Differential Inclusions

Differential inclusions are generalizations of first order, vector differential equations, in which set inclusion replaces equality and the right side is no longer a point valued, but a set valued function (‘multifunction’) of the current time and state. Differential inclusions feature prominently in dynamic optimization. They are used to formulate a dynamic constraint, thereby serving as an alternative to the controlled differential equation framework of the early literature. Even when the original dynamic constraint is of a different nature, it is often helpful (regarding the formulation of hypotheses for existence of minimizers, or under which nonsmooth optimality conditions can be derived) to consider the associated differential inclusion. Many properties of first order differential equations have analogues in the theory of differential inclusions. These include existence of global solutions with a specified initial value, when a Lipschitz continuity hypothesis is imposed on the differential inclusion (strictly speaking, on the associated multifunction), and sensitivity properties of solutions under data perturbations. Differential inclusions fail to have unique solutions even in the simplest cases, however. It is therefore of interest to consider the whole set of solutions for a given initial state and ask: is this set non-empty, is it compact and, if so, w.r.t. what topology, or is it stable under perturbations of the differential inclusion? This chapter provides answers to questions of this nature. The starting point is the generalized Filippov existence theorem, which asserts existence of solutions to Lipschitz continuous differential inclusions. But it supplies additional useful information, in the form of estimates of the distance of a solution from a given arc that is an approximate solution of the differential inclusion. This is followed by the compactness of trajectories theorem, which concerns properties of sequences of arcs with uniformly integrably bounded velocities, approximately satisfying the given differential inclusion with an increasing degree of accuracy. We find that accumulation points of such sequences (in the uniform topology) satisfy, not the nominal differential inclusion but its convexification, in which the values of the associated multifunction are replaced by their convex hulls. We deduce as a straightforward consequence that, under a Lipschitz continuity hypothesis, the set of solutions to a given differential inclusion is closed, provided the differential inclusion is convex (that is, has values convex sets). If the differential inclusion fails to be convex, it can be shown that, nonetheless, an arbitrary solution to the convexified differential inclusion can be approximated uniformly closely (in the uniform norm) by solutions to the original, unconvexified, differential inclusion; this is the relaxation theorem. In the final section of the chapter, we introduce a state constraint. Conditions are given for existence of solutions that satisfy the state constraint. We also provide estimates of the distance of this solution from a nominal solution that violates the state constraint. Distance estimates of this nature are used extensively in the theory of state constrained dynamic optimization, to establish non-degeneracy of optimality conditions and stability of the infimum cost under data perturbations.