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

We extend Legendre condition to a variational problem with inequality state constraints. Since our Legendre-type conditions do not include (x) over do t, they differ from the Legendre-Clebsch condition. They give information a bout the Hesse matrix of the integrand at not only inactive points but also active points. On the other hand, since the inequality state constraints c an be regarded as an infinite number of inequality constraints, they someti mes form an envelope. According to a general theory [9], one has to take th e envelope into consideration when one consider second-order necessary opti mality conditions for an abstract optimization problem with a generalized i nequality constraint, However, we show that we do not need to take it into account when we consider Legendre-type conditions. Finally, we show that an y inequality state constraint forms envelopes except two trivial cases. We prove it by presenting an envelope in a visible form.