential inclusions with fuzzy right-hand side in terms of R - solutions on the finite interval is considered.
Consider the impulsive differential inclusion with the fuzzy right-hand side $$\dot x \in \varepsilon F(t,x) ,\ t \not= t_i,\ x(0)\in X_0,\quad \Delta x \mid _{t=t_i} \in \varepsilon I_i (x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1)$$  where $t\in \mathbb{R}_+$ is time, $x \in \mathbb{R}^n$ is a phase variable, $\varepsilon > 0$ is a small parameter,
$F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n,$ $I_i \colon \mathbb{R}^n \to \mathbb{E}^n$ are fuzzy mappings, moments $t_i$ are enumerated in the increasing order.
Associate with inclusion (1) the following partial averaged differential inclusion $$\dot\xi \in \varepsilon \widetilde F (t, \xi ),\ t \not= s_j ,\ \xi (0) \in X_0,\quad \Delta \xi \vert _{t=s_j} \in \varepsilon K_j (\xi ),\qquad\qquad\qquad\qquad\qquad\qquad\quad (2),$$ where the fuzzy mappings $\widetilde F \colon \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{E}^n ; \quad K_j \colon \mathbb{R} \to \mathbb{E}^n$ satisfy the condition $$\lim _{T \to \infty } \frac 1T D \Big( \int\limits_t^{t+T} F(t,x) dt + \sum_{t \leq t_i < t+T} I_i (x), \int\limits_t^{t+T} \widetilde F(t,x)dt + \sum_{t \leq s_j < t+T} K_j (x) \Big) = 0,\quad\quad (3)$$ moments $s_j$ are enumerated in the increasing order. In the paper is proved the following main theorem:
{\sl Let in the domain $Q = \lbrace t \geq 0 , x \in G \subset \mathbb{R}^n \rbrace$ the following conditions fulfill:
$1)$ fuzzy mappings $F (t,x), \widetilde F(t,x), I_i(x),K_j(x)$
are continuous, uniformly bounded with constant $M$, concave in $x,$ satisfy Lipschitz condition in $x$ with constant $\lambda ;$
$2)$ uniformly with respect to $t, x$ limit (3) exists and $\frac 1T i(t,t+T) \leq d < \infty ,\ \frac 1T j(t,t+T) \leq d < \infty,$
where $i(t,t+T)$ and $j(t,t+T)$ are the quantities of impulse moments $t_i$ and $s_j$ on the interval
$[ t, t+T ]$;
$3)$ {\rm R}-solutions of inclusion (2) for all $X_0 \subset G^{\prime} \subset G$
for $t \in [0,L^{\ast} \varepsilon ^{-1} ]$ belong to the domain $G$ with a $\rho$- neighborhood.
Then for any $\eta > 0$ and $L \in (0,L^{\ast}]$ there exists $\varepsilon _0 (\eta,L) \in (0,\sigma ]$ such that for all $\varepsilon \in (0, \varepsilon _0 ]$ and $t \in [0,L \varepsilon ^{-1}]$ the inequality holds:
$D(R(t, \varepsilon ), \widetilde R (t, \varepsilon)) < \eta,$ where $R(t, \varepsilon), \widetilde R(t, \varepsilon )$ are the {\rm R-} solutions of inclusions (1) and (2), $R(0, \varepsilon ) = \widetilde R (0, \varepsilon).$