The augmented Lagrange method is employed to address the optimal control problem involving pointwise state constraints in parabolic equations. The strong convergence of the primal variables and the weak convergence of the dual variables are rigorously established. The sub-problems arising in the algorithm are solved using the Method of Successive Approximations (MSA), derived from Pontryagin's principle. Numerical experiments are provided to validate the convergence of the proposed algorithm.

翻译：本文采用增广拉格朗日方法处理抛物方程中涉及逐点状态约束的最优控制问题。严格证明了原变量的强收敛性以及对偶变量的弱收敛性。算法中产生的子问题通过源自庞特里亚金原理的逐次逼近法求解。数值实验验证了所提算法的收敛性。