Singular Optimal Control Computation

Abstract

A quadratic functional version of the linear quadratic optimal control problem is employed to develop insights into the computation and theory of singular optimal controls. It is shown that sufficiency, existence, and uniqueness conditions and convergence conditions for gradient-type algorithms require the same basic assumption, namely a positivity assumption on the second-variation operator and an inclusion requirement with regard to the range space of the second variation operator. The theory is interpreted for both the nonsingular and singular problems to show the inherent differences between the two problems. It is shown that the gradient, conjugate gradient, and Davidon-Fletcher-Powell (DFP) algorithms converge if the existence conditions for the optimal control are satisfied, and that the rate of convergence is superlinear for the DFP method applied to nonsingular problems. Operational aspects for improving the rate of convergence on singular problems are discussed along with an informative comparison of the behavior of gradient and accelerated gradient methods on singular problems.