On the structure of attainable sets for generalized differential equations and control systems

AbstractSection 1 deals with the study of properties of the set of solutions of (1) /.x ϵ R(t), x(0) = 0, where the set valued map R is measurable with nonempty compact subsets (of a ball of finite radius in En) as values. This is equivalent to the study of solutions of a linear control system. If Mr ⊂ (L∞[0, T] denotes the set of all measurable selections of R, and for r ϵ MR, (Ir)(t) = ∫0tr(τ)dτ, then I(MR)⊂C[0, T] is the space of all solutions of (1). One type of typical “cost functional” for an associated optimization problem is a continuous map F: C[0, T] → E1. An extension of Aumanns theorem is used, together with the Stone-Weierstrass theorem, to show that the set of F: C[0, T] → E1 such that F(I(MR) is compact is dense in the space of all continuous maps from C[0, T] → E1 with the uniform topology. The implications to optimal control problems are evident.Section 2 deals with the nonlinear problem (2) /.x ϵ R(x), x(0) = 0, where R has values as in Section 1. Using the machinery of Section 1, the existence of solutions of Eq. (2), when R is Lipschitzian, a result of Filippov 1966, is shown to be a trivial consequence of a fixed point theorem for contracting set valued mappings. If R is continuous and convex valued, the fact that the set of solutions in C[0, T] is compact and has compact fixed time cross section (Filippov-Roxin theorem) is also an immediate consequence of, now, the Bohnenblust-Karlin fixed point theorem for set valued maps. The remainder of Section 2 gives an example in which the set of points attainable by solutions of an equation of the form (2), at some time T > 0, is actually open! In fact, in this example R has the control representation R(x) = {ƒ (x, u) : u ϵ U} with U compact and ƒ smooth. To construct this, every point of the boundary of the attainable set of the convexified problem is attainable only as a limit of “chattering solutions” of the original system. This is quite difficult to accomplish (in fact many people conjectured it was not possible)