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 ẋ ∈ εF (t, x), t 6= ti, x(0) ∈ X0, ∆x |t=ti∈ εIi(x), (1) where t ∈ R+ is time, x ∈ R is a phase variable, ε > 0 is a small parameter, F : R+×R → E, Ii : R → E are fuzzy mappings, moments ti are enumerated in the increasing order. Associate with inclusion (1) the following partial averaged differential inclusion ξ̇ ∈ εF̃ (t, ξ), t 6= sj , ξ(0) ∈ X0, ∆ξ|t=sj ∈ εKj(ξ), (2) where the fuzzy mappings F̃ : R+ × R → E; Kj : R→ E satisfy the condition

We develop the ideas of the method of averaging for some classes of fuzzy systems (fuzzy differential equations with delay, fuzzy differerntial equations with pulsed action, fuzzy integral equations, fuzzy differential inclusions, and differential inclusions with fuzzy right-hand sides without and with pulsed action).

We present a survey of the development of ideas of the averaging method for some classes of set-valued impulsive systems (impulsive differential inclusions, impulsive differential equations, and inclusions with Hukuhara derivative; fuzzy impulsive differential equations and inclusions).

We establish theorems on the existence and uniqueness of a solution of the impulsive differential-algebraic equation d/dt[Au(t)] + Bu(t) = f(t,u(t)), where the matrix A may be singular. The results are applied to the theory of electric circuits.

We consider an autonomous evolution inclusion with pulse perturbations at fixed moments of time. Under the conditions of global solvability, we prove the existence of a minimal compact set in the phase space that attracts all trajectories.

We establish conditions for the existence of periodic solutions for systems of differential equations with random right-hand side and random pulse influence at fixed times. We consider the case of small pulse perturbation and weakly nonlinear systems.

We establish sufficient conditions for the stability, asymptotic stability, and instability of invariant sets of discontinuous dynamical systems.

We construct optimal strategies for players and determine the sets of initial positions favorable for one player or another.

We consider the problem of asymptotic stability of the trivial invariant torus of one class of impulsive systems. Sufficient criteria of asymptotic stability are obtained by the method of freezing in one case, and by the direct Lyapunov method for the investigation of stability of solutions of impulsive systems in another case.

We consider the problem of asymptotic stability of the trivial invariant torus of one class of impulsive systems. Sufficient criteria of asymptotic stability are obtained by the method of freezing in one case, and by the direct Lyapunov method for the investigation of stability of solutions of impulsive systems in another case.

We establish conditions of the existence of solutions periodic in t with period T for a weakly nonlinear system of partial differential equations with pulse influence.