Minisymposium Presentation

Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. We devise a mesh-free data-based approximation method for the value-function for high dimensional optimal control problems, which partially mitigates the dimensionality problem. The data comes from open-loop control systems, which are solved via the first-order necessary conditions of the problem, called the Pontryagin’s maximum principle. In this, the most informative initial states for the open-loop process are chosen using a greedy selection strategy. Furthermore, the approximation method is based on a greedy Hermite-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 avoids assembling a large system representing the interpolation conditions. For finite time horizons, convergence of the corresponding scheme can be proven for both the value-function and the surrogate as well as for the optimal vs. the surrogate controlled dynamical system. Experiments support the effectiveness of the scheme, using among others a new academic toy model with an explicit given value function, that may be useful for the community.