The Construction of Differential Inclusions with Prescribed Attainable Sets

For given ε > 0 and a given continuous set-valued mapping t → Z ( t ), t ∈ [ t 0 , θ ], where Z ( t ) ⊂ ℝ n is a compact and convex set for every t ∈ [ t 0 , θ ], a differential inclusion is constructed such that the Hausdorff distance between the attainable set of the constructed differential inclusion at the instant of time t and Z ( t ) is less than ε for every t ∈ [ t 0 , θ ]. The right-hand side of the defined differential inclusion is affine with respect to the phase state vector and satisfies certain conditions which guarantee the existence and extendability of solutions. The solution of the problem is based on the existence of convex extensions of the affine-type convex compact set-valued mappings.