478 CHAPTER 12 12.2 MAIN RESULTS: NECESSARY CONDITIONS OF EULER, LAGRANGE, PONTRYAGIN, AND BELLMAN 12.2.1 Calculus of variations: Euler equation and transversality conditions Let L : R 3 R (L = L(t, x, x)) be a continuous function of three variables. The problem t 1 J(x(·)) = t 0 L(t, x(t), x(t))dt min, x(t i ) = x i , i = 0, 1 (P 12.1 ) is called the simplest problem of the calculus of variations. The problem (P 12.1 ) will be considered in the space C 1 ([t 0 , t 1 ]), equipped with the C 1 -norm · C 1 ([t 0 ,t 1 ]) , defined by x(·) C 1 ([t 0 ,t 1 ]) = max max(|x(t)|, | x(t)|). t[t 0 ,t 1 ] Definition 12.1 A local minimum x(·) of the problem (P 12.1 ) in the space C 1 ([t 0 , t 1 ]) is called a weak minimum of (P 12.1 ). This means that 1