Sensitivity interpretations of the co-state trajectory for opimal control problems with state constraints

Abstract

Sensitivity relations in optimal control identify the costate trajectory and the Hamiltonian, evaluated along a minimizing trajectory, as gradients of the value function. Sensitivity relations for optimal control problems not involving state constraints and formulated in terms of controlled differential equation with smooth data follow easily from standard transversality conditions. In the presence of pathwise state constraints, if the data is nonsmooth or when the dynamic constraint takes the form of a differential inclusion, deriving the sensitivity relations is far from straightforward. We announce both `full' and `partial' sensitivity relations for differential inclusion problems with pathwise state constraints. The partial sensitivity relation identifies the costate with a partial subgradient of the value function with respect to the state, and the full sensitivity relation identifies the costate and the Hamiltonian with a subgradient of the value function with respect to time and state. The partial sensitivity relation is new for state constraint problems. The full sensitivity relation is valid under reduced hypotheses and for a stronger form of necessary conditions, as compared with earlier literature. It is shown for the first time also that the costate arc can be chosen to satisfy the two relations simultaneously. An example illustrates that a costate trajectory may be specially chosen to satisfy the sensitivity relations, and it is possible that some costate trajectories fail to do so.