Numerical methods for optimal control problems with ODE or integral equations

Abstract

An overview of numerical methods for solving optimal control problems described by ODE and integral equations is presented. We consider direct and indirect methods. The finer indirect methods use necessary optimality conditions. Direct methods transform the control problem after discretization to an optimization problem. The nonlinear optimization problem can be solved by means of SQP-methods or gradient methods. Known variants of this method are GESOP and DIRCOL. Then a wave-method is mentioned, in which the state variables are varied at first. The direct methods apply the maximum principle, often it is possible to eliminate the control with the help of the necessary condition. The control problem is transformed to a boundary value problem for the state and the adjoint variable, which is solved by multiple shooting. The iterative procedures of Krylow/Chernousko and Sakawa, respectively, are based on the maximum principle, too. It is referred to the gradient methods described in the monograph of Pytlak and to prox-methods. © Springer-Verlag Berlin Heidelberg 2006.