Relaxation Theory

Abstract

In this chapter we briefly examine the relationship between Convex Integration theory and the Relaxation Theorem, due to A.F. Filippov [13], in Optimal Control theory, and we prove a general C ^r-Relaxation Theorem 10.2. In broadest terms the underlying analytic approximation problem for both the Relaxation Theorem and for Convex Integration theory is the following. Let A ⊂ R ^q and let f: [0,1] → R ^q be a continuous vector valued function which is differentiale a.e. (almost everywhere), such that the derivative f’(t) ∈ Conv Aa.e., where Conv A denotes the convex hull of A in R ^q. Let also e > 0. The problem is to construct a continuous map g: [0,1] → R ^q, differentiable a.e. such that: (i) the derivative g’(t) ∈ A a.e.; (ii) for all t ∈ [0,1], ∥ f - g ∥ < e. Simply put, the problem is to C°-approximate the continuous map f: [0,1] → R ^q, whose derivatives lie in the convex hull of A a.e., by a continuous map g whose derivatives lie in the set A a.e.