Euler Discretization and Inexact Restoration for Optimal Control∗

A computational technique for unconstrained optimal control problems is presented. First an Euler discretization is carried out to obtain a finite-dimensional approximation of the continous-time (infinite-dimensional) problem. Then an inexact restoration (IR) method due to Birgin and Mart́ınez is applied to the discretized problem to find an approximate solution. Convergence of the technique to a solution of the continuous-time problem is facilitated by the convergence of the IR method and the convergence of the discrete (approximate) solution as finer subdivisions are taken. It is shown that a special case of the IR method is equivalent to the projected Newton method for equality constrained quadratic optimization problems. The technique is numerically demonstrated by means of a scalar system and the van der Pol system, and comprehensive comparisons are made with the Newton and projected Newton methods.