On the Convergence of Conjugate Gradient and GMRES Algorithms in the Forward Backward Sweep Method for Optimal Control

ABSTRACT Optimal control problems aim at finding the control function that minimizes a given functional, subject to an initial value problem. The necessary optimality conditions of such problem form a constrained two‐point boundary problem, with state and adjoint ordinary differential equations (ODEs), which can be numerically solved resorting to well‐known time discretization techniques. However, the size of the resulting coupled system of equations depends on the time‐step length, which motivates the use of iterative solution strategies, such as the forward backward sweep method (FBSM). This methodology solves the state and adjoint ODEs forward and backward in time, respectively, while the control is iteratively updated. In this study, we first analyze this strategy when the control variable is updated with the conjugate gradient method. We show that the convergence of this update can be guaranteed only for some time‐discretizations, and that for these choices, the convergence rate depends on the parameters of the functional. Based on these results, we also suggest an alternative update based on adapting the generalized minimal residual method (GMRES), which can be applied to a much wider set of time‐integration combinations. Our analysis shows that the convergence rate of the iterative process is closely related to the time‐discretization employed due to the underlying symmetry of the system of equations. We illustrate our results with some linear and non‐linear problems, which also demonstrate the better convergence properties of the proposed GMRES strategy.