We consider parametrized linear-quadratic optimal control problems and provide their online-efficient solutions by combining greedy reduced basis methods and machine learning algorithms. To this end, we first extend the greedy control algorithm, which builds a reduced basis for the manifold of optimal final time adjoint states, to the setting where the objective functional consists of a penalty term measuring the deviation from a desired state and a term describing the control energy. Afterwards, we apply machine learning surrogates to accelerate the online evaluation of the reduced model. The error estimates proven for the greedy procedure are further transferred to the machine learning models and thus allow for efficient a posteriori error certification. We discuss the computational costs of all considered methods in detail and show by means of two numerical examples the tremendous potential of the proposed methodology.

翻译：我们考虑参数化线性二次型最优控制问题，通过融合贪婪降阶基方法与机器学习算法，提供其在线高效解。为此，首先将贪婪控制算法（该算法为最优终端时刻伴随状态流形构建降阶基）拓展至目标泛函包含惩罚偏离期望状态项与控制能量项的情形。随后应用机器学习代理模型加速降阶模型的在线评估。贪婪过程所证明的误差估计进一步迁移至机器学习模型，从而实现高效的后验误差认证。我们详细讨论了所有考虑方法的计算成本，并通过两个数值示例展示了所提方法的巨大潜力。