Convergence of the costates does not imply convergence of the control

Abstract

Solving an optimal control problem using a digital computer implies discrete approximations. Since the 1960s, there have been well-documented [1–3] naïve applications of Pontryagin’s principle in the discrete domain. Although its incorrect applications continue to this day, the origin of the naïvete is quite understandable because one has a reasonable expectation of the validity of Pontryagin’s principle within a discrete domain. That an application of the Hamiltonian minimization condition is not necessarily valid in a discrete domain [1,4] opens up a vast array of questions in theory and computation [2,5]. These questions continue to dominate the meaning and validity of discrete approximations and computational solutions to optimal control problems [6–10]. Among these questions is the convergence of discrete approximations in optimal control.