Controlled differential equations with a parameter and with multivalued impulses

We study the Cauchy problem for a controlled differential system with a parameter which is an element of some metric space Ξ , containing phase constraints on the control. It is assumed that at the given time instants t k , k = 1, 2,…, p , the solution x is continuous from the left and suffers a discontinuity, the value of which is x t k +0 -x t k , belongs to some non-empty compact set of the space Rn . The notions of an admissible pair of this controlled impulsive system are introduced. The questions of continuity of admissible pairs are considered. Definitions of a priori boundedness and a priori collective boundedness on a given set S ×K (where S ⊂ Rn is a set of initial values, K ⊂ Ξ is a set of parameter values) of the set of phase trajectories are considered. It is proved that if at some point x 0 ,ξ ∈ Rn × Ξ the set of phase trajectories is a priori bounded, then it will be a priori bounded in some neighborhood of this point.

controlled differential system, Cauchy problem, multivalued impulses, differential inclusion, a priori boundedness