Viability and generalized differential
نویسندگان
چکیده
Necessary and sufficient conditions for a set-valued map K : R ։ R to be GDQ-differentiable are given. It is shown that K is GDQ differentiable at t0 if and only if it has a local multiselection that is Cellina continuously approximable and Lipschitz at t0. It is also shown that any minimal GDQ of K at (t0, y0) is a subset of the contingent derivative of K at (t0, y0), evaluated at 1. Then this fact is used to prove a viability theorem that asserts existence of a solution to the initial value problem ẏ(t) ∈ F (t, y(t)), with y(t0) = y0, where F : Gr(K) ։ R n is an orientor field (i.e. multivalued vector field) defined only on GrK and K : T ։ R is a time-varying constraint multifunction. One of the assumptions is GDQ differentiability of K.
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Let K be a multivalued map from the interval [0, 1] to R. On its graph, denoted by Gr(K), another multivalued map F is defined. Its values are closed subsets of R. The map F , called an orientor field, gives rise to a differential inclusion ẋ(t) ∈ F (t, x(t)). The inclusion is called viable, if for every (t0, x0) ∈ Gr(K) there is a global absolutely continuous forward trajectory x : [t0, 1] → R...
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