A procedure to solve optimal control problems numerically by parametrization via runge-kutta-methods

Abstract

In this paper, we consider a class of nonlinear optimal control problems (Bolzaproblems) with constraints of the control vector, initial and boundary conditions of the state vector. The time interval is fixed. We parametrize the control function: we use a fixed uniform partition of the time interval and the control functions are approximated by step func¬tions, where the values of these functions are the optimization variables (parameters). Bat we can approximate the control function by (piecewise defined) polynomials of higher degrees too. In our second example we tested an approach with a polynomials of first degree. To get the approximative values of the state variables we implement the parameters into a integration scheme of Runge-Kutta type. Finally for the optimization the integration scheme directly is combined with a nonlinear programming-solver, for instance an SQP-solver. The existence of an optimal solution is assumed. Convergence properties of this method are not considered in this paper, but will be treated in a forthcoming paper

Keywords:

• Optimal Control
• Bolzaproblem
• Constaints of the Controls and of the State Variables Respectively
• End conditions for the State Variables
• Parametrization
• Direct Implementation of the Parameters into a Solver of Runge-Kutta Type
• Numerical Solution