Second order sufficient conditions and sensitivity analysis for the optimal control of a container crane under state constraints

Stability and sensitivity analysis of parametric control problems has recently been elaborated for optimal control problems subject to pure state constraints. This paper illustrates the numerical aspects of sensitivity analysis for a complex practical example: the optimal control of a container crane with a state constraint on the vertical velocity. The multiple shooting method is used to determine a nominal solution satisfying first order necessary conditions. Second order sufficient conditions are checked by showing that an associated Riccati equation has a bounded solution. Sensitivity differentials of optimal solutions an computed with respect to variations in the swing angle