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

2021 
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 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), $R(0, \varepsilon ) = \widetilde R (0, \varepsilon).$
    • Correction
    • Source
    • Cite
    • Save
    • Machine Reading By IdeaReader
    21
    References
    0
    Citations
    NaN
    KQI
    []