Sensitivity Interpretations of the Costate Variable for Optimal Control Problems with State Constraints

In optimal control theory, it is well known that the co-state arc and the associated maximized Hamiltonian function can be interpreted in terms of gradients of the value function, evaluated along the optimal state trajectory. Such relations have been referred to as ‘sensitivity relations’ in the literature. We provide in this paper new sensitivity relations for state constrained optimal control problems. For the class of optimal control problems considered there is no guarantee that the co-state arc is unique; a key feature of the results is that they assert some choice of co-state arc can be made, for which the sensitivity relations are valid. The proof technique is to introduce an auxiliary optimal control problem that possesses a richer set of control variables than the original problem. The introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer; the extra information thereby obtained yields the desired set of sensitivity relations.