Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems

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 is proposed in a matrix-free way, which circumvents the large-matrix-problem for multivariate Hermite interpolation. For finite time horizons, both convergence of the surrogate to the value function as well as 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 that has a scalable dimension and an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.