Function Space Approach for Gradient Descent in Optimal Control

We develop an iterative numerical method for open-loop optimal control problems based on constrained-gradient descent in function space. The descent algorithm uses a projection of the gradient onto the tangent space of the manifold representing the constraints of the dynamical equation. A key feature of our algorithm is preconditioning of the quadratic portion of the cost functional with a relatively simple procedure. This preconditioning has a simple geometric interpretation, which gives intuitive reasoning for the rapid convergence of the iterations in neighborhoods of local optima. We give two illustrative examples including a continuous stirred-tank chemical reactor, and a bilinear quantum system. The relation of this algorithm to earlier proposed Banach space projection algorithms is outlined.