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 $$\\dot x \\in \\varepsilon F(t,x) ,\\ t \\not= t_i,\\ x(0)\\in X_0,\\quad\\Delta x \\mid _{t=t_i} \\in \\varepsilon I_i (x),\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad (1)$$\xa0where $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), $R(0, \\varepsilon ) = \\widetilde R (0, \\varepsilon).$