Existence Theorems: The Use of Mere Pointwise Convergence on the Trajectories

Abstract

The existence theorems of this chapter are based on pointwise convergence of the trajectories and on the use of Helly’s selection theorem. To present these existence theorems we first state Helly’s theorem in Section 15.1, and then we state and prove some closure theorems based on mere pointwise convergence in Section 15.2 which are similar to those we proved in Section 8.6 for uniform convergence. Corresponding existence theorems for Lagrange problems of the Calculus of Variations and for problems of optimal control are stated and proved in Sections 15.3 and 15.4. Examples are given in Section 15.5.