Feedback Maximum Principle for a Class of Linear Continuity Equations Inspired by Optimal Impulsive Control

The paper deals with an optimal control problem for the simplest version of a “reduced” linear impulsive continuity equation. The latter is introduced in our recent papers as a model of dynamical ensembles enduring jumps, or impulsive ODEs having a probabilistic uncertainty in the initial data. The model, addressed in the manuscript, is, in fact, equivalent to the mentioned impulsive one, while it is stated within the usual, continuous setup. The price for this reduction is the appearance of an integral constraint on control, which makes the problem non-standard. The main focus of our present study is on the theory of so-called feedback necessary optimality conditions, which are one of the recent achievements in the optimal control theory of ODEs. The paradigmatic version of such a condition, called the feedback maximum (or minimum) principle, is formulated with the use of standard constructions of the classical Pontryagin’s Maximum Principle but is shown to strengthen the latter (in particular, for different pathological cases). One of the main advantages of the feedback maximum principle is due to its natural algorithmic property, which enables using it as an iterative numeric algorithm. The paper presents a version of the feedback maximum principle for linear transport equations with integrally bounded controls.