Numerical Solution of Optimal Control Problems by Direct Collocation 1 Statement of Problems

By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a nite dimensional nonlinear program which can be solved by standard SQP-methods 10]. Convergence properties of the discretization are derived. From a solution of this method known as direct collocation, these properties are used to obtain reliable estimates of adjoint variables. In the presence of active state constraints, these estimates can be signiicantly improved by including the switching structure of the state constraint into the optimization procedure. Two numerical examples are presented. Systems governed by ordinary diierential equations arise in many applications as, e. g., in astronautics, aeronautics, robotics, and economics. The task of optimizing these systems leads to the optimal control problems investigated in this paper. The aim is to nd a control vector u(t) and the nal time t f that minimize the functional Ju; t f ] = (x(t f); t f) (1) subject to a system of n nonlinear diierential equations _ x i (t) = f i (4) Here, the l vector of control variables is denoted by u(t) = (u 1 (t); : : : ; u l (t)) T and the n vector of state variables is denoted by x(t) = (x 1 (t); : : : ; x n (t)) T. The functions