We consider non-autonomous calculus of variations problems with a state constraint represented by a given closed set. We prove that if the interior of the Clarke tangent cone of the state constraint set is non-empty (this is the constraint qualification that we suggest here), then the necessary optimality conditions apply in the normal form. We establish normality results for (weak) local minimizers and global minimizers, employing two different approaches and invoking slightly diverse assumptions. More precisely, for the local minimizers result, the Lagrangian is supposed to be Lipschitz with respect to the state variable, and just lower semicontinuous in its third variable. On the other hand, the approach for the global minimizers result (which is simpler) requires the Lagrangian to be convex with respect to its third variable, but the Lipschitz constant of the Lagrangian with respect to the state variable might now depend on time.

中文翻译：

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具有状态约束的变分问题微积分的必要最优条件的正态性

我们考虑具有给定闭合集表示的状态约束的变异问题的非自治演算。我们证明，如果状态约束集的Clarke切线圆锥的内部是非空的（这是我们在此处建议的约束条件），则必要的最优性条件将以正常形式适用。我们采用两种不同的方法并采用略有不同的假设，为（弱）局部极小值和全局极小值建立正态结果。更准确地说，对于局部极小值结果，就状态变量而言，拉格朗日假定为Lipschitz，而在其第三变量中，其准半连续性较低。另一方面，用于全局最小化器结果的方法（更简单）要求Lagrangian关于其第三个变量是凸的，