Themes on Differential Inclusions (

An ordinary differential equation x = f (t, x(t)) (ODE) uniquely assigns the time derivative x (t) = d dt x(t) as a function of t and x. A differential inclusion x ∈ F (t, x(t)), (DI) on the other hand, only requires that the derivative x be inside a given set F (t, x) ⊂ R n. Therefore, given an initial condition x(0) = ¯ x, one can usually find several solutions of (DI). f(x) x _ x _ x x F(x) R(t) x(t) Figure 1: A differential equation (above) and a differential inclusion (below). Here R(t) is the set reached by trajectories of the differential inclusion at time t. As a research field, differential inclusions may be regarded as a gymnasium, where new ideas and techniques can be introduced and developed in a rather elementary setting [?]. Some of these ideas have later found application to other branches of mathematics: control theory, Calculus of Variations, PDEs. A basic case is where the set-valued function F is upper semicontinuous (in practice, this simply means that its graph is closed), and each set F (t, x) is compact and convex. Under these assumptions, many results valid for ODEs with continuous right hand side can be extended to the multivalued case. A direct link between ODEs and differential inclusions is provided by Cellina's approximate selection theorem [7]: Given a multifunction F with closed graph and convex values, for any ε > 0 there exists a continuous function f ε whose graph is contained in an ε-neighborhood of the graph of F .