Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context knowledge by its structure. Especially, the value function surrogate is elegantly enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm allows formulation in a matrix-free way which ensures efficient offline and online evaluation of the surrogate, circumventing the large-matrix problem for multivariate Hermite interpolation. Additionally, an incremental Cholesky factorization is utilized in the offline generation of the surrogate. For finite time horizons, both convergence of the surrogate to the value function and for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model with an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.

用于高维动力系统最优反馈控制的数值方法通常会受到维数灾难的影响。在当前的演示中，我们为最优控制问题的价值函数设计了一种基于无网格数据的近似方法，该方法部分缓解了维数问题。该方法基于贪婪 Hermite 核插值方案，并通过其结构合并上下文知识。特别是，价值函数代理在目标状态下被优雅地强制为 0，非负，并被构造为线性化模型的校正。该算法允许以无矩阵的方式进行制定，从而确保代理的高效离线和在线评估，从而避免了多元 Hermite 插值的大矩阵问题。此外，在代理的离线生成中使用了增量 Cholesky 分解。对于有限时间范围，替代值与价值函数的收敛性以及替代值与最优受控动态系统的收敛性都得到了证明。实验支持该方案的有效性，其中使用了具有明确给定价值函数的新学术模型。对于社区验证其他最优控制方法也可能有用。