On the application of sequential quadratic programming to state-constrained optimal control problems

In this paper a numerical method for the solution of state-constrained optimal control problems is presented. The method is derived from an infinite dimensional analogue of sequential quadratic programming. The main purpose of the paper is to present some theoretical aspects of the method. An experimental numerical implementation of the method is also discussed. An analogue to finite dimensional sequential quadratic programming is developed in Banach spaces. Application to state-constrained control problems follows similar lines as in case of deriving the minimum principle from the abstract necessary conditions for optimality. In the setting of optimal control problems, the analogue to the inversion of the Hessian matrix of the Lagrangian is the solution of a linear multi point boundary value problem. Each iteration of the method involves mainly the solution a linear multi point boundary value problem. Numerically, a collocation method based on collocation with piecewise polynomial functions is proposed for the solution of the linear multi point boundary value problems. The resulting set of linear equations is solved by Gauss elimination. The method is derived considering only constraints of the equality type. Inequality constraints are transformed into equality constraints by means of an active set strategy or by using slackvariables.