Exact Formulas for the Increment of the Cost Functional in Optimal Control of Linear Balance Equation

We study a state-linear optimal control problem for a transport equation with a source term in the space of finite signed Borel measures. For this problem, a version of the classical Pontryagin principle (in the form of the minimum principle) is obtained for the first time. In addition, we propose an approach to enhance the latter based on a certain unconventional procedure of variational analysis, namely, on exact increment formulas, representing the difference in values of the objective functional for any pair of admissible controls, without neglecting residual terms of any expansion. The approach relies on the standard duality and results in a series of necessary optimality conditions of a non-classical, "feedback" type. A constructive consequence of the feedback optimality conditions is a method of successive approximations, devoid of any parameters of "descent depth".