We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$ stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.

翻译：我们提出了一种基于最小二乘有限元方法的分布式最优控制问题数值逼近框架。该方法同时求解状态方程与伴随方程，且对任意相容离散空间的选择均满足$\inf$--$\sup$稳定性。针对控制变量受箱型约束的问题，我们推导了可靠且高效的先验误差估计子，该估计子具有可局部化特性，因此可用于指导自适应算法。对于无约束最优控制问题（即控制空间为希尔伯特空间的情形），我们获得了一种强制性的最小二乘方法，并特别证明了任意离散逼近空间下的拟最优性。针对约束问题，我们推导并分析了一个变分不等式，其中偏微分方程部分采用最小二乘有限元方法处理。我们证明该抽象框架可推广至多类问题，包括标量二阶偏微分方程、Stokes问题以及时空域上的抛物问题。最后给出了若干典型问题的数值算例。