Relaxation Theory
David R. Spring · Birkhäuser Basel eBooks · 1998
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 A a.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. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.