STABILITY OF DIFFERENTIAL-INCLUSIONS WITH MULTIVALUED PERTURBATIONSстатья
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Дата последнего поиска статьи во внешних источниках: 7 декабря 2013 г.
Аннотация: We consider differential inclusions of the form x ˙∈F(t,x)+μG(t,x), where x ˙=dx/dt is the velocity vector from ℝ n , μ is a small parameter, 0≤μ≤e, e>0, F,G are multivalued mappings of the closed domain ����⊂[0,+∞)×ℝ n into the collection Kv(ℝ n ) of compact convex sets of the space ℝ n with the Euclidean norm |·|. Further, ����=ℝ + ×S(ϵ 0 ), ℝ + =[0,+∞), S(ϵ 0 ) is a closed ball from ℝ n of radius ϵ 0 , <·,·> is the inner product in ℝ n ; ρ (a,A) is the distance from the point a to the set A, β (A,B) is the semideviation of the set A from the set B, α (·,·) is the Hausdorff metric in the space of nonempty compact sets ����(ℝ n ) from ℝ n , X(t 0 ,x 0 ,μ) is the collection of all solutions of the inclusion (1) with initial conditions (t 0 ,x 0 )∈����. We assume that the generating inclusion (2) y ˙∈F(t,y) has the trivial solution y≡0, Lyapunov stable, in general, not asymptotically. Naturally, small perturbations μ G may disturb the stability of the differential inclusion (2). In this paper, we obtain sufficient conditions on the multivalued mapping G≠{0}, which ensure the property of uniform μ-stability for the inclusion (1), i.e. (∀ϵ>0)(∃μ 0 >0)(∃δ>0)(∀t 0 ∈ℝ + )(∀x 0 ∈S(δ)), (∀μ∈[0,μ 0 ])(∀x(·)∈X(t 0 ,x 0 ,μ)(∀t≥t 0 )(|x(t)|<ϵ).