On the exact form of V.A. Dykhta's feedback minimum principle in nonlinear control problems

This paper investigates a nonlinear optimal control problem for an ordinary differential equation (in the sense of Bochner) on a Banach space. The problem is posed in the class of conventional controls – measurable, essentially bounded functions of time – and takes the classical Mayer's form with a free right endpoint of the trajectories. It is shown that the increment of the objective functional for such a problem, for any pair of admissible controls, can be represented exactly in terms of the cost function of the reference process – a solution to a linear transport equation. The restriction of this representation to the standard classes of needle-shaped and weak control perturbations plays the role of a functional variation of “infinite order”. A non-canonical necessary condition for optimality follows from the exact formula for the functional increment, which differs from both the Pontryagin principle and known higher-order conditions. This condition can be considered an exact nonlinear form of V.A. Dykhta's feedback minimum principle.