SUMMARY

The focus of this work is on the problem of control selection under uncertainty. In many engineering applications, the control selection that minimizes a cost, or maximizes a score, at a future time or event is sought. However, under uncertainty, the optimization procedure must be performed probabilistically using an expected value. This work introduces an optimal control algorithm in which the Koopman operator is used to solve for the probabilistically optimal input in the presence of initial condition and/or parametric uncertainty. The proposed approach offers unique computational advantages over alternative uncertainty quantification techniques, such as Monte Carlo methods, providing a practical method to compute a probabilistically optimal input. Leveraging kernel-based interpolation methods, the expected value computation may be carried out on large, high-dimensional scattered datasets. In the context of the airdrop problem, these inputs are the optimal package release point, aircraft run-in, and transition altitude. Given an objective function defined over the drop zone and a joint probability density accounting for uncertainty in the system parameters and initial state, the objective function is pulled back to the drop altitude using the Koopman operator, and an expected value is computed with the joint probability density. Simulation examples are presented highlighting performance of the algorithm in real-world scenarios. Results compare favorably with those achieved through deterministic methods.