Adjoint sensitivity analysis of nonstandard objective functions

Adjoint sensitivity analysis (ASA) is an efficient and popular method to calculate the sensitivity of an objective function to changes in the parameters of a dynamic model. ASA has found many applications including parameter estimation, optimal control, and time-dependent sensitivity analysis (TDSA, the sensitivity of an objective function to a time-limited state or parameter perturbation). However, standard ASA assume that the objective functions has the Bolza form typical in optimal control theory, the sum of running and terminal costs. In this paper, we show that this assumption can be relaxed, so that several classes of non-Bolza objective functions of practical interest can still be handled using ASA. These include extremum objectives (e.g., the peak number of active infections during an epidemic) and first passage times (e.g., when the local population density of an invasive organism first exceeds a threshold). Specifically, we show that these and other nonstandard objectives can be replaced by a Bolza objective having exactly the same sensitivities. We illustrate our results using three examples, focusing on applications of ASA to TDSA: a simple epidemic model, a fishery model following a juvenile mass mortality event, and spatial models of invasive species with pulled or pushed invasion fronts.