Rates of Convergence for Discrete Approximations to Unconstrained Control Problems

Convergence rates for the error between the solution to a discrete approximation of a fixed time, unconstrained control problem and the corresponding continuous optimal control are derived for one-step and multistep integration schemes. The convergence rate for multistep schemes depends on the order of the integration scheme and the approximation properties of the discrete costate equation at the right endpoint. Furthermore, the order is 3 and the error in the optimal discrete control exhibits a boundary layer with most of the error concentrated at the right endpoint. For a class of one-step integration schemes satisfying a symmetry condition, second order convergence of the optimal discrete control is both proved and observed experimentally. The computations also indicate that the convergence rate of the optimal discrete state and costate variables equals the order of the integration scheme. By an auxiliary computation, this order can also be recovered for the control approximation. Some numerical examples indicate that the convergence estimates are tight. The question of the best integration scheme for achieving a given order of convergence is examined, and the modified Euler, the 3-point Adams–Moulton and a Runge–Kutta scheme appear to be optimal for orders 2, 3 and 4.