Legendre-type optimality conditions for a variational problem with inequality state constraints (Q1572687)

In optimization problems with infinitely many inequality constraints, the second-order necessary conditions usually contain an extra term which is due to the envelope effect discovered by the second author in 1988 [\textit{H. Kawasaki}, Math. Program., Ser. A 41, No. 1, 73--96 (1988; Zbl 0661.49012)]. In the theory of calculus of variations, the so-called state inequalities form an infinite system of inequalities. The main result of the paper states that, for such problems, the envelope effect does not appear in the Legendre condition, which is a second-order necessary condition for problems of the calculus of variations.