Reformulations of Measure Differential Inclusions and Their Closed Graph Property

New formulations are given for measure differential inclusions, “dx/dt K(t) Rn,” where x(•) is a function of bounded variation and K is a set-valued map with closed convex values and has closed graph. Measure differential inclusions were first named by J. J. Moreau for studying rigid body with impacts, shocks and Coulomb friction and assumed that K(t) is always a cone. The new formulations are used to show that the graph of the solution operator is closed under point- wise convergence of the solutions x(•) and weak* convergence of the differential measures dx, provided that the minimum norm points of K(t) are bounded and the asymptotic cones K(t)∞ are always pointed.