Additional exercises for Part III

Abstract

The chapter consists of several dozen exercises in the calculus of variations, the first of which is the following: Consider the problem

$$\min\;\;\; \int_{ 1}^{\,3} \big\{ \, t \big( x\,' (t)\big)^{2}-x(t)\big\} \, dt\: :\: x\in\, C^{\,2}[\,1,3\,] ,\;\; x(1)\,=\, 0 ,\; \: x(3)\,=-1. $$

• (a) Find the unique admissible extremal x _∗.

• (b) Prove that x _∗ is a global minimizer for the problem.

• (c) Prove that the problem

  $$\min\;\; J(x)\:=\: \int_{-2}^{\,3} \big\{ \, t \big( x\,' (t)\big)^{2}-x(t)\big\} \, dt\: :\: x\in\, C^{\,2}[-2\,,3\,] ,\; x(-2)\,=\, A\,,\; \: x(3)\,=B $$

  admits no local minimizer, regardless of the values of A and B.