Robust Optimal Control using Polynomial Chaos and Adjoints for Systems with Uncertain Inputs

The objective of this note is to show how one can combine Polynomial Chaos Expansions (PCE) and adjoint theory to efficiently obtain sensitivities for robust optimal control. A non-intrusive PCE method is used to analyze the constraint equations for the state (which depends on uncertain inputs), namely the governing equations of the dynamical system. Adjoint solutions are constructed for each of the polynomial basis functions used in the approximate expansion. The combination of the gradient for each basis-adjoint pair is used to form the overall gradient. The resulting gradient can be used to improve an initial guess in an iterative optimization procedure. The repeated use of the non-intrusive PCE method, the adjoint solver and the gradient estimate can be used to determine optimal control laws for the governing system in the presence of uncertainties. The formulation of the optimal control problem is presented in the context of the flow equations where the expected value of a functional is to be minimized. The boundary shape is the control. The associated cost of this approach in an optimization setting is equal to the cost of a PCE analysis (≈ Q deterministic simulations) plus Q (number of unknowns in the PCE expansion) adjoint solves for each iteration of a steepest-descent algorithm. However, this cost can be further reduced for certain objective functions using an intrusive formulation for the adjoint equations.